AI Tools

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  Machine learning

  Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of statistical algorithms that can learn from data and generalize to unseen data, and thus perform tasks without being explicitly programmed. Advances in the field of deep learning have allowed neural networks, a class of statistical algorithms, to surpass many previous machine learning approaches in performance. Statistics and mathematical optimisation methods compose the foundations of machine learning. Data mining is a related field of study, focusing on exploratory data analysis (EDA) through unsupervised learning. From a theoretical viewpoint, probably approximately correct learning provides a mathematical and statistical framework for describing machine learning. Most traditional machine learning and deep learning algorithms can be described as empirical risk minimisation under this framework. == History == The term machine learning was coined in 1959 by Arthur Samuel, an IBM employee and pioneer in the field of computer gaming and artificial intelligence. The synonym self-teaching computers was also used during this time period. The earliest machine learning program was introduced in the 1950s, when Samuel invented a computer program that calculated the chance of winning in checkers for each side, but the history of machine learning is rooted in decades of efforts to study human cognitive processes. In 1949, Canadian psychologist Donald Hebb published the book The Organization of Behavior, in which he introduced a theoretical neural structure formed by certain interactions among nerve cells. The Hebbian theory of neuron interaction set the groundwork for how many machine learning algorithms work, with connected artificial neurons changing the strength of their connections based on data. Other researchers who have studied human cognitive systems contributed to the modern machine learning technologies as well, including Walter Pitts and Warren McCulloch, who proposed the first mathematical model of neural networks including algorithms that mirror human thought processes. By the early 1960s, an experimental "learning machine" with punched tape memory, called Cybertron, had been developed by Raytheon Company to analyse sonar signals, electrocardiograms, and speech patterns using rudimentary reinforcement learning. It was repetitively "trained" by a human operator/teacher to recognise patterns and equipped with a "goof" button to cause it to reevaluate incorrect decisions. A representative book on research into machine learning during the 1960s was Nils Nilsson's book "Learning Machines", dealing mostly with machine learning for pattern classification. Interest related to pattern recognition continued into the 1970s, as described by Duda and Hart in 1973. In 1981, a report was given on using teaching strategies so that an artificial neural network learns to recognise 40 characters (26 letters, 10 digits, and 4 special symbols) from a computer terminal. Tom M. Mitchell provided a widely quoted, more formal definition of the algorithms studied in the machine learning field: "A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P if its performance at tasks in T, as measured by P, improves with experience E." This definition of the tasks in which machine learning is concerned is fundamentally operational rather than defining the field in cognitive terms. This follows Alan Turing's proposal in his paper "Computing Machinery and Intelligence", in which the question, "Can machines think?", is replaced by asking whether machines can convincingly imitate a human in its responses to human-posed questions. In 2014 Ian Goodfellow and others introduced generative adversarial networks (GANs) which could produce realistic synthetic data. By 2016 AlphaGo had won against top human players in Go using reinforcement learning techniques. == Relationships to other fields == === Artificial intelligence === As a scientific endeavour, machine learning grew out of the quest for artificial intelligence (AI). In the early days of AI as an academic discipline, some researchers were interested in having machines learn from data. They attempted to approach the problem with various symbolic methods, as well as what were then termed "neural networks"; these were mostly perceptrons and other models that were later found to be reinventions of the generalised linear models of statistics. Probabilistic reasoning was also employed, especially in automated medical diagnosis. However, an increasing emphasis on the logical, knowledge-based approach caused a rift between AI and machine learning. Probabilistic systems were plagued by theoretical and practical problems of data acquisition and representation. By 1980, expert systems had come to dominate AI, and statistics was out of favour. Work on symbolic/knowledge-based learning continued within AI, leading to inductive logic programming (ILP), but the more statistical line of research was now outside the field of AI proper, in pattern recognition and information retrieval. Neural network research was abandoned by AI and computer science around the same time. This subfield, termed "connectionism", was continued by researchers from other disciplines, including John Hopfield, David Rumelhart, and Geoffrey Hinton. Their main success came in the mid-1980s with the reinvention of backpropagation. Machine learning (ML), reorganised and recognised as its own field, started to flourish in the 1990s. The field changed its goal from achieving artificial intelligence to tackling solvable problems of a practical nature. It shifted focus away from the symbolic approaches it had inherited from AI, and toward methods and models borrowed from statistics, fuzzy logic, and probability theory. === Data compression === === Data mining === Machine learning and data mining often employ the same methods and overlap significantly, but while machine learning focuses on prediction based on known properties learned from the training data, data mining focuses on the discovery of previously unknown properties in the data (this is the analysis step of knowledge discovery in databases). Data mining uses many machine learning methods, but with different goals; on the other hand, machine learning also employs data mining methods as "unsupervised learning" or as a preprocessing step to improve learner accuracy. Much of the confusion between these two research communities comes from the basic assumptions they work with: in machine learning, performance is usually evaluated with respect to the ability to reproduce known knowledge, while in knowledge discovery and data mining (KDD) the key task is the discovery of previously unknown knowledge. Evaluated with respect to known knowledge, an uninformed (unsupervised) method will easily be outperformed by other supervised methods, while in a typical KDD task, supervised methods cannot be used due to the unavailability of training data. Machine learning also has intimate ties to optimization: Many learning problems are formulated as minimisation of some loss function on a training set of examples. Loss functions express the discrepancy between the predictions of the model being trained and the actual problem instances (for example, in classification, one wants to assign a label to instances, and models are trained to correctly predict the preassigned labels of a set of examples). === Generalization === Characterizing the generalisation of various learning algorithms is an active topic of current research, especially for deep learning algorithms. === Statistics === Machine learning and statistics are closely related fields in terms of methods, but distinct in their principal goal: statistics draws population inferences from a sample, while machine learning finds generalisable predictive patterns. Conventional statistical analyses require the a priori selection of a model most suitable for the study data set. In addition, only significant or theoretically relevant variables based on previous experience are included for analysis. In contrast, machine learning is not built on a pre-structured model; rather, the data shape the model by detecting underlying patterns. The more variables (input) used to train the model, the more accurate the ultimate model will be. Leo Breiman distinguished two statistical modelling paradigms: the data model and the algorithmic model, wherein "algorithmic model" means more or less the machine learning algorithms like Random forest. Some statisticians have adopted methods from machine learning, producing the field of statistical learning. === Statistical physics === Analytical and computational techniques derived from deep-rooted physics of disordered systems can be extended to large-scale problems, including machine learning, e.g., to analyse the weight space of deep neural networks. Statistical physics is thus

