AI Code Base

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  Skipper (computer software)

  Skipper is a visualization tool and code/schema generator for PHP ORM frameworks like Doctrine2, Doctrine, Propel, and CakePHP, which are used to create database abstraction layer. Skipper is developed by Czech company Inventic, s.r.o. based in Brno, and was known as ORM Designer prior to rebranding in 2014. == Overview == Generates visual model from the schema definition files Repetitive import/export of schema definitions in supported formats (XML, YML, PHP annotations) Schema definition files are automatically generated from the visual model Visual representation uses ER diagram extended by concepts of inheritance and many-to-many Supports customization using .xml configuration files and JavaScript Does not support direct connections to the database Crude and simplistic visual representation and menus == Architecture == Skipper was built on the Qt framework. Import/export of the schema definitions uses XSL transformations powered by LibXslt library. Imported source files are first converted to XML format: no conversion for XML, simple conversion for YML, creating the Abstract Syntax Tree and its subsequent conversion to XML for PHP annotations. The import/export scripts are configured in JavaScript and can be freely customized. == Supported ORM frameworks == Frameworks supported for visual model and schema files generation: Doctrine2 Doctrine CakePHP == History == Skipper was created as an internal tool for the web applications developed by Inventic. It was first published as a commercial tool under the name ORM Designer in 2009. Application was reworked and optimized in January 2013, and released as ORM Designer 2. In May 2013 ORM Designer became part of the South Moravian Innovation Center Incubator program (support program for innovative technological startups). In June 2014, ORM Designer version 3 was released and rebranded under the name of Skipper

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  Liquid state machine

  A liquid state machine (LSM) is a type of reservoir computer that uses a spiking neural network. An LSM consists of a large collection of units (called nodes, or neurons). Each node receives time varying input from external sources (the inputs) as well as from other nodes. Nodes are randomly connected to each other. The recurrent nature of the connections turns the time varying input into a spatio-temporal pattern of activations in the network nodes. The spatio-temporal patterns of activation are read out by linear discriminant units. The soup of recurrently connected nodes will end up computing a large variety of nonlinear functions on the input. Given a large enough variety of such nonlinear functions, it is theoretically possible to obtain linear combinations (using the read out units) to perform whatever mathematical operation is needed to perform a certain task, such as speech recognition or computer vision. The word liquid in the name comes from the analogy drawn to dropping a stone into a still body of water or other liquid. The falling stone will generate ripples in the liquid. The input (motion of the falling stone) has been converted into a spatio-temporal pattern of liquid displacement (ripples). LSMs have been put forward as a way to explain the operation of brains. LSMs are argued to be an improvement over the theory of artificial neural networks because: Circuits are not hard coded to perform a specific task. Continuous time inputs are handled "naturally". Computations on various time scales can be done using the same network. The same network can perform multiple computations. Criticisms of LSMs as used in computational neuroscience are that LSMs don't actually explain how the brain functions. At best they can replicate some parts of brain functionality. There is no guaranteed way to dissect a working network and figure out how or what computations are being performed. There is very little control over the process. == Universal function approximation == If a reservoir has fading memory and input separability, with help of a readout, it can be proven the liquid state machine is a universal function approximator using Stone–Weierstrass theorem.

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  Nearest neighbor search

  Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point. Closeness is typically expressed in terms of a dissimilarity function: the less similar the objects, the larger the function values. Formally, the nearest neighbor (NN) search problem is defined as follows: given a set S of points in a space M and a query point q ∈ M {\displaystyle q\in M} , find the closest point in S to q. Donald Knuth in volume 3 of The Art of Computer Programming (1973) called it the post-office problem, referring to an application of assigning to a residence the nearest post office. A direct generalization of this problem is a k-NN search, where we need to find the k closest points. Most commonly M is a metric space and dissimilarity is expressed as a distance metric, which is symmetric and satisfies the triangle inequality. Even more common, M is taken to be the d-dimensional vector space where dissimilarity is measured using the Euclidean distance, Manhattan distance or other distance metric. However, the dissimilarity function can be arbitrary. One example is asymmetric Bregman divergence, for which the triangle inequality does not hold. == Applications == The nearest neighbor search problem arises in numerous fields of application, including: Pattern recognition – in particular for optical character recognition Statistical classification – see k-nearest neighbor algorithm Computer vision – for point cloud registration Computational geometry – see Closest pair of points problem Cryptanalysis – for lattice problem Databases – e.g. content-based image retrieval Coding theory – see maximum likelihood decoding Semantic search Vector databases, where nearest-neighbor lookup over embeddings is used to retrieve semantically similar records Retrieval-augmented generation systems, where nearest-neighbor retrieval over embeddings is used to fetch candidate passages or documents before generation Data compression – see MPEG-2 standard Robotic sensing Recommendation systems, e.g. see Collaborative filtering Internet marketing – see contextual advertising and behavioral targeting DNA sequencing Spell checking – suggesting correct spelling Plagiarism detection Similarity scores for predicting career paths of professional athletes. Cluster analysis – assignment of a set of observations into subsets (called clusters) so that observations in the same cluster are similar in some sense, usually based on Euclidean distance Chemical similarity Sampling-based motion planning == Methods == Various solutions to the NNS problem have been proposed. The quality and usefulness of the algorithms are determined by the time complexity of queries as well as the space complexity of any search data structures that must be maintained. The informal observation usually referred to as the curse of dimensionality states that there is no general-purpose exact solution for NNS in high-dimensional Euclidean space using polynomial preprocessing and polylogarithmic search time. === Exact methods === ==== Linear search ==== The simplest solution to the NNS problem is to compute the distance from the query point to every other point in the database, keeping track of the "best so far". This algorithm, sometimes referred to as the naive approach, has a running time of O(dN), where N is the cardinality of S and d is the dimensionality of S. There are no search data structures to maintain, so the linear search has no space complexity beyond the storage of the database. Naive search can, on average, outperform space partitioning approaches on higher dimensional spaces. The absolute distance is not required for distance comparison, only the relative distance. In geometric coordinate systems the distance calculation can be sped up considerably by omitting the square root calculation from the distance calculation between two coordinates. The distance comparison will still yield identical results. ==== Space partitioning ==== Since the 1970s, the branch and bound methodology has been applied to the problem. In the case of Euclidean space, this approach encompasses spatial index or spatial access methods. Several space-partitioning methods have been developed for solving the NNS problem. Perhaps the simplest is the k-d tree, which iteratively bisects the search space into two regions containing half of the points of the parent region. Queries are performed via traversal of the tree from the root to a leaf by evaluating the query point at each split. Depending on the distance specified in the query, neighboring branches that might contain hits may also need to be evaluated. For constant dimension query time, average complexity is O(log N) in the case of randomly distributed points, worst case complexity is O(kN^(1-1/k)) Alternatively the R-tree data structure was designed to support nearest neighbor search in dynamic context, as it has efficient algorithms for insertions and deletions such as the R tree. R-trees can yield nearest neighbors not only for Euclidean distance, but can also be used with other distances. In the case of general metric space, the branch-and-bound approach is known as the metric tree approach. Particular examples include vp-tree and BK-tree methods. Using a set of points taken from a 3-dimensional space and put into a BSP tree, and given a query point taken from the same space, a possible solution to the problem of finding the nearest point-cloud point to the query point is given in the following description of an algorithm. (Strictly speaking, no such point may exist, because it may not be unique. But in practice, usually we only care about finding any one of the subset of all point-cloud points that exist at the shortest distance to a given query point.) The idea is, for each branching of the tree, guess that the closest point in the cloud resides in the half-space containing the query point. This may not be the case, but it is a good heuristic. After having recursively gone through all the trouble of solving the problem for the guessed half-space, now compare the distance returned by this result with the shortest distance from the query point to the partitioning plane. This latter distance is that between the query point and the closest possible point that could exist in the half-space not searched. If this distance is greater than that returned in the earlier result, then clearly there is no need to search the other half-space. If there is such a need, then you must go through the trouble of solving the problem for the other half space, and then compare its result to the former result, and then return the proper result. The performance of this algorithm is nearer to logarithmic time than linear time when the query point is near the cloud, because as the distance between the query point and the closest point-cloud point nears zero, the algorithm needs only perform a look-up using the query point as a key to get the correct result. === Approximation methods === An approximate nearest neighbor search algorithm is allowed to return points whose distance from the query is at most c {\displaystyle c} times the distance from the query to its nearest points. The appeal of this approach is that, in many cases, an approximate nearest neighbor is almost as good as the exact one. In particular, if the distance measure accurately captures the notion of user quality, then small differences in the distance should not matter. ==== Greedy search in proximity neighborhood graphs ==== Proximity graph methods (such as navigable small world graphs and HNSW) are considered the current state-of-the-art for the approximate nearest neighbors search. The methods are based on greedy traversing in proximity neighborhood graphs G ( V , E ) {\displaystyle G(V,E)} in which every point x i ∈ S {\displaystyle x_{i}\in S} is uniquely associated with vertex v i ∈ V {\displaystyle v_{i}\in V} . The search for the nearest neighbors to a query q in the set S takes the form of searching for the vertex in the graph G ( V , E ) {\displaystyle G(V,E)} . The basic algorithm – greedy search – works as follows: search starts from an enter-point vertex v i ∈ V {\displaystyle v_{i}\in V} by computing the distances from the query q to each vertex of its neighborhood { v j : ( v i , v j ) ∈ E } {\displaystyle \{v_{j}:(v_{i},v_{j})\in E\}} , and then finds a vertex with the minimal distance value. If the distance value between the query and the selected vertex is smaller than the one between the query and the current element, then the algorithm moves to the selected vertex, and it becomes new enter-point. The algorithm stops when it reaches a local minimum: a vertex whose neighborhood does not contain a vertex that is closer to the query than the vertex itself. The idea of proximity neighborhood graphs was exploited in multiple publications, including the seminal paper by Arya and Mount, in the VoroNet syst

