AI Detector Zero

AI Detector Zero — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Apache Pig

  Apache Pig is a high-level platform for creating programs that run on Apache Hadoop. The language for this platform is called Pig Latin. Pig can execute its Hadoop jobs in MapReduce, Apache Tez, or Apache Spark. Pig Latin abstracts the programming from the Java MapReduce idiom into a notation which makes MapReduce programming high level, similar to that of SQL for relational database management systems. Pig Latin can be extended using user-defined functions (UDFs) which the user can write in Java, Python, JavaScript, Ruby or Groovy and then call directly from the language. == History == Apache Pig was originally developed at Yahoo Research around 2006 for researchers to have an ad hoc way of creating and executing MapReduce jobs on very large data sets. In 2007, it was moved into the Apache Software Foundation. === Naming === Regarding the naming of the Pig programming language, the name was chosen arbitrarily and stuck because it was memorable, easy to spell, and for novelty. The story goes that the researchers working on the project initially referred to it simply as 'the language'. Eventually they needed to call it something. Off the top of his head, one researcher suggested Pig, and the name stuck. It is quirky yet memorable and easy to spell. While some have hinted that the name sounds coy or silly, it has provided us with an entertaining nomenclature, such as Pig Latin for the language, Grunt for the shell, and PiggyBank for the CPAN-like shared repository. == Example == Below is an example of a "Word Count" program in Pig Latin: The above program will generate parallel executable tasks which can be distributed across multiple machines in a Hadoop cluster to count the number of words in a dataset such as all the webpages on the internet. == Pig vs SQL == In comparison to SQL, Pig has a nested relational model, uses lazy evaluation, uses extract, transform, load (ETL), is able to store data at any point during a pipeline, declares execution plans, supports pipeline splits, thus allowing workflows to proceed along DAGs instead of strictly sequential pipelines. On the other hand, it has been argued DBMSs are substantially faster than the MapReduce system once the data is loaded, but that loading the data takes considerably longer in the database systems. It has also been argued RDBMSs offer out of the box support for column-storage, working with compressed data, indexes for efficient random data access, and transaction-level fault tolerance. Pig Latin is procedural and fits very naturally in the pipeline paradigm while SQL is instead declarative. In SQL users can specify that data from two tables must be joined, but not what join implementation to use (You can specify the implementation of JOIN in SQL, thus "... for many SQL applications the query writer may not have enough knowledge of the data or enough expertise to specify an appropriate join algorithm."). Pig Latin allows users to specify an implementation or aspects of an implementation to be used in executing a script in several ways. In effect, Pig Latin programming is similar to specifying a query execution plan, making it easier for programmers to explicitly control the flow of their data processing task. SQL is oriented around queries that produce a single result. SQL handles trees naturally, but has no built in mechanism for splitting a data processing stream and applying different operators to each sub-stream. Pig Latin script describes a directed acyclic graph (DAG) rather than a pipeline. Pig Latin's ability to include user code at any point in the pipeline is useful for pipeline development. If SQL is used, data must first be imported into the database, and then the cleansing and transformation process can begin.

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  Learning to rank

  Learning to rank (LTR) or machine-learned ranking (MLR) is the application of machine learning, often supervised, semi-supervised or reinforcement learning, in the construction of ranking models for information retrieval and recommender systems. Training data may, for example, consist of lists of items with some partial order specified between items in each list. This order is typically induced by giving a numerical or ordinal score or a binary judgment (e.g. "relevant" or "not relevant") for each item. The goal of constructing the ranking model is to rank new, unseen lists in a similar way to rankings in the training data. == Applications == === In information retrieval === Ranking is a central part of many information retrieval problems, such as document retrieval, collaborative filtering, sentiment analysis, and online advertising. A possible architecture of a machine-learned search engine is shown in the accompanying figure. Training data consists of queries and documents matching them together with the relevance degree of each match. It may be prepared manually by human assessors (or raters, as Google calls them), who check results for some queries and determine relevance of each result. It is not feasible to check the relevance of all documents, and so typically a technique called pooling is used — only the top few documents, retrieved by some existing ranking models are checked. This technique may introduce selection bias. Alternatively, training data may be derived automatically by analyzing clickthrough logs (i.e. search results which got clicks from users), query chains, or such search engines' features as Google's (since-replaced) SearchWiki. Clickthrough logs can be biased by the tendency of users to click on the top search results on the assumption that they are already well-ranked. Training data is used by a learning algorithm to produce a ranking model which computes the relevance of documents for actual queries. Typically, users expect a search query to complete in a short time (such as a few hundred milliseconds for web search), which makes it impossible to evaluate a complex ranking model on each document in the corpus, and so a two-phase scheme is used. First, a small number of potentially relevant documents are identified using simpler retrieval models which permit fast query evaluation, such as the vector space model, Boolean model, weighted AND, or BM25. This phase is called top- k {\displaystyle k} document retrieval and many heuristics were proposed in the literature to accelerate it, such as using a document's static quality score and tiered indexes. In the second phase, a more accurate but computationally expensive machine-learned model is used to re-rank these documents. === In other areas === Learning to rank algorithms have been applied in areas other than information retrieval: In machine translation for ranking a set of hypothesized translations; In computational biology for ranking candidate 3-D structures in protein structure prediction problems; In recommender systems for identifying a ranked list of related news articles to recommend to a user after he or she has read a current news article. == Feature vectors == For the convenience of MLR algorithms, query-document pairs are usually represented by numerical vectors, which are called feature vectors. Such an approach is sometimes called bag of features and is analogous to the bag of words model and vector space model used in information retrieval for representation of documents. Components of such vectors are called features, factors or ranking signals. They may be divided into three groups (features from document retrieval are shown as examples): Query-independent or static features — those features, which depend only on the document, but not on the query. For example, PageRank or document's length. Such features can be precomputed in off-line mode during indexing. They may be used to compute document's static quality score (or static rank), which is often used to speed up search query evaluation. Query-dependent or dynamic features — those features, which depend both on the contents of the document and the query, such as TF-IDF score or other non-machine-learned ranking functions. Query-level features or query features, which depend only on the query. For example, the number of words in a query. Some examples of features, which were used in the well-known LETOR dataset: TF, TF-IDF, BM25, and language modeling scores of document's zones (title, body, anchors text, URL) for a given query; Lengths and IDF sums of document's zones; Document's PageRank, HITS ranks and their variants. Selecting and designing good features is an important area in machine learning, which is called feature engineering. == Evaluation measures == There are several measures (metrics) which are commonly used to judge how well an algorithm is doing on training data and to compare the performance of different MLR algorithms. Often a learning-to-rank problem is reformulated as an optimization problem with respect to one of these metrics. Examples of ranking quality measures: Mean average precision (MAP); DCG and NDCG; Precision@n, NDCG@n, where "@n" denotes that the metrics are evaluated only on top n documents; Mean reciprocal rank; Kendall's tau; Spearman's rho. DCG and its normalized variant NDCG are usually preferred in academic research when multiple levels of relevance are used. Other metrics such as MAP, MRR and precision, are defined only for binary judgments. Recently, there have been proposed several new evaluation metrics which claim to model user's satisfaction with search results better than the DCG metric: Expected reciprocal rank (ERR); Yandex's pfound. Both of these metrics are based on the assumption that the user is more likely to stop looking at search results after examining a more relevant document, than after a less relevant document. == Approaches == Learning to Rank approaches are often categorized using one of three approaches: pointwise (where individual documents are ranked), pairwise (where pairs of documents are ranked into a relative order), and listwise (where an entire list of documents are ordered). Tie-Yan Liu of Microsoft Research Asia has analyzed existing algorithms for learning to rank problems in his book Learning to Rank for Information Retrieval. He categorized them into three groups by their input spaces, output spaces, hypothesis spaces (the core function of the model) and loss functions: the pointwise, pairwise, and listwise approach. In practice, listwise approaches often outperform pairwise approaches and pointwise approaches. This statement was further supported by a large scale experiment on the performance of different learning-to-rank methods on a large collection of benchmark data sets. In this section, without further notice, x {\displaystyle x} denotes an object to be evaluated, for example, a document or an image, f ( x ) {\displaystyle f(x)} denotes a single-value hypothesis, h ( ⋅ ) {\displaystyle h(\cdot )} denotes a bi-variate or multi-variate function and L ( ⋅ ) {\displaystyle L(\cdot )} denotes the loss function. === Pointwise approach === In this case, it is assumed that each query-document pair in the training data has a numerical or ordinal score. Then the learning-to-rank problem can be approximated by a regression problem — given a single query-document pair, predict its score. Formally speaking, the pointwise approach aims at learning a function f ( x ) {\displaystyle f(x)} predicting the real-value or ordinal score of a document x {\displaystyle x} using the loss function L ( f ; x j , y j ) {\displaystyle L(f;x_{j},y_{j})} . A number of existing supervised machine learning algorithms can be readily used for this purpose. Ordinal regression and classification algorithms can also be used in pointwise approach when they are used to predict the score of a single query-document pair, and it takes a small, finite number of values. === Pairwise approach === In this case, the learning-to-rank problem is approximated by a classification problem — learning a binary classifier h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} that can tell which document is better in a given pair of documents. The classifier shall take two documents as its input and the goal is to minimize a loss function L ( h ; x u , x v , y u , v ) {\displaystyle L(h;x_{u},x_{v},y_{u,v})} . The loss function typically reflects the number and magnitude of inversions in the induced ranking. In many cases, the binary classifier h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} is implemented with a scoring function f ( x ) {\displaystyle f(x)} . As an example, RankNet adapts a probability model and defines h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} as the estimated probability of the document x u {\displaystyle x_{u}} has higher quality than x v {\displaystyle x_{v}} : P u , v ( f ) = CDF ( f ( x u ) − f ( x v ) ) , {\displaystyle P_{u,v}(f)={\text{CDF}