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  Ensemble learning

  In statistics and machine learning, ensemble methods use multiple learning algorithms to obtain better predictive performance than could be obtained from any of the constituent learning algorithms alone. Unlike a statistical ensemble in statistical mechanics, which is usually infinite, a machine learning ensemble consists of only a concrete finite set of alternative models, but typically allows for much more flexible structure to exist among those alternatives. == Overview == Supervised learning algorithms search through a hypothesis space to find a suitable hypothesis that will make good predictions with a particular problem. Even if this space contains hypotheses that are very well-suited for a particular problem, it may be very difficult to find a good one. Ensembles combine multiple hypotheses to form one which should be theoretically better. Ensemble learning trains two or more machine learning algorithms on a specific classification or regression task. The algorithms within the ensemble model are generally referred as "base models", "base learners", or "weak learners" in literature. These base models can be constructed using a single modelling algorithm, or several different algorithms. The idea is to train a diverse set of weak models on the same modelling task, such that the outputs of each weak learner have poor predictive ability (i.e., high bias), and among all weak learners, the outcome and error values exhibit high variance. Fundamentally, an ensemble learning model trains at least two high-bias (weak) and high-variance (diverse) models to be combined into a better-performing model. The set of weak models — which would not produce satisfactory predictive results individually — are combined or averaged to produce a single, high performing, accurate, and low-variance model to fit the task as required. Ensemble learning typically refers to bagging (bootstrap aggregating), boosting or stacking/blending techniques to induce high variance among the base models. Bagging creates diversity by generating random samples from the training observations and fitting the same model to each different sample — also known as homogeneous parallel ensembles. Boosting follows an iterative process by sequentially training each base model on the up-weighted errors of the previous base model, producing an additive model to reduce the final model errors — also known as sequential ensemble learning. Stacking or blending consists of different base models, each trained independently (i.e. diverse/high variance) to be combined into the ensemble model — producing a heterogeneous parallel ensemble. Common applications of ensemble learning include random forests (an extension of bagging), Boosted Tree models, and Gradient Boosted Tree Models. Models in applications of stacking are generally more task-specific — such as combining clustering techniques with other parametric and/or non-parametric techniques. Evaluating the prediction of an ensemble typically requires more computation than evaluating the prediction of a single model. In one sense, ensemble learning may be thought of as a way to compensate for poor learning algorithms by performing a lot of extra computation. On the other hand, the alternative is to do a lot more learning with one non-ensemble model. An ensemble may be more efficient at improving overall accuracy for the same increase in compute, storage, or communication resources by using that increase on two or more methods, than would have been improved by increasing resource use for a single method. Fast algorithms such as decision trees are commonly used in ensemble methods (e.g., random forests), although slower algorithms can benefit from ensemble techniques as well. By analogy, ensemble techniques have been used also in unsupervised learning scenarios, for example in consensus clustering or in anomaly detection. == Ensemble theory == Empirically, ensembles tend to yield better results when there is a significant diversity among the models. Many ensemble methods, therefore, seek to promote diversity among the models they combine. Although perhaps non-intuitive, more random algorithms (like random decision trees) can be used to produce a stronger ensemble than very deliberate algorithms (like entropy-reducing decision trees). Using a variety of strong learning algorithms, however, has been shown to be more effective than using techniques that attempt to dumb-down the models in order to promote diversity. It is possible to increase diversity in the training stage of the model using correlation for regression tasks or using information measures such as cross entropy for classification tasks. Theoretically, one can justify the diversity concept because the lower bound of the error rate of an ensemble system can be decomposed into accuracy, diversity, and the other term. === The geometric framework === Ensemble learning, including both regression and classification tasks, can be explained using a geometric framework. Within this framework, the output of each individual classifier or regressor for the entire dataset can be viewed as a point in a multi-dimensional space. Additionally, the target result is also represented as a point in this space, referred to as the "ideal point." The Euclidean distance is used as the metric to measure both the performance of a single classifier or regressor (the distance between its point and the ideal point) and the dissimilarity between two classifiers or regressors (the distance between their respective points). This perspective transforms ensemble learning into a deterministic problem. For example, within this geometric framework, it can be proved that the averaging of the outputs (scores) of all base classifiers or regressors can lead to equal or better results than the average of all the individual models. It can also be proved that if the optimal weighting scheme is used, then a weighted averaging approach can outperform any of the individual classifiers or regressors that make up the ensemble or as good as the best performer at least. == Ensemble size == While the number of component classifiers of an ensemble has a great impact on the accuracy of prediction, there is a limited number of studies addressing this problem. A priori determining of ensemble size and the volume and velocity of big data streams make this even more crucial for online ensemble classifiers. Mostly statistical tests were used for determining the proper number of components. More recently, a theoretical framework suggested that there is an ideal number of component classifiers for an ensemble such that having more or less than this number of classifiers would deteriorate the accuracy. It is called "the law of diminishing returns in ensemble construction." Their theoretical framework shows that using the same number of independent component classifiers as class labels gives the highest accuracy. == Common types of ensembles == === Bayes optimal classifier === The Bayes optimal classifier is a classification technique. It is an ensemble of all the hypotheses in the hypothesis space. On average, no other ensemble can outperform it. The Naive Bayes classifier is a version of this that assumes that the data is conditionally independent on the class and makes the computation more feasible. Each hypothesis is given a vote proportional to the likelihood that the training dataset would be sampled from a system if that hypothesis were true. To facilitate training data of finite size, the vote of each hypothesis is also multiplied by the prior probability of that hypothesis. The Bayes optimal classifier can be expressed with the following equation: y = a r g m a x c j ∈ C ∑ h i ∈ H P ( c j | h i ) P ( T | h i ) P ( h i ) {\displaystyle y={\underset {c_{j}\in C}{\mathrm {argmax} }}\sum _{h_{i}\in H}{P(c_{j}|h_{i})P(T|h_{i})P(h_{i})}} where y {\displaystyle y} is the predicted class, C {\displaystyle C} is the set of all possible classes, H {\displaystyle H} is the hypothesis space, P {\displaystyle P} refers to a probability, and T {\displaystyle T} is the training data. As an ensemble, the Bayes optimal classifier represents a hypothesis that is not necessarily in H {\displaystyle H} . The hypothesis represented by the Bayes optimal classifier, however, is the optimal hypothesis in ensemble space (the space of all possible ensembles consisting only of hypotheses in H {\displaystyle H} ). This formula can be restated using Bayes' theorem, which says that the posterior is proportional to the likelihood times the prior: P ( h i | T ) ∝ P ( T | h i ) P ( h i ) {\displaystyle P(h_{i}|T)\propto P(T|h_{i})P(h_{i})} hence, y = a r g m a x c j ∈ C ∑ h i ∈ H P ( c j | h i ) P ( h i | T ) {\displaystyle y={\underset {c_{j}\in C}{\mathrm {argmax} }}\sum _{h_{i}\in H}{P(c_{j}|h_{i})P(h_{i}|T)}} === Bootstrap aggregating (bagging) === Bootstrap aggregation (bagging) involves training an ensemble on bootstrapped data sets. A bootstrapped set is cr

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  Neural Networks (journal)

  Neural Networks is a monthly peer-reviewed scientific journal and an official journal of the International Neural Network Society, European Neural Network Society, and Japanese Neural Network Society. == History == The journal was established in 1988 and is published by Elsevier. It covers all aspects of research on artificial neural networks. The founding editor-in-chief was Stephen Grossberg (Boston University). The current editors-in-chief are DeLiang Wang (Ohio State University) and Taro Toyoizumi (RIKEN Center for Brain Science). == Abstracting and indexing == The journal is abstracted and indexed in Scopus and the Science Citation Index Expanded. According to the Journal Citation Reports, the journal has a 2022 impact factor of 7.8.

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  Common Voice

  Common Voice is a crowdsourcing project started by Mozilla to create a free and open speech corpus. The project is supported by volunteers who record sample sentences with a microphone and review recordings of other users. The transcribed sentences are collected in a voice database available under the public domain license CC0. This license ensures that developers can use the database for voice-to-text and text-to-voice applications without restrictions or costs. == Aims == Common Voice aims to provide diverse voice samples. According to Mozilla's Katharina Borchert, many existing projects took datasets from public radio or otherwise had datasets that underrepresented both women and people with pronounced accents. == Voice database == The first dataset was released in November 2017. More than 20,000 users worldwide had recorded 500 hours of English sentences. In February 2019, the first batch of languages was released for use. This included 18 languages such as English, French, German and Mandarin Chinese, but also less prevalent languages like Welsh and Kabyle. In total, this included almost 1,400 hours of recorded voice data from more than 42,000 contributors. By July 2020 the database had amassed 7,226 hours of voice recordings in 54 languages, 5,591 hours of which had been verified by volunteers. In May 2021, following the work to add Kinyarwanda, the project received a grant to add Kiswahili. At the beginning of 2022, Bengali.AI partnered with Common Voice to launch the "Bangla Speech Recognition" project that aims to make machines understand the Bangla language. 2000 hours of voice was collected. In September 2022, it was announced that the Twi language of Ghana was the 100th language to be added to the database. As of December 2025, Mozilla Common Voice collects voice data for over 250 languages, with the most hours having been collected in English, Catalan, Kinyarwanda, Belarusian and Esperanto.

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  Perusall

  Perusall is a social web annotation tool intended for use by students at schools and universities. It allows users to annotate the margins of a text in a virtual group setting that is similar to social media—with upvoting, emojis, chat functionality, and notification. It also includes automatic AI grading. == History == Perusall began as a research project at Harvard University. It later became an educational product for students and teachers. As of 2024, Perusall states more than 5 million students have used the tool at over 5,000 educational institutions in 112 countries." == Functionality == Perusall integrates with learning management systems such as Moodle, Canvas and Blackboard to aid with collaborative annotation. The tool supports annotation of a range of media including text, images, equations, videos, PDFs and snapshots of webpages.

