AI Analytics Certification

AI Analytics Certification — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Telebirr

  Telebirr (Amharic: ቴሌብር) is a mobile payment service developed and was launched by Ethio telecom, the state owned telecommunication and Internet service provider in Ethiopia. It took five months to develop the end-to-end service. It facilitates the delivery of cashless transactions. The platform deployed currently has the capacity of processing up to 100 transactions per second (TPS) and can be scaled up to 1000 TPS. The service is accessible via SMS, USSD, and smartphone applications. Telebirr works in five languages. == Services == Though the service is fully accessible for any customer of Ethio telecom, the users need to register through the mobile application called Telebirr or using an authorized agent or Ethio telecom shop or Unstructured Supplementary Service Data (USSD), 127# nationally. However, Telebirr also provides a “quick registration” by using any information that already exists in Ethio telecom's system.

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  Latent Dirichlet allocation

  In natural language processing, latent Dirichlet allocation (LDA) is a generative statistical model that explains how a collection of text documents can be described by a set of unobserved "topics." For example, given a set of news articles, LDA might discover that one topic is characterized by words like "president", "government", and "election", while another is characterized by "team", "game", and "score". It is one of the most common topic models. The LDA model was first presented as a graphical model for population genetics by J. K. Pritchard, M. Stephens and P. Donnelly in 2000. The model was subsequently applied to machine learning by David Blei, Andrew Ng, and Michael I. Jordan in 2003. Although its most frequent application is in modeling text corpora, it has also been used for other problems, such as in clinical psychology, social science, and computational musicology. The core assumption of LDA is that documents are represented as a random mixture of latent topics, and each topic is characterized by a probability distribution over words. The model is a generalization of probabilistic latent semantic analysis (pLSA), differing primarily in that LDA treats the topic mixture as a Dirichlet prior, leading to more reasonable mixtures and less susceptibility to overfitting. Learning the latent topics and their associated probabilities from a corpus is typically done using Bayesian inference, often with methods like Gibbs sampling or variational Bayes. == History == In the context of population genetics, LDA was proposed by J. K. Pritchard, M. Stephens and P. Donnelly in 2000. LDA was applied in machine learning by David Blei, Andrew Ng and Michael I. Jordan in 2003. == Overview == === Population genetics === In population genetics, the model is used to detect the presence of structured genetic variation in a group of individuals. The model assumes that alleles carried by individuals under study have origin in various extant or past populations. The model and various inference algorithms allow scientists to estimate the allele frequencies in those source populations and the origin of alleles carried by individuals under study. The source populations can be interpreted ex-post in terms of various evolutionary scenarios. In association studies, detecting the presence of genetic structure is considered a necessary preliminary step to avoid confounding. === Clinical psychology, mental health, and social science === In clinical psychology research, LDA has been used to identify common themes of self-images experienced by young people in social situations. Other social scientists have used LDA to examine large sets of topical data from discussions on social media (e.g., tweets about prescription drugs). Additionally, supervised Latent Dirichlet Allocation with covariates (SLDAX) has been specifically developed to combine latent topics identified in texts with other manifest variables. This approach allows for the integration of text data as predictors in statistical regression analyses, improving the accuracy of mental health predictions. One of the main advantages of SLDAX over traditional two-stage approaches is its ability to avoid biased estimates and incorrect standard errors, allowing for a more accurate analysis of psychological texts. In the field of social sciences, LDA has proven to be useful for analyzing large datasets, such as social media discussions. For instance, researchers have used LDA to investigate tweets discussing socially relevant topics, like the use of prescription drugs and cultural differences in China. By analyzing these large text corpora, it is possible to uncover patterns and themes that might otherwise go unnoticed, offering valuable insights into public discourse and perception in real time. === Musicology === In the context of computational musicology, LDA has been used to discover tonal structures in different corpora. === Machine learning === One application of LDA in machine learning – specifically, topic discovery, a subproblem in natural language processing – is to discover topics in a collection of documents, and then automatically classify any individual document within the collection in terms of how "relevant" it is to each of the discovered topics. A topic is considered to be a set of terms (i.e., individual words or phrases) that, taken together, suggest a shared theme. For example, in a document collection related to pet animals, the terms dog, spaniel, beagle, golden retriever, puppy, bark, and woof would suggest a DOG_related theme, while the terms cat, siamese, Maine coon, tabby, manx, meow, purr, and kitten would suggest a CAT_related theme. There may be many more topics in the collection – e.g., related to diet, grooming, healthcare, behavior, etc. that we do not discuss for simplicity's sake. (Very common, so called stop words in a language – e.g., "the", "an", "that", "are", "is", etc., – would not discriminate between topics and are usually filtered out by pre-processing before LDA is performed. Pre-processing also converts terms to their "root" lexical forms – e.g., "barks", "barking", and "barked" would be converted to "bark".) If the document collection is sufficiently large, LDA will discover such sets of terms (i.e., topics) based upon the co-occurrence of individual terms, though the task of assigning a meaningful label to an individual topic (i.e., that all the terms are DOG_related) is up to the user, and often requires specialized knowledge (e.g., for collection of technical documents). The LDA approach assumes that: The semantic content of a document is composed by combining one or more terms from one or more topics. Certain terms are ambiguous, belonging to more than one topic, with different probability. (For example, the term training can apply to both dogs and cats, but are more likely to refer to dogs, which are used as work animals or participate in obedience or skill competitions.) However, in a document, the accompanying presence of specific neighboring terms (which belong to only one topic) will disambiguate their usage. Most documents will contain only a relatively small number of topics. In the collection, e.g., individual topics will occur with differing frequencies. That is, they have a probability distribution, so that a given document is more likely to contain some topics than others. Within a topic, certain terms will be used much more frequently than others. In other words, the terms within a topic will also have their own probability distribution. When LDA machine learning is employed, both sets of probabilities are computed during the training phase, using Bayesian methods and an expectation–maximization algorithm. LDA is a generalization of older approach of probabilistic latent semantic analysis (pLSA), The pLSA model is equivalent to LDA under a uniform Dirichlet prior distribution. pLSA relies on only the first two assumptions above and does not care about the remainder. While both methods are similar in principle and require the user to specify the number of topics to be discovered before the start of training (as with k-means clustering) LDA has the following advantages over pLSA: LDA yields better disambiguation of words and a more precise assignment of documents to topics. Computing probabilities allows a "generative" process by which a collection of new "synthetic documents" can be generated that would closely reflect the statistical characteristics of the original collection. Unlike LDA, pLSA is vulnerable to overfitting especially when the size of corpus increases. The LDA algorithm is more readily amenable to scaling up for large data sets using the MapReduce approach on a computing cluster. == Model == With plate notation, which is often used to represent probabilistic graphical models (PGMs), the dependencies among the many variables can be captured concisely. The boxes are "plates" representing replicates, which are repeated entities. The outer plate represents documents, while the inner plate represents the repeated word positions in a given document; each position is associated with a choice of topic and word. The variable names are defined as follows: M denotes the number of documents N is number of words in a given document (document i has N i {\displaystyle N_{i}} words) α is the parameter of the Dirichlet prior on the per-document topic distributions β is the parameter of the Dirichlet prior on the per-topic word distribution θ i {\displaystyle \theta _{i}} is the topic distribution for document i φ k {\displaystyle \varphi _{k}} is the word distribution for topic k z i j {\displaystyle z_{ij}} is the topic for the j-th word in document i w i j {\displaystyle w_{ij}} is the specific word. The fact that W is grayed out means that words w i j {\displaystyle w_{ij}} are the only observable variables, and the other variables are latent variables. As proposed in the original paper, a sparse Dirichlet prior can be used to model the to