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  Physical neural network

  A physical neural network is a type of artificial neural network in which an electrically adjustable material is used to emulate the function of a neural synapse or a higher-order (dendritic) neuron model. "Physical" neural network is used to emphasize the reliance on physical hardware used to emulate neurons as opposed to software-based approaches. More generally the term is applicable to other artificial neural networks in which a memristor or other electrically adjustable resistance material is used to emulate a neural synapse. == Types of physical neural networks == === ADALINE === In the 1960s Bernard Widrow and Ted Hoff developed ADALINE (Adaptive Linear Neuron) which used electrochemical cells called memistors (memory resistors) to emulate synapses of an artificial neuron. The memistors were implemented as 3-terminal devices operating based on the reversible electroplating of copper such that the resistance between two of the terminals is controlled by the integral of the current applied via the third terminal. The ADALINE circuitry was briefly commercialized by the Memistor Corporation in the 1960s enabling some applications in pattern recognition. However, since the memistors were not fabricated using integrated circuit fabrication techniques the technology was not scalable and was eventually abandoned as solid-state electronics became mature. === Analog VLSI === In 1989 Carver Mead published his book Analog VLSI and Neural Systems, which spun off perhaps the most common variant of analog neural networks. The physical realization is implemented in analog VLSI. This is often implemented as field effect transistors in low inversion. Such devices can be modelled as translinear circuits. This is a technique described by Barrie Gilbert in several papers around mid 1970th, and in particular his Translinear Circuits from 1981. With this method circuits can be analyzed as a set of well-defined functions in steady-state, and such circuits assembled into complex networks. === Physical Neural Network === Alex Nugent describes a physical neural network as one or more nonlinear neuron-like nodes used to sum signals and nanoconnections formed from nanoparticles, nanowires, or nanotubes which determine the signal strength input to the nodes. Alignment or self-assembly of the nanoconnections is determined by the history of the applied electric field performing a function analogous to neural synapses. Numerous applications for such physical neural networks are possible. For example, a temporal summation device can be composed of one or more nanoconnections having an input and an output thereof, wherein an input signal provided to the input causes one or more of the nanoconnection to experience an increase in connection strength thereof over time. Another example of a physical neural network is taught by U.S. Patent No. 7,039,619 entitled "Utilized nanotechnology apparatus using a neural network, a solution and a connection gap," which issued to Alex Nugent by the U.S. Patent & Trademark Office on May 2, 2006. A further application of physical neural network is shown in U.S. Patent No. 7,412,428 entitled "Application of hebbian and anti-hebbian learning to nanotechnology-based physical neural networks," which issued on August 12, 2008. Nugent and Molter have shown that universal computing and general-purpose machine learning are possible from operations available through simple memristive circuits operating the AHaH plasticity rule. More recently, it has been argued that also complex networks of purely memristive circuits can serve as neural networks. === Phase change neural network === In 2002, Stanford Ovshinsky described an analog neural computing medium in which phase-change material has the ability to cumulatively respond to multiple input signals. An electrical alteration of the resistance of the phase change material is used to control the weighting of the input signals. === Memristive neural network === Greg Snider of HP Labs describes a system of cortical computing with memristive nanodevices. The memristors (memory resistors) are implemented by thin film materials in which the resistance is electrically tuned via the transport of ions or oxygen vacancies within the film. DARPA's SyNAPSE project has funded IBM Research and HP Labs, in collaboration with the Boston University Department of Cognitive and Neural Systems (CNS), to develop neuromorphic architectures which may be based on memristive systems. === Protonic artificial synapses === In 2022, researchers reported the development of nanoscale brain-inspired artificial synapses, using the ion proton (H+), for 'analog deep learning'.

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  Purged cross-validation

  Purged cross-validation is a variant of k-fold cross-validation designed to prevent look-ahead bias in time series and other structured data, developed in 2017 by Marcos López de Prado at Guggenheim Partners and Cornell University. It is primarily used in financial machine learning to ensure the independence of training and testing samples when labels depend on future events. It provides an alternative to conventional cross-validation and walk-forward backtesting methods, which often yield overly optimistic performance estimates due to information leakage and overfitting. == Motivation == Standard cross-validation assumes that observations are independently and identically distributed (IID), which often does not hold in time series or financial datasets. If the label of a test sample overlaps in time with the features or labels in the training set, the result may be data leakage and overfitting. Purged cross-validation addresses this issue by removing overlapping observations and, optionally, adding a temporal buffer ("embargo") around the test set to further reduce the risk of leakage. The figure below illustrates standard 5 Fold Cross-Validation == Purging == Purging removes from the training set any observation whose timestamp falls within the time range of formation of a label in the test set. This can be the case for train set observations before and after the test set. Their removal ensures that the algorithm cannot learn during train time information that will be used to assess the performance of the algorithm. See the figure below for an illustration of purging. == Embargoing == Embargoing addresses a more subtle form of leakage: even if an observation does not directly overlap the test set, it may still be affected by test events due to market reaction lag or downstream dependencies. To guard against this, a percentage-based embargo is imposed after each test fold. For example, with a 5% embargo and 1000 observations, the 50 observations following each test fold are excluded from training. Unlike purging, embargoing can only occur after the test set. The figure below illustrates the application of embargo: == Applications == Purged and embargoed cross-validation has been useful in: Backtesting of trading strategies Validation of classifiers on labeled event-driven returns Any machine learning task with overlapping label horizons == Example == To illustrate the effect of purging and embargoing, consider the figures below. Both diagrams show the structure of 5-fold cross-validation over a 20-day period. In each row, blue squares indicate training samples and red squares denote test samples. Each label is defined based on the value of the next two observations, hence creating an overlap. If this overlap is left untreated, test set information leaks into the train set. The second figure applies the Purged CV procedure. Notice how purging removes overlapping observations from the training set and the embargo widens the gap between test and training data. This approach ensures that the evaluation more closely resembles a true out-of-sample test and reduces the risk of backtest overfitting. == Combinatorial Purged Cross-Validation == Walk-forward backtesting analysis, another common cross-validation technique in finance, preserves temporal order but evaluates the model on a single sequence of test sets. This leads to high variance in performance estimation, as results are contingent on a specific historical path. Combinatorial Purged Cross-Validation (CPCV) addresses this limitation by systematically constructing multiple train-test splits, purging overlapping samples, and enforcing an embargo period to prevent information leakage. The result is a distribution of out-of-sample performance estimates, enabling robust statistical inference and more realistic assessment of a model's predictive power. === Methodology === CPCV divides a time-series dataset into N sequential, non-overlapping groups. These groups preserve the temporal order of observations. Then, all combinations of k groups (where k < N) are selected as test sets, with the remaining N − k groups used for training. For each combination, the model is trained and evaluated under strict controls to prevent leakage. To eliminate potential contamination between training and test sets, CPCV introduces two additional mechanisms: Purging: Any training observations whose label horizon overlaps with the test period are excluded. This ensures that future information does not influence model training. Embargoing: After the end of each test period, a fixed number of observations (typically a small percentage) are removed from the training set. This prevents leakage due to delayed market reactions or auto-correlated features. Each data point appears in multiple test sets across different combinations. Because test groups are drawn combinatorially, this process produces multiple backtest "paths," each of which simulates a plausible market scenario. From these paths, practitioners can compute a distribution of performance statistics such as the Sharpe ratio, drawdown, or classification accuracy. === Formal definition === Let N be the number of sequential groups into which the dataset is divided, and let k be the number of groups selected as the test set for each split. Then: The number of unique train-test combinations is given by the binomial coefficient: ( N k ) {\displaystyle {\binom {N}{k}}} Each observation is used in k {\displaystyle k} test sets and contributes to φ [ N , k ] {\displaystyle \varphi [N,k]} unique backtest paths: φ [ N , k ] = k N ( N k ) {\displaystyle \varphi [N,k]={\frac {k}{N}}{\binom {N}{k}}} This yields a distribution of performance metrics rather than a single point estimate, making it possible to apply Monte Carlo-based or probabilistic techniques to assess model robustness. === Illustrative example === Consider the case where N = 6 and k = 2. The number of possible test set combinations is ( 6 2 ) = 15 {\displaystyle {\binom {6}{2}}=15} . Each of the six groups appears in five test splits. Consequently, five distinct backtest paths can be constructed, each incorporating one appearance from every group. ==== Test group assignment matrix ==== This table shows the 15 test combinations. An "x" indicates that the corresponding group is included in the test set for that split. ==== Backtest path assignment ==== Each group contributes to five different backtest paths. The number in each cell indicates the path to which the group's result is assigned for that split. === Advantages === Combinatorial Purged Cross-Validation offers several key benefits over conventional methods: It produces a distribution of performance metrics, enabling more rigorous statistical inference. The method systematically eliminates lookahead bias through purging and embargoing. By simulating multiple historical scenarios, it reduces the dependence on any single market regime or realization. It supports high-confidence comparisons between competing models or strategies. CPCV is commonly used in quantitative strategy research, especially for evaluating predictive models such as classifiers, regressors, and portfolio optimizers. It has been applied to estimate realistic Sharpe ratios, assess the risk of overfitting, and support the use of statistical tools such as the Deflated Sharpe Ratio (DSR). === Limitations === The main limitation of CPCV stems from its high computational cost. However, this cost can be managed by sampling a finite number of splits from the space of all possible combinations.