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  Analogical modeling

  Analogical modeling (AM) is a formal theory of exemplar based analogical reasoning, proposed by Royal Skousen, professor of Linguistics and English language at Brigham Young University in Provo, Utah. It is applicable to language modeling and other categorization tasks. Analogical modeling is related to connectionism and nearest neighbor approaches, in that it is data-based rather than abstraction-based; but it is distinguished by its ability to cope with imperfect datasets (such as caused by simulated short term memory limits) and to base predictions on all relevant segments of the dataset, whether near or far. In language modeling, AM has successfully predicted empirically valid forms for which no theoretical explanation was known (see the discussion of Finnish morphology in Skousen et al. 2002). == Implementation == === Overview === An exemplar-based model consists of a general-purpose modeling engine and a problem-specific dataset. Within the dataset, each exemplar (a case to be reasoned from, or an informative past experience) appears as a feature vector: a row of values for the set of parameters that define the problem. For example, in a spelling-to-sound task, the feature vector might consist of the letters of a word. Each exemplar in the dataset is stored with an outcome, such as a phoneme or phone to be generated. When the model is presented with a novel situation (in the form of an outcome-less feature vector), the engine algorithmically sorts the dataset to find exemplars that helpfully resemble it, and selects one, whose outcome is the model's prediction. The particulars of the algorithm distinguish one exemplar-based modeling system from another. In AM, we think of the feature values as characterizing a context, and the outcome as a behavior that occurs within that context. Accordingly, the novel situation is known as the given context. Given the known features of the context, the AM engine systematically generates all contexts that include it (all of its supracontexts), and extracts from the dataset the exemplars that belong to each. The engine then discards those supracontexts whose outcomes are inconsistent (this measure of consistency will be discussed further below), leaving an analogical set of supracontexts, and probabilistically selects an exemplar from the analogical set with a bias toward those in large supracontexts. This multilevel search exponentially magnifies the likelihood of a behavior's being predicted as it occurs reliably in settings that specifically resemble the given context. === Analogical modeling in detail === AM performs the same process for each case it is asked to evaluate. The given context, consisting of n variables, is used as a template to generate 2 n {\displaystyle 2^{n}} supracontexts. Each supracontext is a set of exemplars in which one or more variables have the same values that they do in the given context, and the other variables are ignored. In effect, each is a view of the data, created by filtering for some criteria of similarity to the given context, and the total set of supracontexts exhausts all such views. Alternatively, each supracontext is a theory of the task or a proposed rule whose predictive power needs to be evaluated. It is important to note that the supracontexts are not equal peers one with another; they are arranged by their distance from the given context, forming a hierarchy. If a supracontext specifies all of the variables that another one does and more, it is a subcontext of that other one, and it lies closer to the given context. (The hierarchy is not strictly branching; each supracontext can itself be a subcontext of several others, and can have several subcontexts.) This hierarchy becomes significant in the next step of the algorithm. The engine now chooses the analogical set from among the supracontexts. A supracontext may contain exemplars that only exhibit one behavior; it is deterministically homogeneous and is included. It is a view of the data that displays regularity, or a relevant theory that has never yet been disproven. A supracontext may exhibit several behaviors, but contain no exemplars that occur in any more specific supracontext (that is, in any of its subcontexts); in this case it is non-deterministically homogeneous and is included. Here there is no great evidence that a systematic behavior occurs, but also no counterargument. Finally, a supracontext may be heterogeneous, meaning that it exhibits behaviors that are found in a subcontext (closer to the given context), and also behaviors that are not. Where the ambiguous behavior of the nondeterministically homogeneous supracontext was accepted, this is rejected because the intervening subcontext demonstrates that there is a better theory to be found. The heterogeneous supracontext is therefore excluded. This guarantees that we see an increase in meaningfully consistent behavior in the analogical set as we approach the given context. With the analogical set chosen, each appearance of an exemplar (for a given exemplar may appear in several of the analogical supracontexts) is given a pointer to every other appearance of an exemplar within its supracontexts. One of these pointers is then selected at random and followed, and the exemplar to which it points provides the outcome. This gives each supracontext an importance proportional to the square of its size, and makes each exemplar likely to be selected in direct proportion to the sum of the sizes of all analogically consistent supracontexts in which it appears. Then, of course, the probability of predicting a particular outcome is proportional to the summed probabilities of all the exemplars that support it. (Skousen 2002, in Skousen et al. 2002, pp. 11–25, and Skousen 2003, both passim) === Formulas === Given a context with n {\displaystyle n} elements: total number of pairings: n 2 {\displaystyle n^{2}} number of agreements for outcome i: n i 2 {\displaystyle n_{i}^{2}} number of disagreements for outcome i: n i ( n − n i ) {\displaystyle n_{i}(n-n_{i})} total number of agreements: ∑ n i 2 {\displaystyle \sum {n_{i}^{2}}} total number of disagreements: ∑ n i ( n − n i ) = n 2 − ∑ n i 2 {\displaystyle \sum {n_{i}(n-n_{i})}=n^{2}-\sum {n_{i}^{2}}} === Example === This terminology is best understood through an example. In the example used in the second chapter of Skousen (1989), each context consists of three variables with potential values 0-3 Variable 1: 0,1,2,3 Variable 2: 0,1,2,3 Variable 3: 0,1,2,3 The two outcomes for the dataset are e and r, and the exemplars are: 3 1 0 e 0 3 2 r 2 1 0 r 2 1 2 r 3 1 1 r We define a network of pointers like so: The solid lines represent pointers between exemplars with matching outcomes; the dotted lines represent pointers between exemplars with non-matching outcomes. The statistics for this example are as follows: n = 5 {\displaystyle n=5} n r = 4 {\displaystyle n_{r}=4} n e = 1 {\displaystyle n_{e}=1} total number of pairings: n 2 = 25 {\displaystyle n^{2}=25} number of agreements for outcome r: n r 2 = 16 {\displaystyle n_{r}^{2}=16} number of agreements for outcome e: n e 2 = 1 {\displaystyle n_{e}^{2}=1} number of disagreements for outcome r: n r ( n − n r ) = 4 {\displaystyle n_{r}(n-n_{r})=4} number of disagreements for outcome e: n e ( n − n e ) = 4 {\displaystyle n_{e}(n-n_{e})=4} total number of agreements: n r 2 + n e 2 = 17 {\displaystyle n_{r}^{2}+n_{e}^{2}=17} total number of disagreements: n r ( n − n r ) + n e ( n − n e ) = n 2 − ( n r 2 + n e 2 ) = 8 {\displaystyle n_{r}(n-n_{r})+n_{e}(n-n_{e})=n^{2}-(n_{r}^{2}+n_{e}^{2})=8} uncertainty or fraction of disagreement: 8 / 25 = .32 {\displaystyle 8/25=.32} Behavior can only be predicted for a given context; in this example, let us predict the outcome for the context "3 1 2". To do this, we first find all of the contexts containing the given context; these contexts are called supracontexts. We find the supracontexts by systematically eliminating the variables in the given context; with m variables, there will generally be 2 m {\displaystyle 2^{m}} supracontexts. The following table lists each of the sub- and supracontexts; x means "not x", and - means "anything". These contexts are shown in the venn diagram below: The next step is to determine which exemplars belong to which contexts in order to determine which of the contexts are homogeneous. The table below shows each of the subcontexts, their behavior in terms of the given exemplars, and the number of disagreements within the behavior: Analyzing the subcontexts in the table above, we see that there is only 1 subcontext with any disagreements: "3 1 2", which in the dataset consists of "3 1 0 e" and "3 1 1 r". There are 2 disagreements in this subcontext; 1 pointing from each of the exemplars to the other (see the pointer network pictured above). Therefore, only supracontexts containing this subcontext will contain any disagreements. We use a simple rule to identify the homogeneous supraco

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  Tensor product network

  A tensor product network, in artificial neural networks, is a network that exploits the properties of tensors to model associative concepts such as variable assignment. Orthonormal vectors are chosen to model the ideas (such as variable names and target assignments), and the tensor product of these vectors construct a network whose mathematical properties allow the user to easily extract the association from it.