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  Decision tree pruning

  Pruning is a data compression technique in machine learning and search algorithms that reduces the size of decision trees by removing sections of the tree that are non-critical and redundant to classify instances. Pruning reduces the complexity of the final classifier, and hence improves predictive accuracy by the reduction of overfitting. One of the questions that arises in a decision tree algorithm is the optimal size of the final tree. A tree that is too large risks overfitting the training data and poorly generalizing to new samples. A small tree might not capture important structural information about the sample space. However, it is hard to tell when a tree algorithm should stop because it is impossible to tell if the addition of a single extra node will dramatically decrease error. This problem is known as the horizon effect. A common strategy is to grow the tree until each node contains a small number of instances then use pruning to remove nodes that do not provide additional information. Pruning should reduce the size of a learning tree without reducing predictive accuracy as measured by a cross-validation set. There are many techniques for tree pruning that differ in the measurement that is used to optimize performance. == Techniques == Pruning processes can be divided into two types (pre- and post-pruning). Pre-pruning procedures prevent a complete induction of the training set by replacing a stop () criterion in the induction algorithm (e.g. max. Tree depth or information gain (Attr)> minGain). Pre-pruning methods are considered to be more efficient because they do not induce an entire set, but rather trees remain small from the start. Prepruning methods share a common problem, the horizon effect. This is to be understood as the undesired premature termination of the induction by the stop () criterion. Post-pruning (or just pruning) is the most common way of simplifying trees. Here, nodes and subtrees are replaced with leaves to reduce complexity. Pruning can not only significantly reduce the size but also improve the classification accuracy of unseen objects. It may be the case that the accuracy of the assignment on the train set deteriorates, but the accuracy of the classification properties of the tree increases overall. The procedures are differentiated on the basis of their approach in the tree (top-down or bottom-up). === Bottom-up pruning === These procedures start at the last node in the tree (the lowest point). Following recursively upwards, they determine the relevance of each individual node. If the relevance for the classification is not given, the node is dropped or replaced by a leaf. The advantage is that no relevant sub-trees can be lost with this method. These methods include Reduced Error Pruning (REP), Minimum Cost Complexity Pruning (MCCP), or Minimum Error Pruning (MEP). === Top-down pruning === In contrast to the bottom-up method, this method starts at the root of the tree. Following the structure below, a relevance check is carried out which decides whether a node is relevant for the classification of all n items or not. By pruning the tree at an inner node, it can happen that an entire sub-tree (regardless of its relevance) is dropped. One of these representatives is pessimistic error pruning (PEP), which brings quite good results with unseen items. == Pruning algorithms == === Reduced error pruning === One of the simplest forms of pruning is reduced error pruning. Starting at the leaves, each node is replaced with its most popular class. If the prediction accuracy is not affected then the change is kept. While somewhat naive, reduced error pruning has the advantage of simplicity and speed. === Cost complexity pruning === Cost complexity pruning generates a series of trees ⁠ T 0 … T m {\displaystyle T_{0}\dots T_{m}} ⁠ where ⁠ T 0 {\displaystyle T_{0}} ⁠ is the initial tree and ⁠ T m {\displaystyle T_{m}} ⁠ is the root alone. At step ⁠ i {\displaystyle i} ⁠, the tree is created by removing a subtree from tree ⁠ i − 1 {\displaystyle i-1} ⁠ and replacing it with a leaf node with value chosen as in the tree building algorithm. The subtree that is removed is chosen as follows: Define the error rate of tree ⁠ T {\displaystyle T} ⁠ over data set ⁠ S {\displaystyle S} ⁠ as ⁠ err ⁡ ( T , S ) {\displaystyle \operatorname {err} (T,S)} ⁠. The subtree t {\displaystyle t} that minimizes err ⁡ ( prune ⁡ ( T , t ) , S ) − err ⁡ ( T , S ) | leaves ⁡ ( T ) | − | leaves ⁡ ( prune ⁡ ( T , t ) ) | {\displaystyle {\frac {\operatorname {err} (\operatorname {prune} (T,t),S)-\operatorname {err} (T,S)}{\left\vert \operatorname {leaves} (T)\right\vert -\left\vert \operatorname {leaves} (\operatorname {prune} (T,t))\right\vert }}} is chosen for removal. The function ⁠ prune ⁡ ( T , t ) {\displaystyle \operatorname {prune} (T,t)} ⁠ defines the tree obtained by pruning the subtrees ⁠ t {\displaystyle t} ⁠ from the tree ⁠ T {\displaystyle T} ⁠. Once the series of trees has been created, the best tree is chosen by generalized accuracy as measured by a training set or cross-validation. == Examples == Pruning could be applied in a compression scheme of a learning algorithm to remove the redundant details without compromising the model's performances. In neural networks, pruning removes entire neurons or layers of neurons.

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  Dynamic epistemic logic

  Dynamic epistemic logic (DEL) is a logical framework dealing with knowledge and information change. Typically, DEL focuses on situations involving multiple agents and studies how their knowledge changes when events occur. These events can change factual properties of the actual world (they are called ontic events): for example a red card is painted in blue. They can also bring about changes of knowledge without changing factual properties of the world (they are called epistemic events): for example, a card is revealed publicly (or privately) to be red. Originally, DEL focused on epistemic events. Only some of the basic ideas are present in this entry of the original DEL framework; more details about DEL in general can be found in the references. Due to the nature of its object of study and its abstract approach, DEL is related and has applications to numerous research areas, such as computer science (artificial intelligence), philosophy (formal epistemology), economics (game theory) and cognitive science. In computer science, DEL is for example very much related to multi-agent systems, which are systems where multiple intelligent agents interact and exchange information. As a combination of dynamic logic and epistemic logic, dynamic epistemic logic is a young field of research. It really started in 1989 with Plaza's logic of public announcement. Independently, Gerbrandy and Groeneveld proposed a system dealing moreover with private announcement and that was inspired by the work of Veltman. Another system was proposed by van Ditmarsch whose main inspiration was the Cluedo game. But the most influential and original system was the system proposed by Baltag, Moss and Solecki. This system can deal with all the types of situations studied in the works above and its underlying methodology is conceptually grounded. This entry will present some of its basic ideas. Formally, DEL extends ordinary epistemic logic by the inclusion of event models to describe actions, and a product update operator that defines how epistemic models are updated as the consequence of executing actions described through event models. Epistemic logic will first be recalled. Then, actions and events will enter into the picture and we will introduce the DEL framework. == Epistemic logic == Epistemic logic is a modal logic dealing with the notions of knowledge and belief. As a logic, it is concerned with understanding the process of reasoning about knowledge and belief: which principles relating the notions of knowledge and belief are intuitively plausible? Like epistemology, it stems from the Greek word ϵ π ι σ τ η μ η {\displaystyle \epsilon \pi \iota \sigma \tau \eta \mu \eta } or ‘episteme’ meaning knowledge. Epistemology is nevertheless more concerned with analyzing the very nature and scope of knowledge, addressing questions such as “What is the definition of knowledge?” or “How is knowledge acquired?”. In fact, epistemic logic grew out of epistemology in the Middle Ages thanks to the efforts of Burley and Ockham. The formal work, based on modal logic, that inaugurated contemporary research into epistemic logic dates back only to 1962 and is due to Hintikka. It then sparked in the 1960s discussions about the principles of knowledge and belief and many axioms for these notions were proposed and discussed. For example, the interaction axioms K p → B p {\displaystyle Kp\rightarrow Bp} and B p → K B p {\displaystyle Bp\rightarrow KBp} are often considered to be intuitive principles: if an agent Knows p {\displaystyle p} then (s)he also Believes p {\displaystyle p} , or if an agent Believes p {\displaystyle p} , then (s)he Knows that (s)he Believes p {\displaystyle p} . More recently, these kinds of philosophical theories were taken up by researchers in economics, artificial intelligence and theoretical computer science where reasoning about knowledge is a central topic. Due to the new setting in which epistemic logic was used, new perspectives and new features such as computability issues were then added to the research agenda of epistemic logic. === Syntax === In the sequel, A G T S = { 1 , … , n } {\displaystyle AGTS=\{1,\ldots ,n\}} is a finite set whose elements are called agents and P R O P {\displaystyle PROP} is a set of propositional letters. The epistemic language is an extension of the basic multi-modal language of modal logic with a common knowledge operator C A {\displaystyle C_{A}} and a distributed knowledge operator D A {\displaystyle D_{A}} . Formally, the epistemic language L EL C {\displaystyle {\mathcal {L}}_{\textsf {EL}}^{C}} is defined inductively by the following grammar in BNF: L EL C : ϕ ::= p ∣ ¬ ϕ ∣ ( ϕ ∧ ϕ ) ∣ K j ϕ ∣ C A ϕ ∣ D A ϕ {\displaystyle {\mathcal {L}}_{\textsf {EL}}^{C}:\phi ~~::=~~p~\mid ~\neg \phi ~\mid ~(\phi \land \phi )~\mid ~K_{j}\phi ~\mid ~C_{A}\phi ~\mid ~D_{A}\phi } where p ∈ P R O P {\displaystyle p\in PROP} , j ∈ A G T S {\displaystyle j\in {AGTS}} and A ⊆ A G T S {\displaystyle A\subseteq {AGTS}} . The basic epistemic language L E L {\displaystyle {\mathcal {L}}_{EL}} is the language L E L C {\displaystyle {\mathcal {L}}_{EL}^{C}} without the common knowledge and distributed knowledge operators. The formula ⊥ {\displaystyle \bot } is an abbreviation for ¬ p ∧ p {\displaystyle \neg p\land p} (for a given p ∈ P R O P {\displaystyle p\in PROP} ), ⟨ K j ⟩ ϕ {\displaystyle \langle K_{j}\rangle \phi } is an abbreviation for ¬ K j ¬ ϕ {\displaystyle \neg K_{j}\neg \phi } , E A ϕ {\displaystyle E_{A}\phi } is an abbreviation for ⋀ j ∈ A K j ϕ {\displaystyle \bigwedge \limits _{j\in A}K_{j}\phi } and C ϕ {\displaystyle C\phi } an abbreviation for C A G T S ϕ {\displaystyle C_{AGTS}\phi } . Group notions: general, common and distributed knowledge. In a multi-agent setting there are three important epistemic concepts: general knowledge, distributed knowledge and common knowledge. The notion of common knowledge was first studied by Lewis in the context of conventions. It was then applied to distributed systems and to game theory, where it allows to express that the rationality of the players, the rules of the game and the set of players are commonly known. General knowledge. General knowledge of ϕ {\displaystyle \phi } means that everybody in the group of agents A G T S {\displaystyle {AGTS}} knows that ϕ {\displaystyle \phi } . Formally, this corresponds to the following formula: E ϕ := ⋀ j ∈ A G T S K j ϕ . {\displaystyle E\phi :={\underset {j\in {AGTS}}{\bigwedge }}K_{j}\phi .} Common knowledge. Common knowledge of ϕ {\displaystyle \phi } means that everybody knows ϕ {\displaystyle \phi } but also that everybody knows that everybody knows ϕ {\displaystyle \phi } , that everybody knows that everybody knows that everybody knows ϕ {\displaystyle \phi } , and so on ad infinitum. Formally, this corresponds to the following formula C ϕ := E ϕ ∧ E E ϕ ∧ E E E ϕ ∧ … {\displaystyle C\phi :=E\phi \land EE\phi \land EEE\phi \land \ldots } As we do not allow infinite conjunction the notion of common knowledge will have to be introduced as a primitive in our language. Before defining the language with this new operator, we are going to give an example introduced by Lewis that illustrates the difference between the notions of general knowledge and common knowledge. Lewis wanted to know what kind of knowledge is needed so that the statement p {\displaystyle p} : “every driver must drive on the right” be a convention among a group of agents. In other words, he wanted to know what kind of knowledge is needed so that everybody feels safe to drive on the right. Suppose there are only two agents i {\displaystyle i} and j {\displaystyle j} . Then everybody knowing p {\displaystyle p} (formally E p {\displaystyle Ep} ) is not enough. Indeed, it might still be possible that the agent i {\displaystyle i} considers possible that the agent j {\displaystyle j} does not know p {\displaystyle p} (formally ¬ K i K j p {\displaystyle \neg K_{i}K_{j}p} ). In that case the agent i {\displaystyle i} will not feel safe to drive on the right because he might consider that the agent j {\displaystyle j} , not knowing p {\displaystyle p} , could drive on the left. To avoid this problem, we could then assume that everybody knows that everybody knows that p {\displaystyle p} (formally E E p {\displaystyle EEp} ). This is again not enough to ensure that everybody feels safe to drive on the right. Indeed, it might still be possible that agent i {\displaystyle i} considers possible that agent j {\displaystyle j} considers possible that agent i {\displaystyle i} does not know p {\displaystyle p} (formally ¬ K i K j K i p {\displaystyle \neg K_{i}K_{j}K_{i}p} ). In that case and from i {\displaystyle i} ’s point of view, j {\displaystyle j} considers possible that i {\displaystyle i} , not knowing p {\displaystyle p} , will drive on the left. So from i {\displaystyle i} ’s point of view, j {\displaystyle j} might drive on the left as well (by the same argument as abov