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  Cultural algorithm

  Cultural algorithms (CA) are a branch of evolutionary computation where there is a knowledge component that is called the belief space in addition to the population component. In this sense, cultural algorithms can be seen as an extension to a conventional genetic algorithm. Cultural algorithms were introduced by Reynolds (see references). == Belief space == The belief space of a cultural algorithm is divided into distinct categories. These categories represent different domains of knowledge that the population has of the search space. The belief space is updated after each iteration by the best individuals of the population. The best individuals can be selected using a fitness function that assesses the performance of each individual in population much like in genetic algorithms. === List of belief space categories === Normative knowledge A collection of desirable value ranges for the individuals in the population component e.g. acceptable behavior for the agents in population. Domain specific knowledge Information about the domain of the cultural algorithm problem is applied to. Situational knowledge Specific examples of important events - e.g. successful/unsuccessful solutions Temporal knowledge History of the search space - e.g. the temporal patterns of the search process Spatial knowledge Information about the topography of the search space == Population == The population component of the cultural algorithm is approximately the same as that of the genetic algorithm. == Communication protocol == Cultural algorithms require an interface between the population and belief space. The best individuals of the population can update the belief space via the update function. Also, the knowledge categories of the belief space can affect the population component via the influence function. The influence function can affect population by altering the genome or the actions of the individuals. == Pseudocode for cultural algorithms == Initialize population space (choose initial population) Initialize belief space (e.g. set domain specific knowledge and normative value-ranges) Repeat until termination condition is met Perform actions of the individuals in population space Evaluate each individual by using the fitness function Select the parents to reproduce a new generation of offspring Let the belief space alter the genome of the offspring by using the influence function Update the belief space by using the accept function (this is done by letting the best individuals to affect the belief space) == Applications == Various optimization problems Social simulation Real-parameter optimization

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  Apache Giraph

  Apache Giraph is an Apache project to perform graph processing on big data. Giraph utilizes Apache Hadoop's MapReduce implementation to process graphs. Facebook used Giraph with some performance improvements to analyze one trillion edges using 200 machines in 4 minutes. Giraph is based on a paper published by Google about its own graph processing system called Pregel. It can be compared to other Big Graph processing libraries such as Cassovary. As of September 2023, it is no longer actively developed.

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  Locality-sensitive hashing