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  Principal component analysis

  Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing. The data are linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified. The principal components of a collection of points in a real coordinate space are a sequence of p {\displaystyle p} unit vectors, where the i {\displaystyle i} -th vector is the direction of a line that best fits the data while being orthogonal to the first i − 1 {\displaystyle i-1} vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions (i.e., principal components) constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science. == Overview == When performing PCA, the first principal component of a set of p {\displaystyle p} variables is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through p {\displaystyle p} iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The i {\displaystyle i} -th principal component can be taken as a direction orthogonal to the first i − 1 {\displaystyle i-1} principal components that maximizes the variance of the projected data. For either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset. Robust and L1-norm-based variants of standard PCA have also been proposed. == History == PCA was invented in 1901 by Karl Pearson, as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (invented in the last quarter of the 19th century), eigenvalue decomposition (EVD) of XTX in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's Principal Component Analysis), Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science (Lorenz, 1956), empirical eigenfunction decomposition (Sirovich, 1987), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics. == Intuition == PCA can be thought of as fitting a p-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small. To find the axes of the ellipsoid, we must first center the values of each variable in the dataset on 0 by subtracting the mean of the variable's observed values from each of those values. These transformed values are used instead of the original observed values for each of the variables. Then, we compute the covariance matrix of the data and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then we must normalize each of the orthogonal eigenvectors to turn them into unit vectors. Once this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. This choice of basis will transform the covariance matrix into a diagonalized form, in which the diagonal elements represent the variance of each axis. The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues. Biplots and scree plots (degree of explained variance) are used to interpret findings of the PCA. == Details == PCA is defined as an orthogonal linear transformation on a real inner product space that transforms the data to a new coordinate system such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. Consider an n × p {\displaystyle n\times p} data matrix, X, with column-wise zero empirical mean (the sample mean of each column has been shifted to zero), where each of the n rows represents a different repetition of the experiment, and each of the p columns gives a particular kind of feature (say, the results from a particular sensor). Mathematically, the transformation is defined by a set of size l {\displaystyle l} (where l {\displaystyle l} is usually selected to be strictly less than p {\displaystyle p} to reduce dimensionality) of p {\displaystyle p} -dimensional vectors of weights or coefficients w ( k ) = ( w 1 , … , w p ) ( k ) {\displaystyle \mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)}} that map each row vector x ( i ) = ( x 1 , … , x p ) ( i ) {\displaystyle \mathbf {x} _{(i)}=(x_{1},\dots ,x_{p})_{(i)}} of X to a new vector of principal component scores t ( i ) = ( t 1 , … , t l ) ( i ) {\displaystyle \mathbf {t} _{(i)}=(t_{1},\dots ,t_{l})_{(i)}} , given by t k ( i ) = x ( i ) ⋅ w ( k ) f o r i = 1 , … , n k = 1 , … , l {\displaystyle {t_{k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad \mathrm {for} \qquad i=1,\dots ,n\qquad k=1,\dots ,l} in such a way that the individual variables t 1 , … , t l {\displaystyle t_{1},\dots ,t_{l}} of t considered over the data set successively inherit the maximum possible variance from X, with each coefficient vector w constrained to be a unit vector. The above may equivalently be written in matrix form as T = X W {\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} } where T i k = t k ( i ) {\displaystyle {\mathbf {T} }_{ik}={t_{k}}_{(i)}} , X i j = x j ( i ) {\displaystyle {\mathbf {X} }_{ij}={x_{j}}_{(i)}} , and W j k = w j ( k ) {\displaystyle {\mathbf {W} }_{jk}={w_{j}}_{(k)}} . === First component === In order to maximize variance, the first weight vector w(1) thus has to satisfy w ( 1 ) = arg ⁡ max ‖ w ‖ = 1 { ∑ i ( t 1 ) ( i ) 2 } = arg ⁡ max ‖ w ‖ = 1 { ∑ i ( x ( i ) ⋅ w ) 2 } {\displaystyle \mathbf {w} _{(1)}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}(t_{1})_{(i)}^{2}\right\}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}\left(\mathbf {x} _{(i)}\cdot \mathbf {w} \right)^{2}\right\}} Equivalently, writing this in matrix form gives w ( 1 ) = arg ⁡ max ‖ w ‖ = 1 { ‖ X w ‖ 2 } = arg ⁡ max ‖ w ‖ = 1 { w T X T X w } {\displaystyle \mathbf {w} _{(1)}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {Xw} \right\|^{2}\right\}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} \right\}} Since w(1) has been defined to be a unit vector, it equivalently also satisfies w ( 1 ) = arg ⁡ max { w T X T X w w T w } {\displaystyle \mathbf {w} _{(1)}=\arg \max \left\{{\frac {\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} }{\mathbf {w} ^{\mathsf {T}}\mathbf {w} }}\right\}} The quantity to be maximised can be recognised as a Rayleigh quotient. A standard result for a positive semidefinite matrix such as XTX is that the quotient's maximum possible value is the largest eigenvalue of the matrix, which occurs when w is the corresponding eigenvector. With w(1) found, the first principal component of a data vector

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  Luminance HDR

  Luminance HDR, formerly Qtpfsgui, is graphics software used for the creation and manipulation of high-dynamic-range images. Released under the terms of the GPL, it is available for Linux, Microsoft Windows, and Mac OS X (Intel only). Luminance HDR supports several High Dynamic Range (HDR) as well as Low Dynamic Range (LDR) file formats. == Functionality == Prerequisite of HDR photography are several narrow-range digital images with different exposures. Luminance HDR combines these images and calculates a high-contrast image. In order to view this image on a regular computer monitor, Luminance HDR can convert it into a displayable LDR image format using a variety of methods, such as tone mapping. Currently fifteen different tone mapping operators (algorithms) are available, each one with its tunable parameters. Different image processing techniques can be applied to the generated HDR images, such as resizing, cropping, rotating and a number of projective transformations. The software also provides batch processing functionality for creating HDR images and for tone mapping them in a non-interactive way. A module for copying Exif data among sets of images is also provided. For users who prefers the command line, a non-GUI, non-graphical interface is also available on all supported platforms. A common problem with HDR photography is that images need to be aligned exactly. If the subject is static, this can be achieved using a tripod or a stable surface on which the camera is placed. In the case of image data that does not align exactly, an automatic alignment can be performed using a tool provided by the Hugin project. If this automation doesn't provide the desired result, the user may improve it manually. == Supported formats == HDR images are images with a high dynamic range and, using Luminance HDR, they can be created as well as edited. The following HDR graphic formats are supported: OpenEXR Radiance HDR Tag Image File Format (TIFF) Format: 16 Bit, 32 Bit (Float) and LogLuv Raw PFS native Luminance HDR can create an HDR image from several LDR images and tonemap an HDR into an LDR. The following LDR formats are supported: JPG PNG Portable Pixmap (PPM) Portable Bitmap (PBM) TIFF (8 Bit)

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  Proximal policy optimization