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  Convolution

  In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions f {\displaystyle f} and g {\displaystyle g} that produces a third function f ∗ g {\displaystyle fg} , as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The term convolution refers to both the resulting function and to the process of computing it. The integral is evaluated for all values of shift, producing the convolution function. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). Graphically, it expresses how the 'shape' of one function is modified by the other. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution f ∗ g {\displaystyle fg} differs from cross-correlation f ⋆ g {\displaystyle f\star g} only in that either f ( x ) {\displaystyle f(x)} or g ( x ) {\displaystyle g(x)} is reflected about the y-axis in convolution; thus it is a cross-correlation of g ( − x ) {\displaystyle g(-x)} and f ( x ) {\displaystyle f(x)} , or f ( − x ) {\displaystyle f(-x)} and g ( x ) {\displaystyle g(x)} . For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, computer vision and human vision, geophysics, engineering, physics, and differential equations. The convolution can be defined for functions on Euclidean space and other groups (as algebraic structures). For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing. Computing the inverse of the convolution operation is known as deconvolution. == Definition == The convolution of f {\displaystyle f} and g {\displaystyle g} is written f ∗ g {\displaystyle fg} , denoting the operator with the symbol ∗ {\displaystyle } . It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. As such, it is a particular kind of integral transform: ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau .} An equivalent definition is (see commutativity): ( f ∗ g ) ( t ) := ∫ − ∞ ∞ f ( t − τ ) g ( τ ) d τ . {\displaystyle (fg)(t):=\int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau .} While the symbol t {\displaystyle t} is used above, it need not represent the time domain. At each t {\displaystyle t} , the convolution formula can be described as the area under the function f ( τ ) {\displaystyle f(\tau )} weighted by the function g ( − τ ) {\displaystyle g(-\tau )} shifted by the amount t {\displaystyle t} . As t {\displaystyle t} changes, the weighting function g ( t − τ ) {\displaystyle g(t-\tau )} emphasizes different parts of the input function f ( τ ) {\displaystyle f(\tau )} ; If t {\displaystyle t} is a positive value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted along the τ {\displaystyle \tau } -axis toward the right (toward + ∞ {\displaystyle +\infty } ) by the amount of t {\displaystyle t} , while if t {\displaystyle t} is a negative value, then g ( t − τ ) {\displaystyle g(t-\tau )} is equal to g ( − τ ) {\displaystyle g(-\tau )} that slides or is shifted toward the left (toward − ∞ {\displaystyle -\infty } ) by the amount of | t | {\displaystyle |t|} . For functions f {\displaystyle f} , g {\displaystyle g} supported on only [ 0 , ∞ ) {\displaystyle [0,\infty )} (i.e., zero for negative arguments), the integration limits can be truncated, resulting in: ( f ∗ g ) ( t ) = ∫ 0 t f ( τ ) g ( t − τ ) d τ for f , g : [ 0 , ∞ ) → R . {\displaystyle (fg)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad \ {\text{for }}f,g:[0,\infty )\to \mathbb {R} .} For the multi-dimensional formulation of convolution, see domain of definition (below). === Notation === A common engineering notational convention is: f ( t ) ∗ g ( t ) := ∫ − ∞ ∞ f ( τ ) g ( t − τ ) d τ ⏟ ( f ∗ g ) ( t ) , {\displaystyle f(t)g(t)\mathrel {:=} \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(fg)(t)},} which has to be interpreted carefully to avoid confusion. For instance, f ( t ) ∗ g ( t − t 0 ) {\displaystyle f(t)g(t-t_{0})} is equivalent to ( f ∗ g ) ( t − t 0 ) {\displaystyle (fg)(t-t_{0})} , but f ( t − t 0 ) ∗ g ( t − t 0 ) {\displaystyle f(t-t_{0})g(t-t_{0})} is in fact equivalent to ( f ∗ g ) ( t − 2 t 0 ) {\displaystyle (fg)(t-2t_{0})} . === Relations with other transforms === Given two functions f ( t ) {\displaystyle f(t)} and g ( t ) {\displaystyle g(t)} with bilateral Laplace transforms (two-sided Laplace transform) F ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u {\displaystyle F(s)=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u} and G ( s ) = ∫ − ∞ ∞ e − s v g ( v ) d v {\displaystyle G(s)=\int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v} respectively, the convolution operation ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} can be defined as the inverse Laplace transform of the product of F ( s ) {\displaystyle F(s)} and G ( s ) {\displaystyle G(s)} . More precisely, F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ e − s u f ( u ) d u ⋅ ∫ − ∞ ∞ e − s v g ( v ) d v = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s ( u + v ) f ( u ) g ( v ) d u d v {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }e^{-su}\ f(u)\ {\text{d}}u\cdot \int _{-\infty }^{\infty }e^{-sv}\ g(v)\ {\text{d}}v\\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-s(u+v)}\ f(u)\ g(v)\ {\text{d}}u\ {\text{d}}v\end{aligned}}} Let t = u + v {\displaystyle t=u+v} , then F ( s ) ⋅ G ( s ) = ∫ − ∞ ∞ ∫ − ∞ ∞ e − s t f ( u ) g ( t − u ) d u d t = ∫ − ∞ ∞ e − s t ∫ − ∞ ∞ f ( u ) g ( t − u ) d u ⏟ ( f ∗ g ) ( t ) d t = ∫ − ∞ ∞ e − s t ( f ∗ g ) ( t ) d t . {\displaystyle {\begin{aligned}F(s)\cdot G(s)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{-st}\ f(u)\ g(t-u)\ {\text{d}}u\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}\underbrace {\int _{-\infty }^{\infty }f(u)\ g(t-u)\ {\text{d}}u} _{(fg)(t)}\ {\text{d}}t\\&=\int _{-\infty }^{\infty }e^{-st}(fg)(t)\ {\text{d}}t.\end{aligned}}} Note that F ( s ) ⋅ G ( s ) {\displaystyle F(s)\cdot G(s)} is the bilateral Laplace transform of ( f ∗ g ) ( t ) {\displaystyle (fg)(t)} . A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform). The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. == Visual explanation == == Historical developments == One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754. Also, an expression of the type: ∫ f ( u ) ⋅ g ( x − u ) d u {\displaystyle \int f(u)\cdot g(x-u)\,du} is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, which is the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800. Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, and others. The term itself did not come into wide use until the 1950s or 1960s. Prior to that it was sometimes known as Faltung (which means folding in German), composition product, superposition integral, and Carson's integral. Yet it appears as early as 1903, though the definition is rather unfamiliar in older uses. The operation: ∫ 0 t φ ( s ) ψ ( t − s ) d s , 0 ≤ t < ∞ , {\displaystyle \int _{0}^{t}\varphi (s)\psi (t-s)\,ds,\quad 0\leq t<\infty ,} is a particular case of composition products considered by the Italian mathematician Vito Volterra in 1913. == Circular c

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  Deep tomographic reconstruction

  Deep Tomographic Reconstruction is a set of methods for using deep learning methods to perform tomographic reconstruction of medical and industrial images. It uses artificial intelligence and machine learning, especially deep artificial neural networks or deep learning, to overcome challenges such as measurement noise, data sparsity, image artifacts, and computational inefficiency. This approach has been applied across various imaging modalities, including CT, MRI, PET, SPECT, ultrasound, and optical imaging == Historical background == Traditional tomographic reconstruction relies on analytic methods such as filtered back-projection, or iterative methods which incrementally compute inverse transformations from measurement data (e.g., Radon or Fourier transform data). However, these approaches are not sufficient for certain imaging techniques such as low-dose CT and fast MRI, or scenarios involving metal artifacts and patient motion. == Use in imaging modalities == === Computed tomography (CT) === In CT, deep learning models can be particularly effective in reducing radiation exposure while maintaining image quality. Deep neural networks can also be able to reconstruct images of fair quality from sparsely sampled data without sacrificing diagnostic performance. Deep learning-based generative AI models can reduce CT metal artifacts. === Magnetic resonance imaging (MRI) === In magnetic resonance imaging (MRI), deep learning can lead to reduced MRI motion artifacts, and increased acquisition speed, referred to as fast MRI. Despite suffering from disadvantages such as lower signal-to-noise ratio (SNR), deep learning can enhance image quality in low field MRI, making these systems clinically viable. === Positron emission tomography (PET) and single-photon emission CT (SPECT) === For PET imaging, deep learning models can provide substantial improvements in low-dose imaging and motion artifact correction. Also, deep learning can help SPECT for generation of attenuation background. A notable technique for PET denoising involves integrating MR data through multimodal networks, which use anatomical information from MRI to enhance PET image quality. === Ultrasound imaging === Deep learning can enhance ultrasound imaging by reducing speckle noise and motion blur. For ultrasound beamforming, deep neural networks can allow superior image quality with limited data at high speed. === Optical imaging and microscopy === Diffuse optical tomography, optical coherence tomography and microscopy can be improved by deep neural networks beyond traditional methods. Furthermore, deep learning can also enhance Photoacoustic imaging (see Deep learning in photoacoustic imaging), addressing challenges like high noise, low contrast, and limited resolution. Deep learning has also been applied to label-free live-cell imaging, where convolutional neural networks predict fluorescence labels from transmitted light images, a technique known as in silico labeling. This method can enable high-throughput, non-invasive cell analysis and phenotyping without the need for traditional fluorescent dyes.