  In computer science, locality-sensitive hashing (LSH) is a fuzzy hashing technique that hashes similar input items into the same "buckets" with high probability. The number of buckets is much smaller than the universe of possible input items. Since similar items end up in the same buckets, this technique can be used for data clustering and nearest neighbor search. It differs from conventional hashing techniques in that hash collisions are maximized, not minimized. Alternatively, the technique can be seen as a way to reduce the dimensionality of high-dimensional data; high-dimensional input items can be reduced to low-dimensional versions while preserving relative distances between items. Hashing-based approximate nearest-neighbor search algorithms generally use one of two main categories of hashing methods: either data-independent methods, such as locality-sensitive hashing (LSH); or data-dependent methods, such as locality-preserving hashing (LPH). Locality-preserving hashing was initially devised as a way to facilitate data pipelining in implementations of massively parallel algorithms that use randomized routing and universal hashing to reduce memory contention and network congestion. == Definitions == A finite family F {\displaystyle {\mathcal {F}}} of functions h : M → S {\displaystyle h\colon M\to S} is defined to be an LSH family for a metric space M = ( M , d ) {\displaystyle {\mathcal {M}}=(M,d)} , a threshold r > 0 {\displaystyle r>0} , an approximation factor c > 1 {\displaystyle c>1} , and probabilities p 1 > p 2 {\displaystyle p_{1}>p_{2}} if it satisfies the following condition. For any two points a , b ∈ M {\displaystyle a,b\in M} and a hash function h {\displaystyle h} chosen uniformly at random from F {\displaystyle {\mathcal {F}}} : If d ( a , b ) ≤ r {\displaystyle d(a,b)\leq r} , then h ( a ) = h ( b ) {\displaystyle h(a)=h(b)} (i.e., a and b collide) with probability at least p 1 {\displaystyle p_{1}} , If d ( a , b ) ≥ c r {\displaystyle d(a,b)\geq cr} , then h ( a ) = h ( b ) {\displaystyle h(a)=h(b)} with probability at most p 2 {\displaystyle p_{2}} . Such a family F {\displaystyle {\mathcal {F}}} is called ( r , c r , p 1 , p 2 ) {\displaystyle (r,cr,p_{1},p_{2})} -sensitive. === LSH with respect to a similarity measure === Alternatively it is possible to define an LSH family on a universe of items U endowed with a similarity function ϕ : U × U → [ 0 , 1 ] {\displaystyle \phi \colon U\times U\to [0,1]} . In this setting, a LSH scheme is a family of hash functions H coupled with a probability distribution D over H such that a function h ∈ H {\displaystyle h\in H} chosen according to D satisfies P r [ h ( a ) = h ( b ) ] = ϕ ( a , b ) {\displaystyle Pr[h(a)=h(b)]=\phi (a,b)} for each a , b ∈ U {\displaystyle a,b\in U} . === Amplification === Given a ( d 1 , d 2 , p 1 , p 2 ) {\displaystyle (d_{1},d_{2},p_{1},p_{2})} -sensitive family F {\displaystyle {\mathcal {F}}} , we can construct new families G {\displaystyle {\mathcal {G}}} by either the AND-construction or OR-construction of F {\displaystyle {\mathcal {F}}} . To create an AND-construction, we define a new family G {\displaystyle {\mathcal {G}}} of hash functions g, where each function g is constructed from k random functions h 1 , … , h k {\displaystyle h_{1},\ldots ,h_{k}} from F {\displaystyle {\mathcal {F}}} . We then say that for a hash function g ∈ G {\displaystyle g\in {\mathcal {G}}} , g ( x ) = g ( y ) {\displaystyle g(x)=g(y)} if and only if all h i ( x ) = h i ( y ) {\displaystyle h_{i}(x)=h_{i}(y)} for i = 1 , 2 , … , k {\displaystyle i=1,2,\ldots ,k} . Since the members of F {\displaystyle {\mathcal {F}}} are independently chosen for any g ∈ G {\displaystyle g\in {\mathcal {G}}} , G {\displaystyle {\mathcal {G}}} is a ( d 1 , d 2 , p 1 k , p 2 k ) {\displaystyle (d_{1},d_{2},p_{1}^{k},p_{2}^{k})} -sensitive family. To create an OR-construction, we define a new family G {\displaystyle {\mathcal {G}}} of hash functions g, where each function g is constructed from k random functions h 1 , … , h k {\displaystyle h_{1},\ldots ,h_{k}} from F {\displaystyle {\mathcal {F}}} . We then say that for a hash function g ∈ G {\displaystyle g\in {\mathcal {G}}} , g ( x ) = g ( y ) {\displaystyle g(x)=g(y)} if and only if h i ( x ) = h i ( y ) {\displaystyle h_{i}(x)=h_{i}(y)} for one or more values of i. Since the members of F {\displaystyle {\mathcal {F}}} are independently chosen for any g ∈ G {\displaystyle g\in {\mathcal {G}}} , G {\displaystyle {\mathcal {G}}} is a ( d 1 , d 2 , 1 − ( 1 − p 1 ) k , 1 − ( 1 − p 2 ) k ) {\displaystyle (d_{1},d_{2},1-(1-p_{1})^{k},1-(1-p_{2})^{k})} -sensitive family. == Applications == LSH has been applied to several problem domains, including: Near-duplicate detection Hierarchical clustering Genome-wide association study Image similarity identification VisualRank Gene expression similarity identification Audio similarity identification Nearest neighbor search Audio fingerprint Digital video fingerprinting Shared memory organization in parallel computing Physical data organization in database management systems Training fully connected neural networks Computer security Machine learning == Methods == === Bit sampling for Hamming distance === One of the easiest ways to construct an LSH family is by bit sampling. This approach works for the Hamming distance over d-dimensional vectors { 0 , 1 } d {\displaystyle \{0,1\}^{d}} . Here, the family F {\displaystyle {\mathcal {F}}} of hash functions is simply the family of all the projections of points on one of the d {\displaystyle d} coordinates, i.e., F = { h : { 0 , 1 } d → { 0 , 1 } ∣ h ( x ) = x i for some i ∈ { 1 , … , d } } {\displaystyle {\mathcal {F}}=\{h\colon \{0,1\}^{d}\to \{0,1\}\mid h(x)=x_{i}{\text{ for some }}i\in \{1,\ldots ,d\}\}} , where x i {\displaystyle x_{i}} is the i {\displaystyle i} th coordinate of x {\displaystyle x} . A random function h {\displaystyle h} from F {\displaystyle {\mathcal {F}}} simply selects a random bit from the input point. This family has the following parameters: P 1 = 1 − R / d {\displaystyle P_{1}=1-R/d} , P 2 = 1 − c R / d {\displaystyle P_{2}=1-cR/d} . That is, any two vectors x , y {\displaystyle x,y} with Hamming distance at most R {\displaystyle R} collide under a random h {\displaystyle h} with probability at least P 1 {\displaystyle P_{1}} . Any x , y {\displaystyle x,y} with Hamming distance at least c R {\displaystyle cR} collide with probability at most P 2 {\displaystyle P_{2}} . === Min-wise independent permutations === Suppose U is composed of subsets of some ground set of enumerable items S and the similarity function of interest is the Jaccard index J. If π is a permutation on the indices of S, for A ⊆ S {\displaystyle A\subseteq S} let h ( A ) = min a ∈ A { π ( a ) } {\displaystyle h(A)=\min _{a\in A}\{\pi (a)\}} . Each possible choice of π defines a single hash function h mapping input sets to elements of S. Define the function family H to be the set of all such functions and let D be the uniform distribution. Given two sets A , B ⊆ S {\displaystyle A,B\subseteq S} the event that h ( A ) = h ( B ) {\displaystyle h(A)=h(B)} corresponds exactly to the event that the minimizer of π over A ∪ B {\displaystyle A\cup B} lies inside A ∩ B {\displaystyle A\cap B} . As h was chosen uniformly at random, P r [ h ( A ) = h ( B ) ] = J ( A , B ) {\displaystyle Pr[h(A)=h(B)]=J(A,B)\,} and ( H , D ) {\displaystyle (H,D)\,} define an LSH scheme for the Jaccard index. Because the symmetric group on n elements has size n!, choosing a truly random permutation from the full symmetric group is infeasible for even moderately sized n. Because of this fact, there has been significant work on finding a family of permutations that is "min-wise independent" — a permutation family for which each element of the domain has equal probability of being the minimum under a randomly chosen π. It has been established that a min-wise independent family of permutations is at least of size lcm ⁡ { 1 , 2 , … , n } ≥ e n − o ( n ) {\displaystyle \operatorname {lcm} \{\,1,2,\ldots ,n\,\}\geq e^{n-o(n)}} , and that this bound is tight. Because min-wise independent families are too big for practical applications, two variant notions of min-wise independence are introduced: restricted min-wise independent permutations families, and approximate min-wise independent families. Restricted min-wise independence is the min-wise independence property restricted to certain sets of cardinality at most k. Approximate min-wise independence differs from the property by at most a fixed ε. === Open source methods === ==== Nilsimsa Hash ==== Nilsimsa is a locality-sensitive hashing algorithm used in anti-spam efforts. The goal of Nilsimsa is to generate a hash digest of an email message such that the digests of two similar messages are similar to each other. The paper suggests that the Nilsimsa satisfies three requirements: The digest identifying each message should not

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  Cloud-Based Secure File Transfer

  Cloud-Based Secure File Transfer is a managed or hosted file transfer service that provides cloud storage that can be accessed via SSH File Transfer Protocol (SFTP). These services allow secure, reliable file transfers while offering the scalability, redundancy, and high availability of cloud infrastructure. == Technical overview == The evolution of file transfer protocols began with File Transfer Protocol (FTP) and SSH File Transfer Protocol (SFTP). SFTP offered enhanced security through the use of SSH (Secure Shell) encryption, which addressed many of the security concerns associated with traditional FTP. Over time, as businesses increasingly adopted cloud infrastructure, the demand for services that integrate secure file transfer with cloud storage led to the rise of Cloud-Based Secure File Transfer services. These services combine the benefits of secure, encrypted file transfer with the scalability and flexibility of cloud-based storage systems. Traditional on-premises SFTP typically involves setting up and managing physical or virtual servers to handle file transfers. In contrast, Cloud-Based Secure File Transfer utilizes managed cloud infrastructure, such as AWS EC2, Azure VMs, or Google Cloud, to automate scaling, ensure redundancy, and provide high availability. These cloud environments can be configured to automatically scale with demand, enabling businesses to handle large volumes of data transfers without the need for extensive physical hardware. == Features == Scalability and availability: Cloud-Based Secure File Transfer services are inherently scalable, with features like load balancing, multi-region deployments, and auto-scaling groups that adjust resources in response to traffic spikes. This ensures that the system can handle varying workloads and provides continuous availability, even during high-demand periods. Cost-effectiveness: By eliminating the need for physical infrastructure and reducing ongoing server maintenance costs, Cloud-Based Secure File Transfer services offer significant cost savings compared to traditional on-premises services. Cloud providers typically offer pay-as-you-go pricing models, where users only pay for the resources they use, further optimizing costs. Security and compliance: Cloud-Based Secure File Transfer products offer strong security measures, including end-to-end encryption, key management, detailed logging, and auditing. These services are often compliant with industry regulations such as HIPAA (Health Insurance Portability and Accountability Act), GDPR (General Data Protection Regulation), and SOC 2 (System and Organization Controls), ensuring that data transfers meet necessary security and privacy standards. == Cloud-Based Secure File Transfer providers == == Uses == Cloud-Based Secure File Transfer is used across various industries to securely transfer sensitive data and integrate into business workflows. In healthcare, Cloud-Based Secure File Transfer is essential for securely transferring electronic Protected Health Information (ePHI), ensuring compliance with regulations like HIPAA. In financial institutions, it is used to protect sensitive financial data during transfer, maintaining privacy and security. Data analytics also benefits from Cloud-Based Secure File Transfer, offering a secure and efficient method for transferring large datasets between systems or partners. Technically, Cloud-Based Secure File Transfer is often integrated into enterprise workflows through automated file transfers, using scripting or APIs. It also plays a key role in cloud backup and disaster recovery, ensuring that files are securely transferred and stored in cloud environments, which supports business continuity. However, businesses must address certain implementation challenges. Despite its secure design, Cloud-Based Secure File Transfer is not immune to risks such as misconfigured SSH keys, improper access control, or inadequate encryption. Regular security audits and careful configuration management are necessary to minimize the risk of data breaches. Additionally, integrating Cloud-Based Secure File Transfer with legacy systems can present challenges, such as incompatible APIs or outdated authentication methods. == Comparisons with related technologies == Cloud-Based Secure File Transfer differs from traditional SFTP primarily in its deployment and management model. Traditional SFTP services are typically hosted on-premises or on virtual servers, requiring manual configuration, ongoing infrastructure maintenance, and security management by in-house IT teams. In contrast, Cloud-Based Secure File Transfer is offered as a Software-as-a-Service (SaaS) service, reducing infrastructure overhead by eliminating the need for dedicated hardware or virtual machines. This model simplifies management through centralized web-based interfaces, automated updates, and built-in scalability. While Cloud-Based Secure File Transfer is focused on providing secure file transfers over the SFTP protocol, Managed File Transfer (MFT) platforms generally support a broader range of protocols, including FTP, FTPS, HTTP/S, and AS2. MFT services often include advanced features such as end-to-end encryption, extensive automation, compliance reporting, and integration with enterprise systems. Cloud-Based Secure File Transfer services may offer some of these features but are typically more lightweight and streamlined, targeting organizations seeking a secure and scalable alternative to traditional SFTP without the full suite of MFT capabilities. As such, Cloud-Based Secure File Transfer can be seen as a specialized subset within the broader managed file transfer ecosystem.