  Proximal policy optimization (PPO) is a reinforcement learning (RL) algorithm for training an intelligent agent. Specifically, it is a policy gradient method, often used for deep RL when the policy network is very large. == History == The predecessor to PPO, Trust Region Policy Optimization (TRPO), was published in 2015. It addressed the instability issue of another algorithm, the Deep Q-Network (DQN), by using the trust region method to limit the KL divergence between the old and new policies. However, TRPO uses the Hessian matrix (a matrix of second derivatives) to enforce the trust region, but the Hessian is inefficient for large-scale problems. PPO was published in 2017. It was essentially an approximation of TRPO that does not require computing the Hessian. The KL divergence constraint was approximated by simply clipping the policy gradient. Since 2018, PPO was the default RL algorithm at OpenAI. PPO has been applied to many areas, such as controlling a robotic arm, beating professional players at Dota 2 (OpenAI Five), and playing Atari games. == TRPO == TRPO, the predecessor of PPO, is an on-policy algorithm. It can be used for environments with either discrete or continuous action spaces. The pseudocode is as follows: Input: initial policy parameters θ 0 {\textstyle \theta _{0}} , initial value function parameters ϕ 0 {\textstyle \phi _{0}} Hyperparameters: KL-divergence limit δ {\textstyle \delta } , backtracking coefficient α {\textstyle \alpha } , maximum number of backtracking steps K {\textstyle K} for k = 0 , 1 , 2 , … {\textstyle k=0,1,2,\ldots } do Collect set of trajectories D k = { τ i } {\textstyle {\mathcal {D}}_{k}=\left\{\tau _{i}\right\}} by running policy π k = π ( θ k ) {\textstyle \pi _{k}=\pi \left(\theta _{k}\right)} in the environment. Compute rewards-to-go R ^ t {\textstyle {\hat {R}}_{t}} . Compute advantage estimates, A ^ t {\textstyle {\hat {A}}_{t}} (using any method of advantage estimation) based on the current value function V ϕ k {\textstyle V_{\phi _{k}}} . Estimate policy gradient as g ^ k = 1 | D k | ∑ τ ∈ D k ∑ t = 0 T ∇ θ log ⁡ π θ ( a t ∣ s t ) | θ k A ^ t {\displaystyle {\hat {g}}_{k}=\left.{\frac {1}{\left|{\mathcal {D}}_{k}\right|}}\sum _{\tau \in {\mathcal {D}}_{k}}\sum _{t=0}^{T}\nabla _{\theta }\log \pi _{\theta }\left(a_{t}\mid s_{t}\right)\right|_{\theta _{k}}{\hat {A}}_{t}} Use the conjugate gradient algorithm to compute x ^ k ≈ H ^ k − 1 g ^ k {\displaystyle {\hat {x}}_{k}\approx {\hat {H}}_{k}^{-1}{\hat {g}}_{k}} where H ^ k {\textstyle {\hat {H}}_{k}} is the Hessian of the sample average KL-divergence. Update the policy by backtracking line search with θ k + 1 = θ k + α j 2 δ x ^ k T H ^ k x ^ k x ^ k {\displaystyle \theta _{k+1}=\theta _{k}+\alpha ^{j}{\sqrt {\frac {2\delta }{{\hat {x}}_{k}^{T}{\hat {H}}_{k}{\hat {x}}_{k}}}}{\hat {x}}_{k}} where j ∈ { 0 , 1 , 2 , … K } {\textstyle j\in \{0,1,2,\ldots K\}} is the smallest value which improves the sample loss and satisfies the sample KL-divergence constraint. Fit value function by regression on mean-squared error: ϕ k + 1 = arg ⁡ min ϕ 1 | D k | T ∑ τ ∈ D k ∑ t = 0 T ( V ϕ ( s t ) − R ^ t ) 2 {\displaystyle \phi _{k+1}=\arg \min _{\phi }{\frac {1}{\left|{\mathcal {D}}_{k}\right|T}}\sum _{\tau \in {\mathcal {D}}_{k}}\sum _{t=0}^{T}\left(V_{\phi }\left(s_{t}\right)-{\hat {R}}_{t}\right)^{2}} typically via some gradient descent algorithm. == PPO == The pseudocode is as follows: Input: initial policy parameters θ 0 {\textstyle \theta _{0}} , initial value function parameters ϕ 0 {\textstyle \phi _{0}} for k = 0 , 1 , 2 , … {\textstyle k=0,1,2,\ldots } do Collect set of trajectories D k = { τ i } {\textstyle {\mathcal {D}}_{k}=\left\{\tau _{i}\right\}} by running policy π k = π ( θ k ) {\textstyle \pi _{k}=\pi \left(\theta _{k}\right)} in the environment. Compute rewards-to-go R ^ t {\textstyle {\hat {R}}_{t}} . Compute advantage estimates, A ^ t {\textstyle {\hat {A}}_{t}} (using any method of advantage estimation) based on the current value function V ϕ k {\textstyle V_{\phi _{k}}} . Update the policy by maximizing the PPO-Clip objective: θ k + 1 = arg ⁡ max θ 1 | D k | T ∑ τ ∈ D k ∑ t = 0 T min ( π θ ( a t ∣ s t ) π θ k ( a t ∣ s t ) A π θ k ( s t , a t ) , g ( ϵ , A π θ k ( s t , a t ) ) ) {\displaystyle \theta _{k+1}=\arg \max _{\theta }{\frac {1}{\left|{\mathcal {D}}_{k}\right|T}}\sum _{\tau \in {\mathcal {D}}_{k}}\sum _{t=0}^{T}\min \left({\frac {\pi _{\theta }\left(a_{t}\mid s_{t}\right)}{\pi _{\theta _{k}}\left(a_{t}\mid s_{t}\right)}}A^{\pi _{\theta _{k}}}\left(s_{t},a_{t}\right),\quad g\left(\epsilon ,A^{\pi _{\theta _{k}}}\left(s_{t},a_{t}\right)\right)\right)} typically via stochastic gradient ascent with Adam. Fit value function by regression on mean-squared error: ϕ k + 1 = arg ⁡ min ϕ 1 | D k | T ∑ τ ∈ D k ∑ t = 0 T ( V ϕ ( s t ) − R ^ t ) 2 {\displaystyle \phi _{k+1}=\arg \min _{\phi }{\frac {1}{\left|{\mathcal {D}}_{k}\right|T}}\sum _{\tau \in {\mathcal {D}}_{k}}\sum _{t=0}^{T}\left(V_{\phi }\left(s_{t}\right)-{\hat {R}}_{t}\right)^{2}} typically via some gradient descent algorithm. Like all policy gradient methods, PPO is used for training an RL agent whose actions are determined by a differentiable policy function by gradient ascent. Intuitively, a policy gradient method takes small policy update steps, so the agent can reach higher and higher rewards in expectation. Policy gradient methods may be unstable: A step size that is too big may direct the policy in a suboptimal direction, thus having little possibility of recovery; a step size that is too small lowers the overall efficiency. To solve the instability, PPO implements a clip function that constrains the policy update of an agent from being too large, so that larger step sizes may be used without negatively affecting the gradient ascent process. === Basic concepts === To begin the PPO training process, the agent is set in an environment to perform actions based on its current input. In the early phase of training, the agent can freely explore solutions and keep track of the result. Later, with a certain amount of transition samples and policy updates, the agent will select an action to take by randomly sampling from the probability distribution P ( A | S ) {\displaystyle P(A|S)} generated by the policy network. The actions that are most likely to be beneficial will have the highest probability of being selected from the random sample. After an agent arrives at a different scenario (a new state) by acting, it is rewarded with a positive reward or a negative reward. The objective of an agent is to maximize the cumulative reward signal across sequences of states, known as episodes. === Policy gradient laws: the advantage function === The advantage function (denoted as A {\displaystyle A} ) is central to PPO, as it tries to answer the question of whether a specific action of the agent is better or worse than some other possible action in a given state. By definition, the advantage function is an estimate of the relative value for a selected action. If the output of this function is positive, it means that the action in question is better than the average return, so the possibilities of selecting that specific action will increase. The opposite is true for a negative advantage output. The advantage function can be defined as A = Q − V {\displaystyle A=Q-V} , where Q {\displaystyle Q} is the discounted sum of rewards (the total weighted reward for the completion of an episode) and V {\displaystyle V} is the baseline estimate. Since the advantage function is calculated after the completion of an episode, the program records the outcome of the episode. Therefore, calculating advantage is essentially an unsupervised learning problem. The baseline estimate comes from the value function that outputs the expected discounted sum of an episode starting from the current state. In the PPO algorithm, the baseline estimate will be noisy (with some variance), as it also uses a neural network, like the policy function itself. With Q {\displaystyle Q} and V {\displaystyle V} computed, the advantage function is calculated by subtracting the baseline estimate from the actual discounted return. If A > 0 {\displaystyle A>0} , the actual return of the action is better than the expected return from experience; if A < 0 {\displaystyle A<0} , the actual return is worse. === Ratio function === In PPO, the ratio function ( r t {\displaystyle r_{t}} ) calculates the probability of selecting action a {\displaystyle a} in state s {\displaystyle s} given the current policy network, divided by the previous probability under the old policy. In other words: If r t ( θ ) > 1 {\displaystyle r_{t}(\theta )>1} , where θ {\displaystyle \theta } are the policy network parameters, then selecting action a {\displaystyle a} in state s {\displaystyle s} is more likely based on the current policy than the previous policy. If 0 ≤ r t ( θ ) < 1 {\displaystyle 0\leq r_{t}(\theta )<1} , then selecting actio