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  Quantum artificial life

  Quantum artificial life is the application of quantum algorithms with the ability to simulate biological behavior. Quantum computers offer many potential improvements to processes performed on classical computers, including machine learning and artificial intelligence. Artificial intelligence applications are often inspired by the idea of mimicking human brains through closely related biomimicry. This has been implemented to a certain extent on classical computers (using neural networks), but quantum computers offer many advantages in the simulation of artificial life. Artificial life and artificial intelligence are extremely similar, with minor differences; the goal of studying artificial life is to understand living beings better, while the goal of artificial intelligence is to create intelligent beings. In 2016, Alvarez-Rodriguez et al. developed a proposal for a quantum artificial life algorithm with the ability to simulate life and Darwinian evolution. In 2018, the same research team led by Alvarez-Rodriguez performed the proposed algorithm on the IBM ibmqx4 quantum computer, and received optimistic results. The results accurately simulated a system with the ability to undergo self-replication at the quantum scale. == Artificial life on quantum computers == The growing advancement of quantum computers has led researchers to develop quantum algorithms for simulating life processes. Researchers have designed a quantum algorithm that can accurately simulate Darwinian Evolution. Since the complete simulation of artificial life on quantum computers has only been actualized by one group, this section shall focus on the implementation by Alvarez-Rodriguez, Sanz, Lomata, and Solano on an IBM quantum computer. Individuals were realized as two qubits, one representing the genotype of the individual and the other representing the phenotype. The genotype is copied to transmit genetic information through generations, and the phenotype is dependent on the genetic information as well as the individual's interactions with their environment. In order to set up the system, the state of the genotype is instantiated by some rotation of an ancillary state ( | 0 ⟩ ⟨ 0 | {\displaystyle |0\rangle \langle 0|} ). The environment is a two-dimensional spatial grid occupied by individuals and ancillary states. The environment is divided into cells that are able to possess one or more individuals. Individuals move throughout the grid and occupy cells randomly; when two or more individuals occupy the same cell they interact with each other. === Self replication === The ability to self-replicate is critical for simulating life. Self-replication occurs when the genotype of an individual interacts with an ancillary state, creating a genotype for a new individual; this genotype interacts with a different ancillary state in order to create the phenotype. During this interaction, one would like to copy some information about the initial state into the ancillary state, but by the no cloning theorem, it is impossible to copy an arbitrary unknown quantum state. However, physicists have derived different methods for quantum cloning which does not require the exact copying of an unknown state. The method that has been implemented by Alvarez-Rodriguez et al. is one that involves the cloning of the expectation value of some observable. For a unitary U {\displaystyle U} which copies the expectation value of some set of observables X {\displaystyle {\mathsf {X}}} of state ρ {\displaystyle \rho } into a blank state ρ e {\displaystyle \rho _{e}} , the cloning machine is defined by any ( U , ρ e , X ) {\displaystyle (U,\rho _{e},{\mathsf {X}})} that fulfill the following: ∀ ρ ∀ X ∈ X {\displaystyle \forall \rho \forall X\in {\mathsf {X}}} X ¯ = X 1 ¯ = X 2 ¯ {\displaystyle {\bar {X}}={\bar {X_{1}}}={\bar {X_{2}}}} Where X ¯ {\displaystyle {\bar {X}}} is the mean value of the observable in ρ {\displaystyle \rho } before cloning, X 1 ¯ {\displaystyle {\bar {X_{1}}}} is the mean value of the observable in ρ {\displaystyle \rho } after cloning, and X 2 ¯ {\displaystyle {\bar {X_{2}}}} is the mean value of the observable in ρ e {\displaystyle \rho _{e}} after cloning. Note that the cloning machine has no dependence on ρ {\displaystyle \rho } because we want to be able to clone the expectation of the observables for any initial state. It is important to note that cloning the mean value of the observable transmits more information than is allowed classically. The calculation of the mean value is defined naturally as: X ¯ = T r [ ρ X ] {\displaystyle {\bar {X}}=Tr[\rho X]} , X 1 ¯ = T r [ R X ⊗ I ] {\displaystyle {\bar {X_{1}}}=Tr[RX\otimes I]} , X 2 ¯ = T r [ R I ⊗ X ] {\displaystyle {\bar {X_{2}}}=Tr[RI\otimes X]} where R = U ρ ⊗ ρ e U † {\displaystyle R=U\rho \otimes \rho _{e}U^{\dagger }} The simplest cloning machine clones the expectation value of σ z {\displaystyle \sigma _{z}} in arbitrary state ρ = | ψ ⟩ ⟨ ψ | {\displaystyle \rho =|\psi \rangle \langle \psi |} to ρ e = | 0 ⟩ ⟨ 0 | {\displaystyle \rho _{e}=|0\rangle \langle 0|} using U = C N O T {\displaystyle U=CNOT} . This is the cloning machine implemented for self-replication by Alvarez-Rodriguez et al. The self-replication process clearly only requires interactions between two qubits, and therefore this cloning machine is the only one necessary for self replication. === Interactions === Interactions occur between individuals when the two take up the same space on the environmental grid. The presence of interactions between individuals provides an advantage for shorter-lifespan individuals. When two individuals interact, exchanges of information between the two phenotypes may or may not occur based on their existing values. When both individual's control qubits (genotypes) are alike, no information will be exchanged. When the control qubits differ, the target qubits (phenotype) will be exchanged between the two individuals. This procedure produces a constantly changing predator-prey dynamic in the simulation. Therefore, long-living qubits, with a larger genetic makeup in the simulation, are at a disadvantage. Since information is only exchanged when interacting with an individual of different genetic makeup, the short-lived population has the advantage. === Mutation === Mutations exist in the artificial world with limited probability, equivalent to their occurrence in the real world. There are two ways in which the individual can mutate: through random single qubit rotations and by errors in the self-replication process. There are two different operators that act on the individual and cause mutations. The M operation causes a spontaneous mutation within the individual by rotating a single qubit by parameter θ. The parameter θ is random for each mutation, which creates biodiversity within the artificial environment. The M operation is a unitary matrix which can be described as: M = ( cos ⁡ ( θ ) s i n ( θ ) s i n ( θ ) − c o s ( θ ) ) {\displaystyle M={\begin{pmatrix}\cos(\theta )&sin(\theta )\\sin(\theta )&-cos(\theta )\end{pmatrix}}} The other possible way for mutations to occur is due to errors in the replication process. Due to the no-cloning theorem, it is impossible to produce perfect copies of systems that are originally in unknown quantum states. However, quantum cloning machines make it possible to create imperfect copies of quantum states, in other words, the process introduces some degree of error. The error that exists in current quantum cloning machines is the root cause for the second kind of mutations in the artificial life experiment. The imperfect cloning operation can be seen as: U M ( θ ) = I 4 + 1 2 ( 0 0 0 1 ) ⊗ ( − 1 1 1 − 1 ) ( c o s θ + i s i n θ + 1 ) {\displaystyle U_{M}(\theta )=\mathrm {I} _{4}+{\frac {1}{2}}{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\otimes {\begin{pmatrix}-1&1\\1&-1\end{pmatrix}}(cos\theta +isin\theta +1)} The two kinds of mutations affect the individual differently. While the spontaneous M operation does not affect the phenotype of the individual, the self-replicating error mutation, UM, alters both the genotype of the individual, and its associated lifetime. The presence of mutations in the quantum artificial life experiment is critical for providing randomness and biodiversity. The inclusion of mutations helps to increase the accuracy of the quantum algorithm. === Death === At the instant the individual is created (when the genotype is copied into the phenotype), the phenotype interacts with the environment. As time evolves, the interaction of the individual with the environment simulates aging which eventually leads to the death of the individual. The death of an individual occurs when the expectation value of σ z {\displaystyle \sigma _{z}} is within some ϵ {\displaystyle \epsilon } of 1 in the phenotype, or, equivalently, when ρ p = | 0 ⟩ ⟨ 0 | {\displaystyle \rho _{p}=|0\rangle \langle 0|} The Lindbladian describes the interaction of the individual with the environment: ρ

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  Evolvability (computer science)

  The term evolvability is a framework of computational learning introduced by Leslie Valiant in his paper of the same name. The aim of this theory is to model biological evolution and categorize which types of mechanisms are evolvable. Evolution is an extension of PAC learning and learning from statistical queries. == General framework == Let F n {\displaystyle F_{n}\,} and R n {\displaystyle R_{n}\,} be collections of functions on n {\displaystyle n\,} variables. Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , the goal is to find by local search a representation r ∈ R n {\displaystyle r\in R_{n}} that closely approximates f {\displaystyle f\,} . This closeness is measured by the performance Perf ⁡ ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} of r {\displaystyle r\,} with respect to f {\displaystyle f\,} . As is the case in the biological world, there is a difference between genotype and phenotype. In general, there can be multiple representations (genotypes) that correspond to the same function (phenotype). That is, for some r , r ′ ∈ R n {\displaystyle r,r'\in R_{n}} , with r ≠ r ′ {\displaystyle r\neq r'\,} , still r ( x ) = r ′ ( x ) {\displaystyle r(x)=r'(x)\,} for all x ∈ X n {\displaystyle x\in X_{n}} . However, this need not be the case. The goal then, is to find a representation that closely matches the phenotype of the ideal function, and the spirit of the local search is to allow only small changes in the genotype. Let the neighborhood N ( r ) {\displaystyle N(r)\,} of a representation r {\displaystyle r\,} be the set of possible mutations of r {\displaystyle r\,} . For simplicity, consider Boolean functions on X n = { − 1 , 1 } n {\displaystyle X_{n}=\{-1,1\}^{n}\,} , and let D n {\displaystyle D_{n}\,} be a probability distribution on X n {\displaystyle X_{n}\,} . Define the performance in terms of this. Specifically, Perf ⁡ ( f , r ) = ∑ x ∈ X n f ( x ) r ( x ) D n ( x ) . {\displaystyle \operatorname {Perf} (f,r)=\sum _{x\in X_{n}}f(x)r(x)D_{n}(x).} Note that Perf ⁡ ( f , r ) = Prob ⁡ ( f ( x ) = r ( x ) ) − Prob ⁡ ( f ( x ) ≠ r ( x ) ) . {\displaystyle \operatorname {Perf} (f,r)=\operatorname {Prob} (f(x)=r(x))-\operatorname {Prob} (f(x)\neq r(x)).} In general, for non-Boolean functions, the performance will not correspond directly to the probability that the functions agree, although it will have some relationship. Throughout an organism's life, it will only experience a limited number of environments, so its performance cannot be determined exactly. The empirical performance is defined by Perf s ⁡ ( f , r ) = 1 s ∑ x ∈ S f ( x ) r ( x ) , {\displaystyle \operatorname {Perf} _{s}(f,r)={\frac {1}{s}}\sum _{x\in S}f(x)r(x),} where S {\displaystyle S\,} is a multiset of s {\displaystyle s\,} independent selections from X n {\displaystyle X_{n}\,} according to D n {\displaystyle D_{n}\,} . If s {\displaystyle s\,} is large enough, evidently Perf s ⁡ ( f , r ) {\displaystyle \operatorname {Perf} _{s}(f,r)} will be close to the actual performance Perf ⁡ ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} . Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , initial representation r ∈ R n {\displaystyle r\in R_{n}} , sample size s {\displaystyle s\,} , and tolerance t {\displaystyle t\,} , the mutator Mut ⁡ ( f , r , s , t ) {\displaystyle \operatorname {Mut} (f,r,s,t)} is a random variable defined as follows. Each r ′ ∈ N ( r ) {\displaystyle r'\in N(r)} is classified as beneficial, neutral, or deleterious, depending on its empirical performance. Specifically, r ′ {\displaystyle r'\,} is a beneficial mutation if Perf s ⁡ ( f , r ′ ) − Perf s ⁡ ( f , r ) ≥ t {\displaystyle \operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r)\geq t} ; r ′ {\displaystyle r'\,} is a neutral mutation if − t < Perf s ⁡ ( f , r ′ ) − Perf s ⁡ ( f , r ) < t {\displaystyle -t<\operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r) 0 {\displaystyle \epsilon >0\,} , for all ideal functions f ∈ F n {\displaystyle f\in F_{n}} and representations r 0 ∈ R n {\displaystyle r_{0}\in R_{n}} , with probability at least 1 − ϵ {\displaystyle 1-\epsilon \,} , Perf ⁡ ( f , r g ( n , 1 / ϵ ) ) ≥ 1 − ϵ , {\displaystyle \operatorname {Perf} (f,r_{g(n,1/\epsilon )})\geq 1-\epsilon ,} where the sizes of neighborhoods N ( r ) {\displaystyle N(r)\,} for r ∈ R n {\displaystyle r\in R_{n}\,} are at most p ( n , 1 / ϵ ) {\displaystyle p(n,1/\epsilon )\,} , the sample size is s ( n , 1 / ϵ ) {\displaystyle s(n,1/\epsilon )\,} , the tolerance is t ( 1 / n , ϵ ) {\displaystyle t(1/n,\epsilon )\,} , and the generation size is g ( n , 1 / ϵ ) {\displaystyle g(n,1/\epsilon )\,} . F {\displaystyle F\,} is evolvable over D {\displaystyle D\,} if it is evolvable by some R {\displaystyle R\,} over D {\displaystyle D\,} . F {\displaystyle F\,} is evolvable if it is evolvable over all distributions D {\displaystyle D\,} . == Results == The class of conjunctions and the class of disjunctions are evolvable over the uniform distribution for short conjunctions and disjunctions, respectively. The class of parity functions (which evaluate to the parity of the number of true literals in a given subset of literals) are not evolvable, even for the uniform distribution. Evolvability implies PAC learnability.