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  World Programming System

  The World Programming System, also known as WPS Analytics or WPS, is a software product developed by a company called World Programming (acquired by Altair Engineering). WPS Analytics supports users of mixed ability to access and process data and to perform data science tasks. It has interactive visual programming tools using data workflows, and it has coding tools supporting the use of the SAS language mixed with Python, R and SQL. == About == WPS can use programs written in the language of SAS without the need for translating them into any other language. In this regard WPS is compatible with the SAS system. WPS has a built-in language interpreter able to process the language of SAS and produce similar results. WPS is available to run on z/OS, Windows, macOS, Linux (x86, Armv8 64-bit, IBM Power LE, IBM Z), and AIX. On all supported platforms, programs written in the language of SAS can be executed from a WPS command line interface, often referred to as running in batch mode. WPS can also be used from a graphical user interface known as the WPS Workbench for managing, editing and running programs written in the language of SAS. The WPS Workbench user interface is based on Eclipse. WPS version 4 (released in March 2018) introduced a drag-and-drop workflow canvas providing interactive blocks for data retrieval, blending and preparation, data discovery and profiling, predictive modelling powered by machine learning algorithms, model performance validation and scorecards. WPS version 3 (released in February 2012) provided a new client/server architecture that allows the WPS Workbench GUI to execute SAS programs on remote server installations of WPS in a network or cloud. The resulting output, data sets, logs, etc., can then all be viewed and manipulated from inside the Workbench as if the workloads had been executed locally. SAS programs do not require any special language statements to use this feature. == Summary of main features == Runs on Windows, macOS, z/OS, Linux (x86, Armv8 64-bit, IBM Power LE, IBM Z), and AIX An integrated development environment based on Eclipse for Linux, macOS and Windows. Support for language of SAS elements. Support for the language of SAS Macros. Matrix Programming support using PROC IML. Support for generating band plots, bar charts, box plots, bubble plots, contour plots, dendrogram plots, ellipse plots, fringe plots, heat maps, high-low plots, histograms, loess plots, needle plots, pie charts, penalised b-spline, radar charts, reference lines, scatter plots, series plots, step plots, regression plots and vector plots. Support for statistical procedures ACECLUS, ASSOCRULES, ANOVA, BIN, BOXPLOT, CANCORR, CANDISC, CLUSTER, CORRESP, DISCRIM, DISTANCE, FACTOR, FASTCLUS, FREQ, GAM, GANNO, GENMOD, GLIMMIX, GLM, GLMMOD, GLMSELECT, ICLIFETEST, KDE, LIFEREG, LIFETEST, LOESS, LOGISTIC, MDS, MEANS, MI, MIANALYSE, MIXED, MODECLUS, NESTED, NLIN, NPAR1WAY, PHREG, PLAN, PLS, POWER, PRINCOMP, PROBIT, QUANTREG, RBF, REG, ROBUSTREG, RSREG, SCORE, SEGMENT, SIMNORMAL, STANDARD, STDSIZE, STDRATE, STEPDISC, SUMMARY, SURVEYMEANS, SURVEYSELECT, TPSPLINE, TRANSREG, TREE, TTEST, UNIVARIATE, VARCLUS, VARCOMP Support for time series procedures ARIMA, AUTOREG, ESM, EXPAND, FORECAST, LOAN, SEVERITY, SPECTRA, TIMESERIES, X12 Support for machine learning procedures DECISIONFOREST, DECISIONTREE, GMM, MLP, OPTIMALBIN, SEGMENT, SVM Support for ODS. Reads and writes SAS datasets (compressed or uncompressed). Access: Actian Matrix (previously known as ParAccel), DASD, DB2, Excel, Greenplum, Hadoop, Informix, Kognitio Archived 2012-08-24 at the Wayback Machine, MariaDB, MySQL, Netezza, ODBC, OLEDB, Oracle, PostgreSQL, SAND, Snowflake, SPSS/PSPP, SQL Server, Sybase, Sybase IQ, Teradata, VSAM, Vertica and XML. Support for SAS Tape Format. Direct output of reports to CSV, PDF and HTML. Support to connect WPS systems programmatically, remote submit parts of a program to execute on connected remote servers, upload and download data between the connected systems. Support for Hadoop Support for R Support for Python == Industry recognition == Gartner recognized World Programming in their Cool Vendors in Data Science, 2014 Report. == Lawsuit == In 2010 World Programming defended its use of the language of SAS in the High Court of England and Wales in SAS Institute Inc. v World Programming Ltd. The software was the subject of a lawsuit by SAS Institute. The EU Court of Justice ruled in favor of World Programming, stating that the copyright protection does not extend to the software functionality, the programming language used and the format of the data files used by the program. It stated that there is no copyright infringement when a company which does not have access to the source code of a program studies, observes and tests that program to create another program with the same functionality.

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  Teaching dimension

  In computational learning theory, the teaching dimension of a concept class C is defined to be max c ∈ C { w C ( c ) } {\displaystyle \max _{c\in C}\{w_{C}(c)\}} , where w C ( c ) {\displaystyle {w_{C}(c)}} is the minimum size of a witness set for c in C. Intuitively, this measures the number of instances that are needed to identify a concept in the class, using supervised learning with examples provided by a helpful teacher who is trying to convey the concept as succinctly as possible. This definition was formulated in 1995 by Sally Goldman and Michael Kearns, based on earlier work by Goldman, Ron Rivest, and Robert Schapire. The teaching dimension of a finite concept class can be used to give a lower and an upper bound on the membership query cost of the concept class. In Stasys Jukna's book "Extremal Combinatorics", a lower bound is given for the teaching dimension in general: Let C be a concept class over a finite domain X. If the size of C is greater than 2 k ( | X | k ) , {\displaystyle 2^{k}{|X| \choose k},} then the teaching dimension of C is greater than k. However, there are more specific teaching models that make assumptions about teacher or learner, and can get lower values for the teaching dimension. For instance, several models are the classical teaching (CT) model, the optimal teacher (OT) model, recursive teaching (RT), preference-based teaching (PBT), and non-clashing teaching (NCT).