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  Premature convergence

  Premature convergence is an unwanted effect in evolutionary algorithms (EA), a metaheuristic that mimics the basic principles of biological evolution as a computer algorithm for solving an optimization problem. The effect means that the population of an EA has converged too early, resulting in being suboptimal. In this context, the parental solutions, through the aid of genetic operators, are not able to generate offspring that are superior to, or outperform, their parents. Premature convergence is a common problem found in evolutionary algorithms, as it leads to a loss, or convergence of, a large number of alleles, subsequently making it very difficult to search for a specific gene in which the alleles were present. An allele is considered lost if, in a population, a gene is present, where all individuals are sharing the same value for that particular gene. An allele is, as defined by De Jong, considered to be a converged allele, when 95% of a population share the same value for a certain gene. == Strategies for preventing premature convergence == Strategies to regain genetic variation can be: a mating strategy called incest prevention, uniform crossover, mimicking sexual selection, favored replacement of similar individuals (preselection or crowding), segmentation of individuals of similar fitness (fitness sharing), increasing population size niche and specie The genetic variation can also be regained by mutation though this process is highly random. A general strategy to reduce the risk of premature convergence is to use structured populations instead of the commonly used panmictic ones. == Identification of the occurrence of premature convergence == It is hard to determine when premature convergence has occurred, and it is equally hard to predict its presence in the future. One measure is to use the difference between the average and maximum fitness values, as used by Patnaik & Srinivas, to then vary the crossover and mutation probabilities. Population diversity is another measure which has been extensively used in studies to measure premature convergence. However, although it has been widely accepted that a decrease in the population diversity directly leads to premature convergence, there have been little studies done on the analysis of population diversity. In other words, by using the term population diversity, the argument for a study in preventing premature convergence lacks robustness, unless specified what their definition of population diversity is. There are models to counter the effect and risk of premature convergence that do not compromise core GA parameters like population size, mutation rate, and other core mechanisms. These models were inspired by biological ecology, where genetic interactions are limited by external mechanisms such as spatial topologies or speciation. These ecological models, such as the Eco-GA, adopt diffusion-based strategies to improve the robustness of GA runs and increase the likelihood of reaching near-global optima. == Causes for premature convergence == There are a number of presumed or hypothesized causes for the occurrence of premature convergence. === Self-adaptive mutations === Rechenberg introduced the idea of self-adaptation of mutation distributions in evolution strategies. According to Rechenberg, the control parameters for these mutation distributions evolved internally through self-adaptation, rather than predetermination. He called it the 1/5-success rule of evolution strategies (1 + 1)-ES: The step size control parameter would be increased by some factor if the relative frequency of positive mutations through a determined period of time is larger than 1/5, vice versa if it is smaller than 1/5. Self-adaptive mutations may very well be one of the causes for premature convergence. Accurately locating of optima can be enhanced by self-adaptive mutation, as well as accelerating the search for this optima. This has been widely recognized, though the mechanism's underpinnings of this have been poorly studied, as it is often unclear whether the optima is found locally or globally. Self-adaptive methods can cause global convergence to global optimum, provided that the selection methods used are using elitism, as well as that the rule of self-adaptation doesn't interfere with the mutation distribution, which has the property of ensuring a positive minimum probability when hitting a random subset. This is for non-convex objective functions with sets that include bounded lower levels of non-zero measurements. A study by Rudolph suggests that self-adaption mechanisms among elitist evolution strategies do resemble the 1/5-success rule, and could very well get caught by a local optimum that include a positive probability. === Panmictic populations === Most EAs use unstructured or panmictic populations where basically every individual in the population is eligible for mate selection based on fitness. Thus, The genetic information of an only slightly better individual can spread in a population within a few generations, provided that no better other offspring is produced during this time. Especially in comparatively small populations, this can quickly lead to a loss of genotypic diversity and thus to premature convergence. A well-known countermeasure is to switch to alternative population models which introduce substructures into the population that preserve genotypic diversity over a longer period of time and thus counteract the tendency towards premature convergence. This has been shown for various EAs such as genetic algorithms, the evolution strategy, other EAs or memetic algorithms.

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  Vanishing gradient problem

  In machine learning, the vanishing gradient problem is the problem of greatly diverging gradient magnitudes between earlier and later layers encountered when training neural networks with backpropagation. In such methods, neural network weights are updated proportional to their partial derivative of the loss function. As the number of forward propagation steps in a network increases, for instance due to greater network depth, the gradients of earlier weights are calculated with increasingly many multiplications. These multiplications shrink the gradient magnitude. Consequently, the gradients of earlier weights will be exponentially smaller than the gradients of later weights. This difference in gradient magnitude might introduce instability in the training process, slow it, or halt it entirely. For instance, consider the hyperbolic tangent activation function. The gradients of this function are in range [0,1]. The product of repeated multiplication with such gradients decreases exponentially. The inverse problem, when weight gradients at earlier layers get exponentially larger, is called the exploding gradient problem. Backpropagation allowed researchers to train supervised deep artificial neural networks from scratch, initially with little success. Hochreiter's diplom thesis of 1991 formally identified the reason for this failure in the "vanishing gradient problem", which not only affects many-layered feedforward networks, but also recurrent networks. The latter are trained by unfolding them into very deep feedforward networks, where a new layer is created for each time-step of an input sequence processed by the network (the combination of unfolding and backpropagation is termed backpropagation through time). == Prototypical models == This section is based on the paper On the difficulty of training Recurrent Neural Networks by Pascanu, Mikolov, and Bengio. === Recurrent network model === A generic recurrent network has hidden states h 1 , h 2 , … {\displaystyle h_{1},h_{2},\dots } , inputs u 1 , u 2 , … {\displaystyle u_{1},u_{2},\dots } , and outputs x 1 , x 2 , … {\displaystyle x_{1},x_{2},\dots } . Let it be parameterized by θ {\displaystyle \theta } , so that the system evolves as ( h t , x t ) = F ( h t − 1 , u t , θ ) {\displaystyle (h_{t},x_{t})=F(h_{t-1},u_{t},\theta )} Often, the output x t {\displaystyle x_{t}} is a function of h t {\displaystyle h_{t}} , as some x t = G ( h t ) {\displaystyle x_{t}=G(h_{t})} . The vanishing gradient problem already presents itself clearly when x t = h t {\displaystyle x_{t}=h_{t}} , so we simplify our notation to the special case with: x t = F ( x t − 1 , u t , θ ) {\displaystyle x_{t}=F(x_{t-1},u_{t},\theta )} Now, take its differential: d x t = ∇ θ F ( x t − 1 , u t , θ ) d θ + ∇ x F ( x t − 1 , u t , θ ) d x t − 1 = ∇ θ F ( x t − 1 , u t , θ ) d θ + ∇ x F ( x t − 1 , u t , θ ) [ ∇ θ F ( x t − 2 , u t − 1 , θ ) d θ + ∇ x F ( x t − 2 , u t − 1 , θ ) d x t − 2 ] ⋮ = [ ∇ θ F ( x t − 1 , u t , θ ) + ∇ x F ( x t − 1 , u t , θ ) ∇ θ F ( x t − 2 , u t − 1 , θ ) + ⋯ ] d θ {\displaystyle {\begin{aligned}dx_{t}&=\nabla _{\theta }F(x_{t-1},u_{t},\theta )d\theta +\nabla _{x}F(x_{t-1},u_{t},\theta )dx_{t-1}\\&=\nabla _{\theta }F(x_{t-1},u_{t},\theta )d\theta +\nabla _{x}F(x_{t-1},u_{t},\theta )\left[\nabla _{\theta }F(x_{t-2},u_{t-1},\theta )d\theta +\nabla _{x}F(x_{t-2},u_{t-1},\theta )dx_{t-2}\right]\\&\;\;\vdots \\&=\left[\nabla _{\theta }F(x_{t-1},u_{t},\theta )+\nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{\theta }F(x_{t-2},u_{t-1},\theta )+\cdots \right]d\theta \end{aligned}}} Training the network requires us to define a loss function to be minimized. Let it be L ( x T , u 1 , … , u T ) {\displaystyle L(x_{T},u_{1},\dots ,u_{T})} , then minimizing it by gradient descent gives Δ θ = − η ⋅ [ ∇ x L ( x T ) ( ∇ θ F ( x t − 1 , u t , θ ) + ∇ x F ( x t − 1 , u t , θ ) ∇ θ F ( x t − 2 , u t − 1 , θ ) + ⋯ ) ] T {\displaystyle \Delta \theta =-\eta \cdot \left[\nabla _{x}L(x_{T})\left(\nabla _{\theta }F(x_{t-1},u_{t},\theta )+\nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{\theta }F(x_{t-2},u_{t-1},\theta )+\cdots \right)\right]^{T}} where η {\displaystyle \eta } is the learning rate. The vanishing/exploding gradient problem appears because there are repeated multiplications, of the form ∇ x F ( x t − 1 , u t , θ ) ∇ x F ( x t − 2 , u t − 1 , θ ) ∇ x F ( x t − 3 , u t − 2 , θ ) ⋯ {\displaystyle \nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{x}F(x_{t-2},u_{t-1},\theta )\nabla _{x}F(x_{t-3},u_{t-2},\theta )\cdots } ==== Example: recurrent network with sigmoid activation ==== For a concrete example, consider a typical recurrent network defined by x t = F ( x t − 1 , u t , θ ) = W rec σ ( x t − 1 ) + W in u t + b {\displaystyle x_{t}=F(x_{t-1},u_{t},\theta )=W_{\text{rec}}\sigma (x_{t-1})+W_{\text{in}}u_{t}+b} where θ = ( W rec , W in ) {\displaystyle \theta =(W_{\text{rec}},W_{\text{in}})} is the network parameter, σ {\displaystyle \sigma } is the sigmoid activation function, applied to each vector coordinate separately, and b {\displaystyle b} is the bias vector. Then, ∇ x F ( x t − 1 , u t , θ ) = W rec diag ⁡ ( σ ′ ( x t − 1 ) ) {\displaystyle \nabla _{x}F(x_{t-1},u_{t},\theta )=W_{\text{rec}}\operatorname {diag} (\sigma '(x_{t-1}))} , and so ∇ x F ( x t − 1 , u t , θ ) ∇ x F ( x t − 2 , u t − 1 , θ ) ⋯ ∇ x F ( x t − k , u t − k + 1 , θ ) = W rec diag ⁡ ( σ ′ ( x t − 1 ) ) W rec diag ⁡ ( σ ′ ( x t − 2 ) ) ⋯ W rec diag ⁡ ( σ ′ ( x t − k ) ) {\displaystyle {\begin{aligned}&\nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{x}F(x_{t-2},u_{t-1},\theta )\cdots \nabla _{x}F(x_{t-k},u_{t-k+1},\theta )\\&=W_{\text{rec}}\operatorname {diag} (\sigma '(x_{t-1}))W_{\text{rec}}\operatorname {diag} (\sigma '(x_{t-2}))\cdots W_{\text{rec}}\operatorname {diag} (\sigma '(x_{t-k}))\end{aligned}}} Since | σ ′ | ≤ 1 {\displaystyle \left|\sigma '\right|\leq 1} , the operator norm of the above multiplication is bounded above by ‖ W rec ‖ k {\displaystyle \left\|W_{\text{rec}}\right\|^{k}} . So if the spectral radius of W rec {\displaystyle W_{\text{rec}}} is γ < 1 {\displaystyle \gamma <1} , then at large k {\displaystyle k} , the above multiplication has operator norm bounded above by γ k → 0 {\displaystyle \gamma ^{k}\to 0} . This is the prototypical vanishing gradient problem. The effect of a vanishing gradient is that the network cannot learn long-range effects. Recall Equation (loss differential): ∇ θ L = ∇ x L ( x T , u 1 , … , u T ) [ ∇ θ F ( x t − 1 , u t , θ ) + ∇ x F ( x t − 1 , u t , θ ) ∇ θ F ( x t − 2 , u t − 1 , θ ) + ⋯ ] {\displaystyle \nabla _{\theta }L=\nabla _{x}L(x_{T},u_{1},\dots ,u_{T})\left[\nabla _{\theta }F(x_{t-1},u_{t},\theta )+\nabla _{x}F(x_{t-1},u_{t},\theta )\nabla _{\theta }F(x_{t-2},u_{t-1},\theta )+\cdots \right]} The components of ∇ θ F ( x , u , θ ) {\displaystyle \nabla _{\theta }F(x,u,\theta )} are just components of σ ( x ) {\displaystyle \sigma (x)} and u {\displaystyle u} , so if u t , u t − 1 , … {\displaystyle u_{t},u_{t-1},\dots } are bounded, then ‖ ∇ θ F ( x t − k − 1 , u t − k , θ ) ‖ {\displaystyle \left\|\nabla _{\theta }F(x_{t-k-1},u_{t-k},\theta )\right\|} is also bounded by some M > 0 {\displaystyle M>0} , and so the terms in ∇ θ L {\displaystyle \nabla _{\theta }L} decay as M γ k {\displaystyle M\gamma ^{k}} . This means that, effectively, ∇ θ L {\displaystyle \nabla _{\theta }L} is affected only by the first O ( γ − 1 ) {\displaystyle O(\gamma ^{-1})} terms in the sum. If γ ≥ 1 {\displaystyle \gamma \geq 1} , the above analysis does not quite work. For the prototypical exploding gradient problem, the next model is clearer. === Dynamical systems model === Following (Doya, 1993), consider this one-neuron recurrent network with sigmoid activation: x t + 1 = ( 1 − ε ) x t + ε σ ( w x t + b ) + ε w ′ u t {\displaystyle x_{t+1}=(1-\varepsilon )x_{t}+\varepsilon \sigma (wx_{t}+b)+\varepsilon w'u_{t}} At the small ε {\displaystyle \varepsilon } limit, the dynamics of the network becomes d x d t = − x ( t ) + σ ( w x ( t ) + b ) + w ′ u ( t ) {\displaystyle {\frac {dx}{dt}}=-x(t)+\sigma (wx(t)+b)+w'u(t)} Consider first the autonomous case, with u = 0 {\displaystyle u=0} . Set w = 5.0 {\displaystyle w=5.0} , and vary b {\displaystyle b} in [ − 3 , − 2 ] {\displaystyle [-3,-2]} . As b {\displaystyle b} decreases, the system has 1 stable point, then has 2 stable points and 1 unstable point, and finally has 1 stable point again. Explicitly, the stable points are ( x , b ) = ( x , ln ⁡ ( x 1 − x ) − 5 x ) {\displaystyle (x,b)=\left(x,\ln \left({\frac {x}{1-x}}\right)-5x\right)} . Now consider Δ x ( T ) Δ x ( 0 ) {\displaystyle {\frac {\Delta x(T)}{\Delta x(0)}}} and Δ x ( T ) Δ b {\displaystyle {\frac {\Delta x(T)}{\Delta b}}} , where T {\displaystyle T} is large enough that the system has settled into one of the stable points. If ( x ( 0 ) , b ) {\displaystyle (x(0),b)} puts the system very close to an unstable point, then a tiny variation in x ( 0 ) {\displaystyle x(0)} or b {\displaystyle b} wo