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  Geometric hashing

  In computer science, geometric hashing is a method for efficiently finding two-dimensional objects represented by discrete points that have undergone an affine transformation, though extensions exist to other object representations and transformations. In an off-line step, the objects are encoded by treating each pair of points as a geometric basis. The remaining points can be represented in an invariant fashion with respect to this basis using two parameters. For each point, its quantized transformed coordinates are stored in the hash table as a key, and indices of the basis points as a value. Then a new pair of basis points is selected, and the process is repeated. In the on-line (recognition) step, randomly selected pairs of data points are considered as candidate bases. For each candidate basis, the remaining data points are encoded according to the basis and possible correspondences from the object are found in the previously constructed table. The candidate basis is accepted if a sufficiently large number of the data points index a consistent object basis. Geometric hashing was originally suggested in computer vision for object recognition in 2D and 3D, but later was applied to different problems such as structural alignment of proteins. == Geometric hashing in computer vision == Geometric hashing is a method used for object recognition. Let’s say that we want to check if a model image can be seen in an input image. This can be accomplished with geometric hashing. The method could be used to recognize one of the multiple objects in a base, in this case the hash table should store not only the pose information but also the index of object model in the base. === Example === For simplicity, this example will not use too many point features and assume that their descriptors are given by their coordinates only (in practice local descriptors such as SIFT could be used for indexing). ==== Training Phase ==== Find the model's feature points. Assume that 5 feature points are found in the model image with the coordinates ( 12 , 17 ) ; {\displaystyle (12,17);} ( 45 , 13 ) ; {\displaystyle (45,13);} ( 40 , 46 ) ; {\displaystyle (40,46);} ( 20 , 35 ) ; {\displaystyle (20,35);} ( 35 , 25 ) {\displaystyle (35,25)} , see the picture. Introduce a basis to describe the locations of the feature points. For 2D space and similarity transformation the basis is defined by a pair of points. The point of origin is placed in the middle of the segment connecting the two points (P2, P4 in our example), the x ′ {\displaystyle x'} axis is directed towards one of them, the y ′ {\displaystyle y'} is orthogonal and goes through the origin. The scale is selected such that absolute value of x ′ {\displaystyle x'} for both basis points is 1. Describe feature locations with respect to that basis, i.e. compute the projections to the new coordinate axes. The coordinates should be discretised to make recognition robust to noise, we take the bin size 0.25. We thus get the coordinates ( − 0.75 , − 1.25 ) ; {\displaystyle (-0.75,-1.25);} ( 1.00 , 0.00 ) ; {\displaystyle (1.00,0.00);} ( − 0.50 , 1.25 ) ; {\displaystyle (-0.50,1.25);} ( − 1.00 , 0.00 ) ; {\displaystyle (-1.00,0.00);} ( 0.00 , 0.25 ) {\displaystyle (0.00,0.25)} Store the basis in a hash table indexed by the features (only transformed coordinates in this case). If there were more objects to match with, we should also store the object number along with the basis pair. Repeat the process for a different basis pair (Step 2). It is needed to handle occlusions. Ideally, all the non-colinear pairs should be enumerated. We provide the hash table after two iterations, the pair (P1, P3) is selected for the second one. Hash Table: Most hash tables cannot have identical keys mapped to different values. So in real life one won’t encode basis keys (1.0, 0.0) and (-1.0, 0.0) in a hash table. ==== Recognition Phase ==== Find interesting feature points in the input image. Choose an arbitrary basis. If there isn't a suitable arbitrary basis, then it is likely that the input image does not contain the target object. Describe coordinates of the feature points in the new basis. Quantize obtained coordinates as it was done before. Compare all the transformed point features in the input image with the hash table. If the point features are identical or similar, then increase the count for the corresponding basis (and the type of object, if any). For each basis such that the count exceeds a certain threshold, verify the hypothesis that it corresponds to an image basis chosen in Step 2. Transfer the image coordinate system to the model one (for the supposed object) and try to match them. If successful, the object is found. Otherwise, go back to Step 2. === Finding mirrored pattern === It seems that this method is only capable of handling scaling, translation, and rotation. However, the input image may contain the object in mirror transform. Therefore, geometric hashing should be able to find the object, too. There are two ways to detect mirrored objects. For the vector graph, make the left side positive, and the right side negative. Multiplying the x position by -1 will give the same result. Use 3 points for the basis. This allows detecting mirror images (or objects). Actually, using 3 points for the basis is another approach for geometric hashing. === Geometric hashing in higher-dimensions === Similar to the example above, hashing applies to higher-dimensional data. For three-dimensional data points, three points are also needed for the basis. The first two points define the x-axis, and the third point defines the y-axis (with the first point). The z-axis is perpendicular to the created axis using the right-hand rule. Notice that the order of the points affects the resulting basis

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  Problem solving

  Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to get from point A to B) to complex issues in business and technical fields. The former is an example of simple problem solving (SPS) addressing one issue, whereas the latter is complex problem solving (CPS) with multiple interrelated obstacles. Another classification of problem-solving tasks is into well-defined problems with specific obstacles and goals, and ill-defined problems in which the current situation is troublesome but it is not clear what kind of resolution to aim for. Similarly, one may distinguish formal or fact-based problems requiring psychometric intelligence, versus socio-emotional problems which depend on the changeable emotions of individuals or groups, such as tactful behavior, fashion, or gift choices. Solutions require sufficient resources and knowledge to attain the goal. Professionals such as lawyers, doctors, programmers, and consultants are largely problem solvers for issues that require technical skills and knowledge beyond general competence. Many businesses have found profitable markets by recognizing a problem and creating a solution: the more widespread and inconvenient the problem, the greater the opportunity to develop a scalable solution. There are many specialized problem-solving techniques and methods in fields such as science, engineering, business, medicine, mathematics, computer science, philosophy, and social organization. The mental techniques to identify, analyze, and solve problems are studied in psychology and cognitive sciences. Also widely researched are the mental obstacles that prevent people from finding solutions; problem-solving impediments include confirmation bias, mental set, and functional fixedness. == Definition == The term problem solving has a slightly different meaning depending on the discipline. For instance, it is a mental process in psychology and a computerized process in computer science. There are two different types of problems: ill-defined and well-defined; different approaches are used for each. Well-defined problems have specific end goals and clearly expected solutions, while ill-defined problems do not. Well-defined problems allow for more initial planning than ill-defined problems. Solving problems sometimes involves dealing with pragmatics (the way that context contributes to meaning) and semantics (the interpretation of the problem). The ability to understand what the end goal of the problem is, and what rules could be applied, represents the key to solving the problem. Sometimes a problem requires abstract thinking or coming up with a creative solution. Problem solving has two major domains: mathematical problem solving and personal problem solving. Each concerns some difficulty or barrier that is encountered. === Psychology === Problem solving in psychology refers to the process of finding solutions to problems encountered in life. Solutions to these problems are usually situation- or context-specific. The process starts with problem finding and problem shaping, in which the problem is discovered and simplified. The next step is to generate possible solutions and evaluate them. Finally a solution is selected to be implemented and verified. Problems have an end goal to be reached; how you get there depends upon problem orientation (problem-solving coping style and skills) and systematic analysis. Mental health professionals study the human problem-solving processes using methods such as introspection, behaviorism, simulation, computer modeling, and experiment. Social psychologists look into the person-environment relationship aspect of the problem and independent and interdependent problem-solving methods. Problem solving has been defined as a higher-order cognitive process and intellectual function that requires the modulation and control of more routine or fundamental skills. Empirical research shows many different strategies and factors influence everyday problem solving. Rehabilitation psychologists studying people with frontal lobe injuries have found that deficits in emotional control and reasoning can be re-mediated with effective rehabilitation and could improve the capacity of injured persons to resolve everyday problems. Interpersonal everyday problem solving is dependent upon personal motivational and contextual components. One such component is the emotional valence of "real-world" problems, which can either impede or aid problem-solving performance. Researchers have focused on the role of emotions in problem solving, demonstrating that poor emotional control can disrupt focus on the target task, impede problem resolution, and lead to negative outcomes such as fatigue, depression, and inertia. In conceptualization,human problem solving consists of two related processes: problem orientation, and the motivational/attitudinal/affective approach to problematic situations and problem-solving skills. People's strategies cohere with their goals and stem from the process of comparing oneself with others. === Cognitive sciences === Among the first experimental psychologists to study problem solving were the Gestaltists in Germany, such as Karl Duncker in The Psychology of Productive Thinking (1935). Perhaps best known is the work of Allen Newell and Herbert A. Simon. Experiments in the 1960s and early 1970s asked participants to solve relatively simple, well-defined, but not previously seen laboratory tasks. These simple problems, such as the Tower of Hanoi, admitted optimal solutions that could be found quickly, allowing researchers to observe the full problem-solving process. Researchers assumed that these model problems would elicit the characteristic cognitive processes by which more complex "real world" problems are solved. An outstanding problem-solving technique found by this research is the principle of decomposition. === Computer science === Much of computer science and artificial intelligence involves designing automated systems to solve a specified type of problem: to accept input data and calculate a correct or adequate response, reasonably quickly. Algorithms are recipes or instructions that direct such systems, written into computer programs. Steps for designing such systems include problem determination, heuristics, root cause analysis, de-duplication, analysis, diagnosis, and repair. Analytic techniques include linear and nonlinear programming, queuing systems, and simulation. A large, perennial obstacle is to find and fix errors in computer programs: debugging. === Logic === Formal logic concerns issues like validity, truth, inference, argumentation, and proof. In a problem-solving context, it can be used to formally represent a problem as a theorem to be proved, and to represent the knowledge needed to solve the problem as the premises to be used in a proof that the problem has a solution. The use of computers to prove mathematical theorems using formal logic emerged as the field of automated theorem proving in the 1950s. It included the use of heuristic methods designed to simulate human problem solving, as in the Logic Theory Machine, developed by Allen Newell, Herbert A. Simon and J. C. Shaw, as well as algorithmic methods such as the resolution principle developed by John Alan Robinson. In addition to its use for finding proofs of mathematical theorems, automated theorem-proving has also been used for program verification in computer science. In 1958, John McCarthy proposed the advice taker, to represent information in formal logic and to derive answers to questions using automated theorem-proving. An important step in this direction was made by Cordell Green in 1969, who used a resolution theorem prover for question-answering and for such other applications in artificial intelligence as robot planning. The resolution theorem-prover used by Cordell Green bore little resemblance to human problem solving methods. In response to criticism of that approach from researchers at MIT, Robert Kowalski developed logic programming and SLD resolution, which solves problems by problem decomposition. He has advocated logic for both computer and human problem solving and computational logic to improve human thinking. === Engineering === When products or processes fail, problem solving techniques can be used to develop corrective actions that can be taken to prevent further failures. Such techniques can also be applied to a product or process prior to an actual failure event—to predict, analyze, and mitigate a potential problem in advance. Techniques such as failure mode and effects analysis can proactively reduce the likelihood of problems. In either the reactive or the proactive case, it is necessary to build a causal explanation through a process of diagnosis. In deriving an explanation of effects in terms of causes, abduction generates new ideas or hypothes