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  Random forest

  Random forests or random decision forests is an ensemble learning method for classification, regression and other tasks that works by creating a multitude of decision trees during training. For classification tasks, the output of the random forest is the class selected by most trees. For regression tasks, the output is the average of the predictions of the trees. Random forests correct for decision trees' habit of overfitting to their training set. The first algorithm for random decision forests was created in 1995 by Tin Kam Ho using the random subspace method, which, in Ho's formulation, is a way to implement the "stochastic discrimination" approach to classification proposed by Eugene Kleinberg. An extension of the algorithm was developed by Leo Breiman and Adele Cutler, who registered "Random Forests" as a trademark in 2006 (as of 2019, owned by Minitab, Inc.). The extension combines Breiman's "bagging" idea and random selection of features, introduced first by Ho and later independently by Amit and Geman in order to construct a collection of decision trees with controlled variance. == History == The general method of random decision forests was first proposed by Salzberg and Heath in 1993, with a method that used a randomized decision tree algorithm to create multiple trees and then combine them using majority voting. This idea was developed further by Ho in 1995. Ho established that forests of trees splitting with oblique hyperplanes can gain accuracy as they grow without suffering from overtraining, as long as the forests are randomly restricted to be sensitive to only selected feature dimensions. A subsequent work along the same lines concluded that other splitting methods behave similarly, as long as they are randomly forced to be insensitive to some feature dimensions. This observation that a more complex classifier (a larger forest) gets more accurate nearly monotonically is in sharp contrast to the common belief that the complexity of a classifier can only grow to a certain level of accuracy before being hurt by overfitting. The explanation of the forest method's resistance to overtraining can be found in Kleinberg's theory of stochastic discrimination. The early development of Breiman's notion of random forests was influenced by the work of Amit and Geman who introduced the idea of searching over a random subset of the available decisions when splitting a node, in the context of growing a single tree. The idea of random subspace selection from Ho was also influential in the design of random forests. This method grows a forest of trees, and introduces variation among the trees by projecting the training data into a randomly chosen subspace before fitting each tree or each node. Finally, the idea of randomized node optimization, where the decision at each node is selected by a randomized procedure, rather than a deterministic optimization was first introduced by Thomas G. Dietterich. The proper introduction of random forests was made in a paper by Leo Breiman, that has become one of the world's most cited papers. This paper describes a method of building a forest of uncorrelated trees using a CART like procedure, combined with randomized node optimization and bagging. In addition, this paper combines several ingredients, some previously known and some novel, which form the basis of the modern practice of random forests, in particular: Using out-of-bag error as an estimate of the generalization error. Measuring variable importance through permutation. The report also offers the first theoretical result for random forests in the form of a bound on the generalization error which depends on the strength of the trees in the forest and their correlation. == Algorithm == === Preliminaries: decision tree learning === Decision trees are a popular method for various machine learning tasks. Tree learning is almost "an off-the-shelf procedure for data mining", say Hastie et al., "because it is invariant under scaling and various other transformations of feature values, is robust to inclusion of irrelevant features, and produces inspectable models. However, they are seldom accurate". In particular, trees that are grown very deep tend to learn highly irregular patterns: they overfit their training sets, i.e. have low bias, but very high variance. Random forests are a way of averaging multiple deep decision trees, trained on different parts of the same training set, with the goal of reducing the variance. This comes at the expense of a small increase in the bias and some loss of interpretability, but generally greatly boosts the performance in the final model. === Bagging === The training algorithm for random forests applies the general technique of bootstrap aggregating, or bagging, to tree learners. Given a training set X = x1, ..., xn with responses Y = y1, ..., yn, bagging repeatedly (B times) selects a random sample with replacement of the training set and fits trees to these samples: After training, predictions for unseen samples x' can be made by averaging the predictions from all the individual regression trees on x': f ^ = 1 B ∑ b = 1 B f b ( x ′ ) {\displaystyle {\hat {f}}={\frac {1}{B}}\sum _{b=1}^{B}f_{b}(x')} or by taking the plurality vote in the case of classification trees. This bootstrapping procedure leads to better model performance because it decreases the variance of the model, without increasing the bias. This means that while the predictions of a single tree are highly sensitive to noise in its training set, the average of many trees is not, as long as the trees are not correlated. Simply training many trees on a single training set would give strongly correlated trees (or even the same tree many times, if the training algorithm is deterministic); bootstrap sampling is a way of de-correlating the trees by showing them different training sets. Additionally, an estimate of the uncertainty of the prediction can be made as the standard deviation of the predictions from all the individual regression trees on x′: σ = ∑ b = 1 B ( f b ( x ′ ) − f ^ ) 2 B − 1 . {\displaystyle \sigma ={\sqrt {\frac {\sum _{b=1}^{B}(f_{b}(x')-{\hat {f}})^{2}}{B-1}}}.} The number B of samples (equivalently, of trees) is a free parameter. Typically, a few hundred to several thousand trees are used, depending on the size and nature of the training set. B can be optimized using cross-validation, or by observing the out-of-bag error: the mean prediction error on each training sample xi, using only the trees that did not have xi in their bootstrap sample. The training and test error tend to level off after some number of trees have been fit. === From bagging to random forests === The above procedure describes the original bagging algorithm for trees. Random forests also include another type of bagging scheme: they use a modified tree learning algorithm that selects, at each candidate split in the learning process, a random subset of the features. This process is sometimes called "feature bagging". The reason for doing this is the correlation of the trees in an ordinary bootstrap sample: if one or a few features are very strong predictors for the response variable (target output), these features will be selected in many of the B trees, causing them to become correlated. An analysis of how bagging and random subspace projection contribute to accuracy gains under different conditions is given by Ho. Typically, for a classification problem with p {\displaystyle p} features, p {\displaystyle {\sqrt {p}}} (rounded down) features are used in each split. For regression problems the inventors recommend p / 3 {\displaystyle p/3} (rounded down) with a minimum node size of 5 as the default. In practice, the best values for these parameters should be tuned on a case-to-case basis for every problem. === ExtraTrees === Adding one further step of randomization yields extremely randomized trees, or ExtraTrees. As with ordinary random forests, they are an ensemble of individual trees, but there are two main differences: (1) each tree is trained using the whole learning sample (rather than a bootstrap sample), and (2) the top-down splitting is randomized: for each feature under consideration, a number of random cut-points are selected, instead of computing the locally optimal cut-point (based on, e.g., information gain or the Gini impurity). The values are chosen from a uniform distribution within the feature's empirical range (in the tree's training set). Then, of all the randomly chosen splits, the split that yields the highest score is chosen to split the node. Similar to ordinary random forests, the number of randomly selected features to be considered at each node can be specified. Default values for this parameter are p {\displaystyle {\sqrt {p}}} for classification and p {\displaystyle p} for regression, where p {\displaystyle p} is the number of features in the model. === Random forests for high-dimensional data === The basic random forest procedure may