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  Globetrooper

  Globetrooper is a free travel app known for assisting travelers in finding partners for group trips and world adventures. Globetrooper offers a free social travel platform that helps people find travel partners. == History == Globetrooper was developed and released in 2010 by a couple; Todd Sullivan and Lauren McLeod who are two travel-minded individuals that wanted to make it easier for travelers to plan a journey and see the world. With their backgrounds in business, software & design, and a love for travel, both left the corporate world and launched Globetrooper on Lauren’s birthday 28 March 2010. Globetrooper was first launched as an information portal with a view to making it more social, but after some months, the content quickly grew and changed to the ‘travel partner’ concept.

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  Algorithmic learning theory

  Algorithmic learning theory is a mathematical framework for analyzing machine learning problems and algorithms. Synonyms include formal learning theory and algorithmic inductive inference. Algorithmic learning theory is different from statistical learning theory in that it does not make use of statistical assumptions and analysis. Both algorithmic and statistical learning theory are concerned with machine learning and can thus be viewed as branches of computational learning theory. == Distinguishing characteristics == Unlike statistical learning theory and most statistical theory in general, algorithmic learning theory does not assume that data are random samples, that is, that data points are independent of each other. This makes the theory suitable for domains where observations are (relatively) noise-free but not random, such as language learning and automated scientific discovery. The fundamental concept of algorithmic learning theory is learning in the limit: as the number of data points increases, a learning algorithm should converge to a correct hypothesis on every possible data sequence consistent with the problem space. This is a non-probabilistic version of statistical consistency, which also requires convergence to a correct model in the limit, but allows a learner to fail on data sequences with probability measure 0 . Algorithmic learning theory investigates the learning power of Turing machines. Other frameworks consider a much more restricted class of learning algorithms than Turing machines, for example, learners that compute hypotheses more quickly, for instance in polynomial time. An example of such a framework is probably approximately correct learning . == Learning in the limit == The concept was introduced in E. Mark Gold's seminal paper "Language identification in the limit". The objective of language identification is for a machine running one program to be capable of developing another program by which any given sentence can be tested to determine whether it is "grammatical" or "ungrammatical". The language being learned need not be English or any other natural language - in fact the definition of "grammatical" can be absolutely anything known to the tester. In Gold's learning model, the tester gives the learner an example sentence at each step, and the learner responds with a hypothesis, which is a suggested program to determine grammatical correctness. It is required of the tester that every possible sentence (grammatical or not) appears in the list eventually, but no particular order is required. It is required of the learner that at each step the hypothesis must be correct for all the sentences so far. A particular learner is said to be able to "learn a language in the limit" if there is a certain number of steps beyond which its hypothesis no longer changes. At this point it has indeed learned the language, because every possible sentence appears somewhere in the sequence of inputs (past or future), and the hypothesis is correct for all inputs (past or future), so the hypothesis is correct for every sentence. The learner is not required to be able to tell when it has reached a correct hypothesis, all that is required is that it be true. Gold showed that any language which is defined by a Turing machine program can be learned in the limit by another Turing-complete machine using enumeration. This is done by the learner testing all possible Turing machine programs in turn until one is found which is correct so far - this forms the hypothesis for the current step. Eventually, the correct program will be reached, after which the hypothesis will never change again (but note that the learner does not know that it won't need to change). Gold also showed that if the learner is given only positive examples (that is, only grammatical sentences appear in the input, not ungrammatical sentences), then the language can only be guaranteed to be learned in the limit if there are only a finite number of possible sentences in the language (this is possible if, for example, sentences are known to be of limited length). Language identification in the limit is a highly abstract model. It does not allow for limits of runtime or computer memory which can occur in practice, and the enumeration method may fail if there are errors in the input. However the framework is very powerful, because if these strict conditions are maintained, it allows the learning of any program known to be computable. This is because a Turing machine program can be written to mimic any program in any conventional programming language. See Church-Turing thesis. == Other identification criteria == Learning theorists have investigated other learning criteria, such as the following. Efficiency: minimizing the number of data points required before convergence to a correct hypothesis. Mind Changes: minimizing the number of hypothesis changes that occur before convergence. Mind change bounds are closely related to mistake bounds that are studied in statistical learning theory. Kevin Kelly has suggested that minimizing mind changes is closely related to choosing maximally simple hypotheses in the sense of Occam’s Razor. == Annual conference == Since 1990, there is an International Conference on Algorithmic Learning Theory (ALT), called Workshop in its first years (1990–1997). Between 1992 and 2016, proceedings were published in the LNCS series. Starting from 2017, they are published by the Proceedings of Machine Learning Research. The 34th conference will be held in Singapore in Feb 2023. The topics of the conference cover all of theoretical machine learning, including statistical and computational learning theory, online learning, active learning, reinforcement learning, and deep learning.