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  Sycophancy (artificial intelligence)

  In the field of artificial intelligence, sycophancy is a tendency of large language models (LLMs) and other AI assistants to tailor their responses to what they predict the user wants to hear rather than to what is accurate or warranted. The behavior takes several forms: an assistant may agree with a user's stated opinion even when the user is mistaken; it may abandon a correct answer after a challenge such as "are you sure?"; it may validate beliefs, decisions or self-presentation regardless of merit; or it may praise the user, their work or their ideas in unwarranted terms. The word is borrowed from the ordinary English term for fawning flattery, and is used in AI alignment and AI safety research to describe a class of misalignment failures associated with training on human feedback. Researchers at Anthropic first documented the behavior systematically in 2022. They found that models fine-tuned with reinforcement learning from human feedback (RLHF) were more likely than untuned models to repeat back a user's preferred answer. A 2023 follow-up paper, "Towards Understanding Sycophancy in Language Models", showed that five frontier assistants from OpenAI, Anthropic and Meta all exhibited the behavior, and traced its origin to biases in the human preference data used during training. Later work documented sycophancy in mathematics, medicine, academic peer review and other domains, and identified a broader category called "social sycophancy" affecting an assistant's emotional and interpersonal responses. The issue drew widespread public attention in April 2025 after OpenAI rolled back an update to its GPT-4o model. Users had reported that the assistant praised dangerous decisions, endorsed delusional thinking and offered exaggerated compliments for trivial prompts. OpenAI's post-mortem attributed the change in behavior to an additional training signal based on user thumbs-up and thumbs-down feedback. That episode, together with reporting in The New York Times, Rolling Stone and elsewhere on users drawn into delusional thinking through prolonged chatbot interaction, has been cited in litigation and in academic studies as evidence that sycophancy poses risks to user well-being. Proposed mitigations include fine-tuning on synthetic data that rewards disagreement with incorrect user statements, editing the small subset of model parameters causally responsible for the behavior, changes to the dialogue or system prompt, and benchmarks designed to surface sycophantic behavior before models are released. == Causes == The dominant explanation points to RLHF, the standard technique for aligning chat assistants with user expectations. Human annotators rank candidate model responses; a reward model is trained to predict those rankings; and the language model is then optimized against the reward model. Because human raters tend to prefer outputs that confirm their existing beliefs or flatter their work, the pipeline systematically rewards responses that agree with the annotator. Perez and colleagues at Anthropic published the first large-scale empirical evidence of the effect in 2022. They reported that RLHF training increased the probability that a model would repeat back a dialog user's preferred answer, and that larger models exhibited the behavior more strongly. Sharma and colleagues, the following year, went further and examined Anthropic's own preference data directly. Both the human raters and the reward models trained on their judgments preferred convincingly written sycophantic responses to truthful ones at a non-negligible rate. Wei and co-authors at Google DeepMind found similar results in the PaLM family, observing that both model scale and instruction tuning increased sycophancy on opinion questions. The behavior is often classified as a form of reward hacking, in which an optimization process exploits a flaw in its reward signal rather than achieving the intended objective. OpenAI's post-mortem of the April 2025 GPT-4o incident identified a more specific mechanism. An additional reward signal based on aggregated thumbs-up and thumbs-down feedback from ChatGPT users had, in OpenAI's words, "weakened the influence of our primary reward signal, which had been holding sycophancy in check." Separately, an Anthropic interpretability paper from 2025 located a linear direction in a model's internal activations corresponding to sycophantic behavior, and showed that such "persona vectors" could be used to flag sycophancy-inducing training data and to steer models away from the trait at inference time. == Measurement == The Anthropic team released SycophancyEval with its 2023 paper, supplying test sets for each of the four canonical behaviors. Two further benchmarks from Stanford followed in 2025. SycEval, applied to mathematical and medical reasoning tasks, reported an overall sycophancy rate of 58 per cent across the GPT-4o, Claude and Gemini models tested. ELEPHANT, aimed at social sycophancy, found that the eleven LLMs evaluated affirmed posts that the Reddit community r/AmITheAsshole had judged inappropriate in 42 per cent of cases, and preserved a user's face 45 percentage points more often than human respondents did. Domain-specific benchmarks have followed. BrokenMath tests robustness to plausible-looking but false mathematical claims drawn from competition problems, and reports that the best evaluated model was sycophantic in 29 per cent of cases. SYCON-Bench measures how many dialogue turns are required before a model abandons a correct position. Visual sycophancy in multimodal models has been examined with MM-SY and PENDULUM. A 2026 study by researchers at the Massachusetts Institute of Technology reported that personalization features, which adapt assistants to individual users over repeated sessions, can intensify social sycophancy. == Notable incidents == === GPT-4o rollback (April 2025) === On 25 April 2025, OpenAI completed the rollout of an update to GPT-4o, the default model used in ChatGPT at the time. Within days, users reported that the assistant had begun praising trivial messages in extravagant terms, endorsing impulsive or dangerous decisions, and reinforcing strong emotional statements without pushback. Widely shared examples included the model congratulating a user who reported stopping prescribed psychiatric medication, and praising a business plan to sell "shit on a stick" as venture-capital ready. OpenAI's chief executive, Sam Altman, wrote on 27 April that recent updates had made the model "too sycophant-y and annoying" and said fixes were in progress. The company began reverting the update on 28 April and completed the rollback for free users by 30 April. Two post-mortems followed: a short note on 29 April and a longer technical follow-up, "Expanding on what we missed with sycophancy", on 2 May. Both attributed the regression to a new training signal based on user thumbs-up and thumbs-down feedback, to inadequate pre-launch evaluation for sycophantic drift, and to the dismissal of qualitative concerns raised by internal testers before release. Reporting in CNN, Fortune and Bloomberg News treated the incident as a turning point in public awareness of the problem. === Chatbot-related psychological harm === From mid-2025 onward, news reports began to link sycophantic chatbot behavior to acute psychological harm. In June 2025, The New York Times technology reporter Kashmir Hill published an investigation centered on Eugene Torres, a Manhattan accountant with no history of mental illness, who developed a sustained delusional episode after a series of conversations with ChatGPT about simulation theory. According to the article, the assistant encouraged Torres to stop taking prescribed medication, to cut off friends and family, and at one point told him that he could fly from a nineteen-story building if he "truly believed". Futurism and Rolling Stone ran parallel investigations documenting other cases in which heavy use of ChatGPT had been associated with delusional thinking, involuntary commitment or, in at least one case, the death of a user with a pre-existing psychiatric diagnosis. A 2026 paper by researchers at the Massachusetts Institute of Technology and the University of Washington put forward a formal Bayesian model. It showed that even an ideally rational user could be drawn into what the authors call "delusional spiraling" when interacting with a sufficiently sycophantic assistant, and that the effect was not eliminated by suppressing hallucinations or by warning users in advance. The lawsuit Raine v. OpenAI, filed in San Francisco Superior Court in August 2025 by the parents of a sixteen-year-old who had died by suicide, alleges that "heightened sycophancy" was a design feature of ChatGPT that contributed to their son's death; it is the first wrongful-death suit against a large language-model provider. === Wider commentary === Mainstream coverage in outlets including The New York Times, The Washington Pos