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  AI-complete

  In the field of artificial intelligence (AI), tasks that are hypothesized to require artificial general intelligence to solve are informally known as AI-complete or AI-hard. Calling a problem AI-complete reflects the belief that it cannot be solved by a simple specific algorithm. Prior to 2013, problems supposed to be AI-complete included computer vision, natural language understanding, and dealing with unexpected circumstances while solving any real-world problem. AI-complete tasks were notably considered useful for distinguishing humans from automated agents, as CAPTCHAs aim to do. == History == The term was coined by Fanya Montalvo by analogy with NP-complete and NP-hard in complexity theory, which formally describes the most famous class of difficult problems. Early uses of the term are in Erik Mueller's 1987 PhD dissertation and in Eric Raymond's 1991 Jargon File. Expert systems, that were popular in the 1980s, were able to solve very simple and/or restricted versions of AI-complete problems, but never in their full generality. When AI researchers attempted to "scale up" their systems to handle more complicated, real-world situations, the programs tended to become excessively brittle without commonsense knowledge or a rudimentary understanding of the situation: they would fail as unexpected circumstances outside of its original problem context would begin to appear. When human beings are dealing with new situations in the world, they are helped by their awareness of the general context: they know what the things around them are, why they are there, what they are likely to do and so on. They can recognize unusual situations and adjust accordingly. Expert systems lacked this adaptability and were brittle when facing new situations. DeepMind published a work in May 2022 in which they trained a single model to do several things at the same time. The model, named Gato, can "play Atari, caption images, chat, stack blocks with a real robot arm and much more, deciding based on its context whether to output text, joint torques, button presses, or other tokens." Similarly, some tasks once considered to be AI-complete, like machine translation, are among the capabilities of large language models. == AI-complete problems == AI-complete problems have been hypothesized to include: AI peer review (composite natural language understanding, automated reasoning, automated theorem proving, formalized logic expert system) Bongard problems Computer vision (and subproblems such as object recognition) Natural language understanding (and subproblems such as text mining, machine translation, and word-sense disambiguation) Autonomous driving Dealing with unexpected circumstances while solving any real world problem, whether navigation, planning, or even the kind of reasoning done by expert systems. == Formalization == Computational complexity theory deals with the relative computational difficulty of computable functions. By definition, it does not cover problems whose solution is unknown or has not been characterized formally. Since many AI problems have no formalization yet, conventional complexity theory does not enable a formal definition of AI-completeness. == Research == Roman Yampolskiy suggests that a problem C {\displaystyle C} is AI-Complete if it has two properties: It is in the set of AI problems (Human Oracle-solvable). Any AI problem can be converted into C {\displaystyle C} by some polynomial time algorithm. On the other hand, a problem H {\displaystyle H} is AI-Hard if and only if there is an AI-Complete problem C {\displaystyle C} that is polynomial time Turing-reducible to H {\displaystyle H} . This also gives as a consequence the existence of AI-Easy problems, that are solvable in polynomial time by a deterministic Turing machine with an oracle for some problem. Yampolskiy has also hypothesized that the Turing Test is a defining feature of AI-completeness. Groppe and Jain classify problems which require artificial general intelligence to reach human-level machine performance as AI-complete, while only restricted versions of AI-complete problems can be solved by the current AI systems. For Šekrst, getting a polynomial solution to AI-complete problems would not necessarily be equal to solving the issue of artificial general intelligence, while emphasizing the lack of computational complexity research being the limiting factor towards achieving artificial general intelligence. For Kwee-Bintoro and Velez, solving AI-complete problems would have strong repercussions on society.

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  Multiple correspondence analysis

  In statistics, multiple correspondence analysis (MCA) is a data analysis technique for nominal categorical data, used to detect and represent underlying structures in a data set. It does this by representing data as points in a low-dimensional Euclidean space. The procedure thus appears to be the counterpart of principal component analysis for categorical data. MCA can be viewed as an extension of simple correspondence analysis (CA) in that it is applicable to a large set of categorical variables. == As an extension of correspondence analysis == MCA is performed by applying the CA algorithm to either an indicator matrix (also called complete disjunctive table – CDT) or a Burt table formed from these variables. An indicator matrix is an individuals × variables matrix, where the rows represent individuals and the columns are dummy variables representing categories of the variables. Analyzing the indicator matrix allows the direct representation of individuals as points in geometric space. The Burt table is the symmetric matrix of all two-way cross-tabulations between the categorical variables, and has an analogy to the covariance matrix of continuous variables. Analyzing the Burt table is a more natural generalization of simple correspondence analysis, and individuals or the means of groups of individuals can be added as supplementary points to the graphical display. In the indicator matrix approach, associations between variables are uncovered by calculating the chi-square distance between different categories of the variables and between the individuals (or respondents). These associations are then represented graphically as "maps", which eases the interpretation of the structures in the data. Oppositions between rows and columns are then maximized, in order to uncover the underlying dimensions best able to describe the central oppositions in the data. As in factor analysis or principal component analysis, the first axis is the most important dimension, the second axis the second most important, and so on, in terms of the amount of variance accounted for. The number of axes to be retained for analysis is determined by calculating modified eigenvalues. == Details == Since MCA is adapted to draw statistical conclusions from categorical variables (such as multiple choice questions), the first thing one needs to do is to transform quantitative data (such as age, size, weight, day time, etc) into categories (using for instance statistical quantiles). When the dataset is completely represented as categorical variables, one is able to build the corresponding so-called complete disjunctive table. We denote this table X {\displaystyle X} . If I {\displaystyle I} persons answered a survey with J {\displaystyle J} multiple choices questions with 4 answers each, X {\displaystyle X} will have I {\displaystyle I} rows and 4 J {\displaystyle 4J} columns. More theoretically, assume X {\displaystyle X} is the completely disjunctive table of I {\displaystyle I} observations of K {\displaystyle K} categorical variables. Assume also that the k {\displaystyle k} -th variable have J k {\displaystyle J_{k}} different levels (categories) and set J = ∑ k = 1 K J k {\displaystyle J=\sum _{k=1}^{K}J_{k}} . The table X {\displaystyle X} is then a I × J {\displaystyle I\times J} matrix with all coefficient being 0 {\displaystyle 0} or 1 {\displaystyle 1} . Set the sum of all entries of X {\displaystyle X} to be N {\displaystyle N} and introduce Z = X / N {\displaystyle Z=X/N} . In an MCA, there are also two special vectors: first r {\displaystyle r} , that contains the sums along the rows of Z {\displaystyle Z} , and c {\displaystyle c} , that contains the sums along the columns of Z {\displaystyle Z} . Note D r = diag ( r ) {\displaystyle D_{r}={\text{diag}}(r)} and D c = diag ( c ) {\displaystyle D_{c}={\text{diag}}(c)} , the diagonal matrices containing r {\displaystyle r} and c {\displaystyle c} respectively as diagonal. With these notations, computing an MCA consists essentially in the singular value decomposition of the matrix: M = D r − 1 / 2 ( Z − r c T ) D c − 1 / 2 {\displaystyle M=D_{r}^{-1/2}(Z-rc^{T})D_{c}^{-1/2}} The decomposition of M {\displaystyle M} gives you P {\displaystyle P} , Δ {\displaystyle \Delta } and Q {\displaystyle Q} such that M = P Δ Q T {\displaystyle M=P\Delta Q^{T}} with P, Q two unitary matrices and Δ {\displaystyle \Delta } is the generalized diagonal matrix of the singular values (with the same shape as Z {\displaystyle Z} ). The positive coefficients of Δ 2 {\displaystyle \Delta ^{2}} are the eigenvalues of Z {\displaystyle Z} . The interest of MCA comes from the way observations (rows) and variables (columns) in Z {\displaystyle Z} can be decomposed. This decomposition is called a factor decomposition. The coordinates of the observations in the factor space are given by F = D r − 1 / 2 P Δ {\displaystyle F=D_{r}^{-1/2}P\Delta } The i {\displaystyle i} -th rows of F {\displaystyle F} represent the i {\displaystyle i} -th observation in the factor space. And similarly, the coordinates of the variables (in the same factor space as observations!) are given by G = D c − 1 / 2 Q Δ {\displaystyle G=D_{c}^{-1/2}Q\Delta } == Recent works and extensions == In recent years, several students of Jean-Paul Benzécri have refined MCA and incorporated it into a more general framework of data analysis known as geometric data analysis. This involves the development of direct connections between simple correspondence analysis, principal component analysis and MCA with a form of cluster analysis known as Euclidean classification. Two extensions have great practical use. It is possible to include, as active elements in the MCA, several quantitative variables. This extension is called factor analysis of mixed data (see below). Very often, in questionnaires, the questions are structured in several issues. In the statistical analysis it is necessary to take into account this structure. This is the aim of multiple factor analysis which balances the different issues (i.e. the different groups of variables) within a global analysis and provides, beyond the classical results of factorial analysis (mainly graphics of individuals and of categories), several results (indicators and graphics) specific of the group structure. == Application fields == In the social sciences, MCA is arguably best known for its application by Pierre Bourdieu, notably in his books La Distinction, Homo Academicus and The State Nobility. Bourdieu argued that there was an internal link between his vision of the social as spatial and relational --– captured by the notion of field, and the geometric properties of MCA. Sociologists following Bourdieu's work most often opt for the analysis of the indicator matrix, rather than the Burt table, largely because of the central importance accorded to the analysis of the 'cloud of individuals'. == Multiple correspondence analysis and principal component analysis == MCA can also be viewed as a PCA applied to the complete disjunctive table. To do this, the CDT must be transformed as follows. Let y i k {\displaystyle y_{ik}} denote the general term of the CDT. y i k {\displaystyle y_{ik}} is equal to 1 if individual i {\displaystyle i} possesses the category k {\displaystyle k} and 0 if not. Let denote p k {\displaystyle p_{k}} , the proportion of individuals possessing the category k {\displaystyle k} . The transformed CDT (TCDT) has as general term: x i k = y i k / p k − 1 {\displaystyle x_{ik}=y_{ik}/p_{k}-1} The unstandardized PCA applied to TCDT, the column k {\displaystyle k} having the weight p k {\displaystyle p_{k}} , leads to the results of MCA. This equivalence is fully explained in a book by Jérôme Pagès. It plays an important theoretical role because it opens the way to the simultaneous treatment of quantitative and qualitative variables. Two methods simultaneously analyze these two types of variables: factor analysis of mixed data and, when the active variables are partitioned in several groups: multiple factor analysis. This equivalence does not mean that MCA is a particular case of PCA as it is not a particular case of CA. It only means that these methods are closely linked to one another, as they belong to the same family: the factorial methods. == Software == There are numerous software of data analysis that include MCA, such as STATA and SPSS. The R package FactoMineR also features MCA. This software is related to a book describing the basic methods for performing MCA . There is also a Python package for [1] which works with numpy array matrices; the package has not been implemented yet for Spark dataframes.