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  Consensus clustering

  Consensus clustering is a method of aggregating (potentially conflicting) results from multiple clustering algorithms. Also called cluster ensembles or aggregation of clustering (or partitions), it refers to the situation in which a number of different (input) clusterings have been obtained for a particular dataset and it is desired to find a single (consensus) clustering which is a better fit in some sense than the existing clusterings. Consensus clustering is thus the problem of reconciling clustering information about the same data set coming from different sources or from different runs of the same algorithm. When cast as an optimization problem, consensus clustering is known as median partition, and has been shown to be NP-complete, even when the number of input clusterings is three. Consensus clustering for unsupervised learning is analogous to ensemble learning in supervised learning. == Issues with existing clustering techniques == Current clustering techniques do not address all the requirements adequately. Dealing with large number of dimensions and large number of data items can be problematic because of time complexity; Effectiveness of the method depends on the definition of "distance" (for distance-based clustering) If an obvious distance measure doesn't exist, we must "define" it, which is not always easy, especially in multidimensional spaces. The result of the clustering algorithm (that, in many cases, can be arbitrary itself) can be interpreted in different ways. == Justification for using consensus clustering == There are potential shortcomings for all existing clustering techniques. This may cause interpretation of results to become difficult, especially when there is no knowledge about the number of clusters. Clustering methods are also very sensitive to the initial clustering settings, which can cause non-significant data to be amplified in non-reiterative methods. An extremely important issue in cluster analysis is the validation of the clustering results, that is, how to gain confidence about the significance of the clusters provided by the clustering technique (cluster numbers and cluster assignments). Lacking an external objective criterion (the equivalent of a known class label in supervised analysis), this validation becomes somewhat elusive. Iterative descent clustering methods, such as the SOM and k-means clustering circumvent some of the shortcomings of hierarchical clustering by providing for univocally defined clusters and cluster boundaries. Consensus clustering provides a method that represents the consensus across multiple runs of a clustering algorithm, to determine the number of clusters in the data, and to assess the stability of the discovered clusters. The method can also be used to represent the consensus over multiple runs of a clustering algorithm with random restart (such as K-means, model-based Bayesian clustering, SOM, etc.), so as to account for its sensitivity to the initial conditions. It can provide data for a visualization tool to inspect cluster number, membership, and boundaries. However, they lack the intuitive and visual appeal of hierarchical clustering dendrograms, and the number of clusters must be chosen a priori. == The Monti consensus clustering algorithm == The Monti consensus clustering algorithm is one of the most popular consensus clustering algorithms and is used to determine the number of clusters, K {\displaystyle K} . Given a dataset of N {\displaystyle N} total number of points to cluster, this algorithm works by resampling and clustering the data, for each K {\displaystyle K} and a N × N {\displaystyle N\times N} consensus matrix is calculated, where each element represents the fraction of times two samples clustered together. A perfectly stable matrix would consist entirely of zeros and ones, representing all sample pairs always clustering together or not together over all resampling iterations. The relative stability of the consensus matrices can be used to infer the optimal K {\displaystyle K} . More specifically, given a set of points to cluster, D = { e 1 , e 2 , . . . e N } {\displaystyle D=\{e_{1},e_{2},...e_{N}\}} , let D 1 , D 2 , . . . , D H {\displaystyle D^{1},D^{2},...,D^{H}} be the list of H {\displaystyle H} perturbed (resampled) datasets of the original dataset D {\displaystyle D} , and let M h {\displaystyle M^{h}} denote the N × N {\displaystyle N\times N} connectivity matrix resulting from applying a clustering algorithm to the dataset D h {\displaystyle D^{h}} . The entries of M h {\displaystyle M^{h}} are defined as follows: M h ( i , j ) = { 1 , if points i and j belong to the same cluster 0 , otherwise {\displaystyle M^{h}(i,j)={\begin{cases}1,&{\text{if}}{\text{ points i and j belong to the same cluster}}\\0,&{\text{otherwise}}\end{cases}}} Let I h {\displaystyle I^{h}} be the N × N {\displaystyle N\times N} identicator matrix where the ( i , j ) {\displaystyle (i,j)} -th entry is equal to 1 if points i {\displaystyle i} and j {\displaystyle j} are in the same perturbed dataset D h {\displaystyle D^{h}} , and 0 otherwise. The indicator matrix is used to keep track of which samples were selected during each resampling iteration for the normalisation step. The consensus matrix C {\displaystyle C} is defined as the normalised sum of all connectivity matrices of all the perturbed datasets and a different one is calculated for every K {\displaystyle K} . C ( i , j ) = ( ∑ h = 1 H M h ( i , j ) ∑ h = 1 H I h ( i , j ) ) {\displaystyle C(i,j)=\left({\frac {\textstyle \sum _{h=1}^{H}M^{h}(i,j)\displaystyle }{\sum _{h=1}^{H}I^{h}(i,j)}}\right)} That is the entry ( i , j ) {\displaystyle (i,j)} in the consensus matrix is the number of times points i {\displaystyle i} and j {\displaystyle j} were clustered together divided by the total number of times they were selected together. The matrix is symmetric and each element is defined within the range [ 0 , 1 ] {\displaystyle [0,1]} . A consensus matrix is calculated for each K {\displaystyle K} to be tested, and the stability of each matrix, that is how far the matrix is towards a matrix of perfect stability (just zeros and ones) is used to determine the optimal K {\displaystyle K} . One way of quantifying the stability of the K {\displaystyle K} th consensus matrix is examining its CDF curve (see below). == Over-interpretation potential of the Monti consensus clustering algorithm == Monti consensus clustering can be a powerful tool for identifying clusters, but it needs to be applied with caution as shown by Şenbabaoğlu et al. It has been shown that the Monti consensus clustering algorithm is able to claim apparent stability of chance partitioning of null datasets drawn from a unimodal distribution, and thus has the potential to lead to over-interpretation of cluster stability in a real study. If clusters are not well separated, consensus clustering could lead one to conclude apparent structure when there is none, or declare cluster stability when it is subtle. Identifying false positive clusters is a common problem throughout cluster research, and has been addressed by methods such as SigClust and the GAP-statistic. However, these methods rely on certain assumptions for the null model that may not always be appropriate. Şenbabaoğlu et al demonstrated the original delta K metric to decide K {\displaystyle K} in the Monti algorithm performed poorly, and proposed a new superior metric for measuring the stability of consensus matrices using their CDF curves. In the CDF curve of a consensus matrix, the lower left portion represents sample pairs rarely clustered together, the upper right portion represents those almost always clustered together, whereas the middle segment represent those with ambiguous assignments in different clustering runs. The proportion of ambiguous clustering (PAC) score measure quantifies this middle segment; and is defined as the fraction of sample pairs with consensus indices falling in the interval (u1, u2) ∈ [0, 1] where u1 is a value close to 0 and u2 is a value close to 1 (for instance u1=0.1 and u2=0.9). A low value of PAC indicates a flat middle segment, and a low rate of discordant assignments across permuted clustering runs. One can therefore infer the optimal number of clusters by the K {\displaystyle K} value having the lowest PAC. == Related work == Clustering ensemble (Strehl and Ghosh): They considered various formulations for the problem, most of which reduce the problem to a hyper-graph partitioning problem. In one of their formulations they considered the same graph as in the correlation clustering problem. The solution they proposed is to compute the best k-partition of the graph, which does not take into account the penalty for merging two nodes that are far apart. Clustering aggregation (Fern and Brodley): They applied the clustering aggregation idea to a collection of soft clusterings they obtained by random projections. They used an agglomerative algorithm

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