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  Cross-entropy method

  The cross-entropy (CE) method is a Monte Carlo method for importance sampling and optimization. It is applicable to both combinatorial and continuous problems, with either a static or noisy objective. The method approximates the optimal importance sampling estimator by repeating two phases: Draw a sample from a probability distribution. Minimize the cross-entropy between this distribution and a target distribution to produce a better sample in the next iteration. Reuven Rubinstein developed the method in the context of rare-event simulation, where tiny probabilities must be estimated, for example in network reliability analysis, queueing models, or performance analysis of telecommunication systems. The method has also been applied to the traveling salesman, quadratic assignment, DNA sequence alignment, max-cut and buffer allocation problems. == Estimation via importance sampling == Consider the general problem of estimating the quantity ℓ = E u [ H ( X ) ] = ∫ H ( x ) f ( x ; u ) d x {\displaystyle \ell =\mathbb {E} _{\mathbf {u} }[H(\mathbf {X} )]=\int H(\mathbf {x} )\,f(\mathbf {x} ;\mathbf {u} )\,{\textrm {d}}\mathbf {x} } , where H {\displaystyle H} is some performance function and f ( x ; u ) {\displaystyle f(\mathbf {x} ;\mathbf {u} )} is a member of some parametric family of distributions. Using importance sampling this quantity can be estimated as ℓ ^ = 1 N ∑ i = 1 N H ( X i ) f ( X i ; u ) g ( X i ) {\displaystyle {\hat {\ell }}={\frac {1}{N}}\sum _{i=1}^{N}H(\mathbf {X} _{i}){\frac {f(\mathbf {X} _{i};\mathbf {u} )}{g(\mathbf {X} _{i})}}} , where X 1 , … , X N {\displaystyle \mathbf {X} _{1},\dots ,\mathbf {X} _{N}} is a random sample from g {\displaystyle g\,} . For positive H {\displaystyle H} , the theoretically optimal importance sampling density (PDF) is given by g ∗ ( x ) = H ( x ) f ( x ; u ) / ℓ {\displaystyle g^{}(\mathbf {x} )=H(\mathbf {x} )f(\mathbf {x} ;\mathbf {u} )/\ell } . This, however, depends on the unknown ℓ {\displaystyle \ell } . The CE method aims to approximate the optimal PDF by adaptively selecting members of the parametric family that are closest (in the Kullback–Leibler sense) to the optimal PDF g ∗ {\displaystyle g^{}} . == Generic CE algorithm == Choose initial parameter vector v ( 0 ) {\displaystyle \mathbf {v} ^{(0)}} ; set t = 1. Generate a random sample X 1 , … , X N {\displaystyle \mathbf {X} _{1},\dots ,\mathbf {X} _{N}} from f ( ⋅ ; v ( t − 1 ) ) {\displaystyle f(\cdot ;\mathbf {v} ^{(t-1)})} Solve for v ( t ) {\displaystyle \mathbf {v} ^{(t)}} , where v ( t ) = argmax v ⁡ 1 N ∑ i = 1 N H ( X i ) f ( X i ; u ) f ( X i ; v ( t − 1 ) ) log ⁡ f ( X i ; v ) {\displaystyle \mathbf {v} ^{(t)}=\mathop {\textrm {argmax}} _{\mathbf {v} }{\frac {1}{N}}\sum _{i=1}^{N}H(\mathbf {X} _{i}){\frac {f(\mathbf {X} _{i};\mathbf {u} )}{f(\mathbf {X} _{i};\mathbf {v} ^{(t-1)})}}\log f(\mathbf {X} _{i};\mathbf {v} )} If convergence is reached then stop; otherwise, increase t by 1 and reiterate from step 2. In several cases, the solution to step 3 can be found analytically. Situations in which this occurs are When f {\displaystyle f\,} belongs to the natural exponential family When f {\displaystyle f\,} is discrete with finite support When H ( X ) = I { x ∈ A } {\displaystyle H(\mathbf {X} )=\mathrm {I} _{\{\mathbf {x} \in A\}}} and f ( X i ; u ) = f ( X i ; v ( t − 1 ) ) {\displaystyle f(\mathbf {X} _{i};\mathbf {u} )=f(\mathbf {X} _{i};\mathbf {v} ^{(t-1)})} , then v ( t ) {\displaystyle \mathbf {v} ^{(t)}} corresponds to the maximum likelihood estimator based on those X k ∈ A {\displaystyle \mathbf {X} _{k}\in A} . == Continuous optimization—example == The same CE algorithm can be used for optimization, rather than estimation. Suppose the problem is to maximize some function S {\displaystyle S} , for example, S ( x ) = e − ( x − 2 ) 2 + 0.8 e − ( x + 2 ) 2 {\displaystyle S(x)={\textrm {e}}^{-(x-2)^{2}}+0.8\,{\textrm {e}}^{-(x+2)^{2}}} . To apply CE, one considers first the associated stochastic problem of estimating P θ ( S ( X ) ≥ γ ) {\displaystyle \mathbb {P} _{\boldsymbol {\theta }}(S(X)\geq \gamma )} for a given level γ {\displaystyle \gamma \,} , and parametric family { f ( ⋅ ; θ ) } {\displaystyle \left\{f(\cdot ;{\boldsymbol {\theta }})\right\}} , for example the 1-dimensional Gaussian distribution, parameterized by its mean μ t {\displaystyle \mu _{t}\,} and variance σ t 2 {\displaystyle \sigma _{t}^{2}} (so θ = ( μ , σ 2 ) {\displaystyle {\boldsymbol {\theta }}=(\mu ,\sigma ^{2})} here). Hence, for a given γ {\displaystyle \gamma \,} , the goal is to find θ {\displaystyle {\boldsymbol {\theta }}} so that D K L ( I { S ( x ) ≥ γ } ‖ f θ ) {\displaystyle D_{\mathrm {KL} }({\textrm {I}}_{\{S(x)\geq \gamma \}}\|f_{\boldsymbol {\theta }})} is minimized. This is done by solving the sample version (stochastic counterpart) of the KL divergence minimization problem, as in step 3 above. It turns out that parameters that minimize the stochastic counterpart for this choice of target distribution and parametric family are the sample mean and sample variance corresponding to the elite samples, which are those samples that have objective function value ≥ γ {\displaystyle \geq \gamma } . The worst of the elite samples is then used as the level parameter for the next iteration. This yields the following randomized algorithm that happens to coincide with the so-called Estimation of Multivariate Normal Algorithm (EMNA), an estimation of distribution algorithm. === Pseudocode === // Initialize parameters μ := −6 σ2 := 100 t := 0 maxits := 100 N := 100 Ne := 10 // While maxits not exceeded and not converged while t < maxits and σ2 > ε do // Obtain N samples from current sampling distribution X := SampleGaussian(μ, σ2, N) // Evaluate objective function at sampled points S := exp(−(X − 2) ^ 2) + 0.8 exp(−(X + 2) ^ 2) // Sort X by objective function values in descending order X := sort(X, S) // Update parameters of sampling distribution via elite samples μ := mean(X(1:Ne)) σ2 := var(X(1:Ne)) t := t + 1 // Return mean of final sampling distribution as solution return μ == Related methods == Simulated annealing Genetic algorithms Harmony search Estimation of distribution algorithm Tabu search Natural Evolution Strategy Ant colony optimization algorithms

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  Multimodal representation learning

  Multimodal representation learning is a subfield of representation learning focused on integrating and interpreting information from different modalities, such as text, images, audio, or video, by projecting them into a shared latent space. This allows for semantically similar content across modalities to be mapped to nearby points within that space, facilitating a unified understanding of diverse data types. By automatically learning meaningful features from each modality and capturing their inter-modal relationships, multimodal representation learning enables a unified representation that enhances performance in cross-media analysis tasks such as video classification, event detection, and sentiment analysis. It also supports cross-modal retrieval and translation, including image captioning, video description, and text-to-image synthesis. == Motivation == The primary motivations for multimodal representation learning arise from the inherent nature of real-world data and the limitations of unimodal approaches. Since multimodal data offers complementary and supplementary information about an object or event from different perspectives, it is more informative than relying on a single modality. A key motivation is to narrow the heterogeneity gap that exists between different modalities by projecting their features into a shared semantic subspace. This allows semantically similar content across modalities to be represented by similar vectors, facilitating the understanding of relationships and correlations between them. Multimodal representation learning aims to leverage the unique information provided by each modality to achieve a more comprehensive and accurate understanding of concepts. These unified representations are crucial for improving performance in various cross-media analysis tasks such as video classification, event detection, and sentiment analysis. They also enable cross-modal retrieval, allowing users to search and retrieve content across different modalities. Additionally, it facilitates cross-modal translation, where information can be converted from one modality to another, as seen in applications like image captioning and text-to-image synthesis. The abundance of ubiquitous multimodal data in real-world applications, including understudied areas like healthcare, finance, and human-computer interaction (HCI), further motivates the development of effective multimodal representation learning techniques. == Approaches and methods == === Canonical-correlation analysis based methods === Canonical-correlation analysis (CCA) was first introduced in 1936 by Harold Hotelling and is a fundamental approach for multimodal learning. CCA aims to find linear relationships between two sets of variables. Given two data matrices X ∈ R n × p {\displaystyle X\in \mathbb {R} ^{n\times p}} and Y ∈ R n × q {\displaystyle Y\in \mathbb {R} ^{n\times q}} representing different modalities, CCA finds projection vectors w x ∈ R p {\displaystyle w_{x}\in \mathbb {R} ^{p}} and w y ∈ R q {\displaystyle w_{y}\in \mathbb {R} ^{q}} that maximizes the correlation between the projected variables: ρ = max w x , w y w x ⊤ Σ x y w y w x ⊤ Σ x x w x w y ⊤ Σ y y w y {\displaystyle \rho =\max _{w_{x},w_{y}}{\frac {w_{x}^{\top }\Sigma _{xy}w_{y}}{{\sqrt {w_{x}^{\top }\Sigma _{xx}w_{x}}}{\sqrt {w_{y}^{\top }\Sigma _{yy}w_{y}}}}}} such that Σ x x {\displaystyle \Sigma _{xx}} and Σ y y {\displaystyle \Sigma _{yy}} are the within-modality covariance matrices, and Σ x y {\displaystyle \Sigma _{xy}} is the between-modality covariance matrix. However, standard CCA is limited by its linearity, which led to the development of nonlinear extensions, such as kernel CCA and deep CCA. ==== Kernel CCA ==== Kernel canonical correlation analysis (KCCA) extends traditional CCA to capture nonlinear relationships between modalities by implicitly mapping the data into high dimensional feature spaces using kernel functions. Given kernel functions K x {\displaystyle K_{x}} and K y {\displaystyle K_{y}} with corresponding Gram matrices K x ∈ R n × n {\displaystyle K_{x}\in \mathbb {R} ^{n\times n}} and K y ∈ R n × n {\displaystyle K_{y}\in \mathbb {R} ^{n\times n}} , KCCA seeks coefficients α {\displaystyle \alpha } and β {\displaystyle \beta } that maximize: ρ = max α , β α ⊤ K x K y β α ⊤ K x 2 α β ⊤ K y 2 β {\displaystyle \rho =\max _{\alpha ,\beta }{\frac {\alpha ^{\top }K_{x}Ky\beta }{{\sqrt {\alpha ^{\top }K_{x}^{2}\alpha }}{\sqrt {\beta ^{\top }K_{y}^{2}\beta }}}}} To prevent overfitting, regularization terms are typically added, resulting in: ρ = max α , β α T K x K y β α T ( K x 2 + λ x K x ) α β T ( K y 2 + λ y K y ) β {\displaystyle \rho =\max _{\alpha ,\beta }{\frac {\alpha ^{T}K_{x}K_{y}\beta }{{\sqrt {\alpha ^{T}\left(K_{x}^{2}+\lambda _{x}K_{x}\right)\alpha }}{\sqrt {\;\beta ^{T}\left(K_{y}^{2}+\lambda _{y}K_{y}\right)\beta }}}}} where λ x {\displaystyle \lambda _{x}} and λ y {\displaystyle \lambda _{y}} are regularization parameters. KCCA has proven effective for tasks such as cross-modal retrieval and semantic analysis, though it faces computational challenges with large datasets due to its O ( n 2 ) {\displaystyle O(n^{2})} memory requirement for sorting kernel matrices. KCCA was proposed independently by several researchers. ==== Deep CCA ==== Deep canonical correlation analysis (DCCA), introduced in 2013, employs neural networks to learn nonlinear transformations for maximizing the correlation between modalities. DCCA uses separate neural networks f x {\displaystyle f_{x}} and f y {\displaystyle f_{y}} for each modality to transform the original data before applying CCA: max W x , W y , θ x , θ y corr ⁡ ( f x ( X ; θ x ) , f y ( Y ; θ y ) ) {\displaystyle \max _{W_{x},W_{y},\theta _{x},\theta _{y}}\operatorname {corr} \left(f_{x}(X;\theta _{x}),f_{y}(Y;\theta _{y})\right)} where θ x {\displaystyle \theta _{x}} and θ y {\displaystyle \theta _{y}} represent the parameters of the neural networks, and W x {\displaystyle W_{x}} and W y {\displaystyle W_{y}} are the CCA projection matrices. The correlation objective is computed as: corr ⁡ ( H x , H y ) = tr ⁡ ( T − 1 / 2 H x T H y S − 1 / 2 ) {\displaystyle \operatorname {corr} (H_{x},H_{y})=\operatorname {tr} \left(T^{-1/2}H_{x}^{T}H_{y}S^{-1/2}\right)} where H x = f x ( X ) {\displaystyle H_{x}=f_{x}(X)} and H y = f y ( Y ) {\displaystyle H_{y}=f_{y}(Y)} are the network outputs, T = H x T H x + r x I {\displaystyle T=H_{x}^{T}H_{x}+r_{x}I} , S = H y T H y + r y I {\displaystyle S=H_{y}^{T}H_{y}+r_{y}I} and r x , r y {\displaystyle r_{x},r_{y}} are the regularization parameters. DCCA overcomes the limitations of linear CCA and kernel CCA by learning complex nonlinear relationships while maintaining computational efficiency for large datasets through mini-batch optimization. === Graph-based methods === Graph-based approaches for multimodal representation learning leverage graph structure to model relationships between entities across different modalities. These methods typically represent each modality as a graph and then learn embedding that preserve cross-modal similarities, enabling more effective joint representation of heterogeneous data. One such method is cross-modal graph neural networks (CMGNNs) that extend traditional graph neural networks (GNNs) to handle data from multiple modalities by constructing graphs that capture both intra-modal and inter-modal relationships. These networks model interactions across modalities by representing them as nodes and their relationships as edges. Other graph-based methods include Probabilistic Graphical Models (PGMs) such as deep belief networks (DBN) and deep Boltzmann machines (DBM). These models can learn a joint representation across modalities, for instance, a multimodal DBN achieves this by adding a shared restricted Boltzmann Machine (RBM) hidden layer on top of modality-specific DBNs. Additionally, the structure of data in some domains like Human-Computer Interaction (HCI), such as the view hierarchy of app screens, can potentially be modeled using graph-like structures. The field of graph representation learning is also relevant, with ongoing progress in developing evaluation benchmarks. === Diffusion maps === Another set of methods relevant to multimodal representation learning are based on diffusion maps and their extensions to handle multiple modalities. ==== Multi-view diffusion maps ==== Multi-view diffusion maps address the challenge of achieving multi-view dimensionality reduction by effectively utilizing the availability of multiple views to extract a coherent low-dimensional representation of the data. The core idea is to exploit both the intrinsic relations within each view and the mutual relations between the different views, defining a cross-view model where a random walk process implicitly hops between objects in different views. A multi-view kernel matrix is constructed by combining these relations, defining a cross-view diffusion process and associ