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  Dynamic time warping

  In time series analysis, dynamic time warping (DTW) is an algorithm for measuring similarity between two temporal sequences, which may vary in speed. For instance, similarities in walking could be detected using DTW, even if one person was walking faster than the other, or if there were accelerations and decelerations during the course of an observation. DTW has been applied to temporal sequences of video, audio, and graphics data — indeed, any data that can be turned into a one-dimensional sequence can be analyzed with DTW. A well-known application has been automatic speech recognition, to cope with different speaking speeds. Other applications include speaker recognition and online signature recognition. It can also be used in partial shape matching applications. In general, DTW is a method that calculates an optimal match between two given sequences (e.g. time series) with certain restriction and rules: Every index from the first sequence must be matched with one or more indices from the other sequence, and vice versa The first index from the first sequence must be matched with the first index from the other sequence (but it does not have to be its only match) The last index from the first sequence must be matched with the last index from the other sequence (but it does not have to be its only match) The mapping of the indices from the first sequence to indices from the other sequence must be monotonically increasing, and vice versa, i.e. if j > i {\displaystyle j>i} are indices from the first sequence, then there must not be two indices l > k {\displaystyle l>k} in the other sequence, such that index i {\displaystyle i} is matched with index l {\displaystyle l} and index j {\displaystyle j} is matched with index k {\displaystyle k} , and vice versa We can plot each match between the sequences 1 : M {\displaystyle 1:M} and 1 : N {\displaystyle 1:N} as a path in a M × N {\displaystyle M\times N} matrix from ( 1 , 1 ) {\displaystyle (1,1)} to ( M , N ) {\displaystyle (M,N)} , such that each step is one of ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) {\displaystyle (0,1),(1,0),(1,1)} . In this formulation, we see that the number of possible matches is the Delannoy number. The optimal match is denoted by the match that satisfies all the restrictions and the rules and that has the minimal cost, where the cost is computed as the sum of absolute differences, for each matched pair of indices, between their values. The sequences are "warped" non-linearly in the time dimension to determine a measure of their similarity independent of certain non-linear variations in the time dimension. This sequence alignment method is often used in time series classification. Although DTW measures a distance-like quantity between two given sequences, it doesn't guarantee the triangle inequality to hold. In addition to a similarity measure between the two sequences (a so called "warping path" is produced), by warping according to this path the two signals may be aligned in time. The signal with an original set of points X(original), Y(original) is transformed to X(warped), Y(warped). This finds applications in genetic sequence and audio synchronisation. In a related technique sequences of varying speed may be averaged using this technique see the average sequence section. This is conceptually very similar to the Needleman–Wunsch algorithm. == Implementation == This example illustrates the implementation of the dynamic time warping algorithm when the two sequences s and t are strings of discrete symbols. For two symbols x and y, d ( x , y ) {\displaystyle d(x,y)} is a distance between the symbols, e.g., d ( x , y ) = | x − y | {\displaystyle d(x,y)=|x-y|} . int DTWDistance(s: array [1..n], t: array [1..m]) { DTW := array [0..n, 0..m] for i := 0 to n for j := 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := 1 to m cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } where DTW[i, j] is the distance between s[1:i] and t[1:j] with the best alignment. We sometimes want to add a locality constraint. That is, we require that if s[i] is matched with t[j], then | i − j | {\displaystyle |i-j|} is no larger than w, a window parameter. We can easily modify the above algorithm to add a locality constraint (differences marked). However, the above given modification works only if | n − m | {\displaystyle |n-m|} is no larger than w, i.e. the end point is within the window length from diagonal. In order to make the algorithm work, the window parameter w must be adapted so that | n − m | ≤ w {\displaystyle |n-m|\leq w} (see the line marked with () in the code). int DTWDistance(s: array [1..n], t: array [1..m], w: int) { DTW := array [0..n, 0..m] w := max(w, abs(n-m)) // adapt window size () for i := 0 to n for j:= 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) DTW[i, j] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } == Warping properties == The DTW algorithm produces a discrete matching between existing elements of one series to another. In other words, it does not allow time-scaling of segments within the sequence. Other methods allow continuous warping. For example, Correlation Optimized Warping (COW) divides the sequence into uniform segments that are scaled in time using linear interpolation, to produce the best matching warping. The segment scaling causes potential creation of new elements, by time-scaling segments either down or up, and thus produces a more sensitive warping than DTW's discrete matching of raw elements. == Complexity == The time complexity of the DTW algorithm is O ( N M ) {\displaystyle O(NM)} , where N {\displaystyle N} and M {\displaystyle M} are the lengths of the two input sequences. The 50 years old quadratic time bound was broken in 2016: an algorithm due to Gold and Sharir enables computing DTW in O ( N 2 / log ⁡ log ⁡ N ) {\displaystyle O({N^{2}}/\log \log N)} time and space for two input sequences of length N {\displaystyle N} . This algorithm can also be adapted to sequences of different lengths. Despite this improvement, it was shown that a strongly subquadratic running time of the form O ( N 2 − ϵ ) {\displaystyle O(N^{2-\epsilon })} for some ϵ > 0 {\displaystyle \epsilon >0} cannot exist unless the Strong exponential time hypothesis fails. While the dynamic programming algorithm for DTW requires O ( N M ) {\displaystyle O(NM)} space in a naive implementation, the space consumption can be reduced to O ( min ( N , M ) ) {\displaystyle O(\min(N,M))} using Hirschberg's algorithm. == Fast computation == Fast techniques for computing DTW include PrunedDTW, SparseDTW, FastDTW, and the MultiscaleDTW. A common task, retrieval of similar time series, can be accelerated by using lower bounds such as LB_Keogh, LB_Improved, or LB_Petitjean. However, the Early Abandon and Pruned DTW algorithm reduces the degree of acceleration that lower bounding provides and sometimes renders it ineffective. In a survey, Wang et al. reported slightly better results with the LB_Improved lower bound than the LB_Keogh bound, and found that other techniques were inefficient. Subsequent to this survey, the LB_Enhanced bound was developed that is always tighter than LB_Keogh while also being more efficient to compute. LB_Petitjean is the tightest known lower bound that can be computed in linear time. == Average sequence == Averaging for dynamic time warping is the problem of finding an average sequence for a set of sequences. NLAAF is an exact method to average two sequences using DTW. For more than two sequences, the problem is related to that of multiple alignment and requires heuristics. DBA is currently a reference method to average a set of sequences consistently with DTW. COMASA efficiently randomizes the search for the average sequence, using DBA as a local optimization process. == Supervised learning == A nearest-neighbour classifier can achieve state-of-the-art performance when using dynamic time warping as a distance measure. == Amerced Dynamic Time Warping == Amerced Dynamic Time Warping (ADTW) is a variant of DTW designed to better control DTW's permissiveness in the alignments that it allows. The windows that classical DTW uses to constrain alignments introduce a step function. Any warping of the path is allowed within the window and none beyond it. In contrast, ADTW employs an additive penalty that is incurred each time that the path is warped. Any amount of warping is allowed, but each warping action incurs a direct penalty. ADTW significantly outperforms DTW with windowing when applied as a nearest neighbor classifier on a set of benchmark time series classification tasks. == Alternative approaches == In functional data analysis, time series are regarde

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