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  Leakage (machine learning)

  In statistics and machine learning, leakage (also known as data leakage or target leakage) refers to the use of information during model training that would not be available at prediction time. This results in overly optimistic performance estimates, as the model appears to perform better during evaluation than it actually would in a production environment. Leakage is often subtle and indirect, making it difficult to detect and eliminate. It can lead a statistician or modeler to select a suboptimal model, which may be outperformed by a leakage-free alternative. == Leakage modes == Leakage can occur at multiple stages of the machine learning workflow. Broadly, its sources can be divided into two categories: those arising from features and those arising from training examples. === Feature leakage === Feature or column-wise leakage is caused by the inclusion of columns which are one of the following: a duplicate label, a proxy for the label, or the label itself. These features, known as anachronisms, will not be available when the model is used for predictions, and result in leakage if included when the model is trained. For example, including a "MonthlySalary" column when predicting "YearlySalary"; or "MinutesLate" when predicting "IsLate". === Training example leakage === Row-wise leakage is caused by improper sharing of information between rows of data. Types of row-wise leakage include: Premature featurization; leaking from premature featurization before Cross-validation/Train/Test split (must fit MinMax/ngrams/etc on only the train split, then transform the test set) Duplicate rows between train/validation/test (for example, oversampling a dataset to pad its size before splitting; or, different rotations/augmentations of a single image; bootstrap sampling before splitting; or duplicating rows to up sample the minority class) Non-independent and identically distributed random (non-IID) data Time leakage (for example, splitting a time-series dataset randomly instead of newer data in test set using a train/test split or rolling-origin cross-validation) Group leakage—not including a grouping split column (for example, Andrew Ng's group had 100k x-rays of 30k patients, meaning ~3 images per patient. The paper used random splitting instead of ensuring that all images of a patient were in the same split. Hence the model partially memorized the patients instead of learning to recognize pneumonia in chest x-rays.) A 2023 review found data leakage to be "a widespread failure mode in machine-learning (ML)-based science", having affected at least 294 academic publications across 17 disciplines, and causing a potential reproducibility crisis. == Detection == Data leakage in machine learning can be detected through various methods, focusing on performance analysis, feature examination, data auditing, and model behavior analysis. Performance-wise, unusually high accuracy or significant discrepancies between training and test results often indicate leakage. Inconsistent cross-validation outcomes may also signal issues. Feature examination involves scrutinizing feature importance rankings and ensuring temporal integrity in time series data. A thorough audit of the data pipeline is crucial, reviewing pre-processing steps, feature engineering, and data splitting processes. Detecting duplicate entries across dataset splits is also important. For language models, the Min-K% method can detect the presence of data in a pretraining dataset. It presents a sentence suspected to be present in the pretraining dataset, and computes the log-likelihood of each token, then compute the average of the lowest K of these. If this exceeds a threshold, then the sentence is likely present. This method is improved by comparing against a baseline of the mean and variance. Analyzing model behavior can reveal leakage. Models relying heavily on counter-intuitive features or showing unexpected prediction patterns warrant investigation. Performance degradation over time when tested on new data may suggest earlier inflated metrics due to leakage. Advanced techniques include backward feature elimination, where suspicious features are temporarily removed to observe performance changes. Using a separate hold-out dataset for final validation before deployment is advisable.

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  Wetware computer

  A wetware computer is an organic computer (which can also be known as an artificial organic brain or a neurocomputer) composed of organic material "wetware" such as "living" neurons. Wetware computers composed of neurons are different than conventional computers because they use biological materials, and offer the possibility of substantially more energy-efficient computing. While a wetware computer is still largely conceptual, there has been limited success with construction and prototyping, which has acted as a proof of the concept's realistic application to computing in the future. The most notable prototypes have stemmed from the research completed by biological engineer William Ditto during his time at the Georgia Institute of Technology. His work constructing a simple neurocomputer capable of basic addition from leech neurons in 1999 was a significant discovery for the concept. This research was a primary example driving interest in creating these artificially constructed, but still organic brains. == Origins and theoretical foundations == The term wetware came from cyberpunk fiction, notably through Gibson's Neuromancer, but was quickly taken up in scientific literature to explain computation by biological material. Theories of early biological computation borrowed from Alan Turing's morphogenesis model, which showed that chemical interactions could produce complex patterns without centralized control. Hopfield's associative memory networks also provided a foundation for biological information systems with fault tolerance and self-organization. == Major characteristics and processes == Biological wetware systems demonstrate dynamic reconfigurability underpinned by neuroplasticity and enable continuous learning and adaptation. Reaction-diffusion-based computing and molecular logic gates allow spatially parallel information processing unachievable in conventional systems. These systems also show fault tolerance and self-repair at the cellular and network level. The development of cerebral organoids—miniature lab-grown brains—demonstrates spontaneous learning behavior and suggests biological tissue as a viable computational substrate. == Overview == The concept of wetware is an application of specific interest to the field of computer manufacturing. Moore's law, which states that the number of transistors which can be placed on a silicon chip is doubled roughly every two years, has acted as a goal for the industry for decades, but as the size of computers continues to decrease, the ability to meet this goal has become more difficult, threatening to reach a plateau. Due to the difficulty in reducing the size of computers because of size limitations of transistors and integrated circuits, wetware provides an unconventional alternative. A wetware computer composed of neurons is an ideal concept because, unlike conventional materials which operate in binary (on/off), a neuron can shift between thousands of states, constantly altering its chemical conformation, and redirecting electrical pulses through over 200,000 channels in any of its many synaptic connections. Because of this large difference in the possible settings for any one neuron, compared to the binary limitations of conventional computers, the space limitations are far fewer. == Background == The concept of wetware is distinct and unconventional and draws slight resonance with both hardware and software from conventional computers. While hardware is understood as the physical architecture of traditional computational devices, comprising integrated circuits and supporting infrastructure, software represents the encoded architecture of storage and instructions. Wetware is a separate concept that uses the formation of organic molecules, mostly complex cellular structures (such as neurons), to create a computational device such as a computer. In wetware, the ideas of hardware and software are intertwined and interdependent. The molecular and chemical composition of the organic or biological structure would represent not only the physical structure of the wetware but also the software, being continually reprogrammed by the discrete shifts in electrical pulses and chemical concentration gradients as the molecules change their structures to communicate signals. The responsiveness of a cell, proteins, and molecules to changing conformations, both within their structures and around them, ties the idea of internal programming and external structure together in a way that is alien to the current model of conventional computer architecture. The structure of wetware represents a model where the external structure and internal programming are interdependent and unified; meaning that changes to the programming or internal communication between molecules of the device would represent a physical change in the structure. The dynamic nature of wetware borrows from the function of complex cellular structures in biological organisms. The combination of "hardware" and "software" into one dynamic, and interdependent system which uses organic molecules and complexes to create an unconventional model for computational devices is a specific example of applied biorobotics. === The cell as a model of wetware === Cells in many ways can be seen as their form of naturally occurring wetware, similar to the concept that the human brain is the preexisting model system for complex wetware. In his book Wetware: A Computer in Every Living Cell (2009) Dennis Bray explains his theory that cells, which are the most basic form of life, are just a highly complex computational structure, like a computer. To simplify one of his arguments a cell can be seen as a type of computer, using its structured architecture. In this architecture, much like a traditional computer, many smaller components operate in tandem to receive input, process the information, and compute an output. In an overly simplified, non-technical analysis, cellular function can be broken into the following components: Information and instructions for execution are stored as DNA in the cell, RNA acts as a source for distinctly encoded input, processed by ribosomes and other transcription factors to access and process the DNA and to output a protein. Bray's argument in favor of viewing cells and cellular structures as models of natural computational devices is important when considering the more applied theories of wetware to biorobotics. === Biorobotics === Wetware and biorobotics are closely related concepts, which both borrow from similar overall principles. A biorobotic structure can be defined as a system modeled from a preexisting organic complex or model such as cells (neurons) or more complex structures like organs (brain) or whole organisms. Unlike wetware, the concept of biorobotics is not always a system composed of organic molecules, but instead could be composed of conventional material which is designed and assembled in a structure similar or derived from a biological model. Biorobotics have many applications and are used to address the challenges of conventional computer architecture. Conceptually, designing a program, robot, or computational device after a preexisting biological model such as a cell, or even a whole organism, provides the engineer or programmer the benefits of incorporating into the structure the evolutionary advantages of the model. == Effects on users == Wetware technologies such as BCIs and neuromorphic chips offer new possibilities for user autonomy. For those with disabilities, such systems could restore motor or sensory functions and enhance quality of life. However, these technologies raise ethical questions: cognitive privacy, consent over biological data, and risk of exploitation. Without proper oversight, wetware technologies may also widen inequality, favoring those with access to cognitive enhancements. Open governance frameworks and ethical AI design grounded in neuro ethics will be essential. With the development of wetware devices, disparities in access could exacerbate social inequalities, benefiting those who have resources to enhance cognitive or physical abilities. It is necessary to create strong ethical frameworks, inclusive development practices, and open systems of governance to reduce risks and make sure that wetware advances are beneficial to all segments of society. == Applications and goals == === Basic neurocomputer composed of leech neurons === In 1999 William Ditto and his team of researchers at Georgia Institute of Technology and Emory University created a basic form of a wetware computer capable of simple addition by harnessing leech neurons. Leeches were used as a model organism due to the large size of their neuron, and the ease associated with their collection and manipulation. However, these results have never been published in a peer-reviewed journal, prompting questions about the validity of the claims. The computer was able to complete basic addition through electrical probes

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