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  UIMA

  UIMA ( yoo-EE-mə), short for Unstructured Information Management Architecture, is an OASIS standard for content analytics, originally developed at IBM. It provides a component software architecture for the development, discovery, composition, and deployment of multi-modal analytics for the analysis of unstructured information and integration with search technologies. == Structure == The UIMA architecture can be thought of in four dimensions: It specifies component interfaces in an analytics pipeline. It describes a set of design patterns. It suggests two data representations: an in-memory representation of annotations for high-performance analytics and an XML representation of annotations for integration with remote web services. It suggests development roles allowing tools to be used by users with diverse skills. == Implementations and uses == Apache UIMA, a reference implementation of UIMA, is maintained by the Apache Software Foundation. UIMA is used in a number of software projects: IBM Research's Watson uses UIMA for analyzing unstructured data. The Clinical Text Analysis and Knowledge Extraction System (Apache cTAKES) is a UIMA-based system for information extraction from medical records. DKPro Core is a collection of reusable UIMA components for general-purpose natural language processing.

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  Feature selection

  In machine learning, feature selection is the process of selecting a subset of relevant features (variables, predictors) for use in model construction. Feature selection techniques are used for several reasons: simplification of models to make them easier to interpret, shorter training times, to avoid the curse of dimensionality, improve the compatibility of the data with a certain learning model class, to encode inherent symmetries present in the input space. The central premise when using feature selection is that data sometimes contains features that are redundant or irrelevant, and can thus be removed without incurring much loss of information. Redundancy and irrelevance are two distinct notions, since one relevant feature may be redundant in the presence of another relevant feature with which it is strongly correlated. Feature extraction creates new features from functions of the original features, whereas feature selection finds a subset of the features. Feature selection techniques are often used in domains where there are many features and comparatively few samples (data points). == Introduction == A feature selection algorithm can be seen as the combination of a search technique for proposing new feature subsets, along with an evaluation measure which scores the different feature subsets. The simplest algorithm is to test each possible subset of features finding the one which minimizes the error rate. This is an exhaustive search of the space, and is computationally intractable for all but the smallest of feature sets. The choice of evaluation metric heavily influences the algorithm, and it is these evaluation metrics which distinguish between the three main categories of feature selection algorithms: wrappers, filters and embedded methods. Wrapper methods use a predictive model to score feature subsets. Each new subset is used to train a model, which is tested on a hold-out set. Counting the number of mistakes made on that hold-out set (the error rate of the model) gives the score for that subset. As wrapper methods train a new model for each subset, they are very computationally intensive, but usually provide the best performing feature set for that particular type of model or typical problem. Filter methods use a proxy measure instead of the error rate to score a feature subset. This measure is chosen to be fast to compute, while still capturing the usefulness of the feature set. Common measures include the mutual information, the pointwise mutual information, Pearson product-moment correlation coefficient, Relief-based algorithms, and inter/intra class distance or the scores of significance tests for each class/feature combinations. Filters are usually less computationally intensive than wrappers, but they produce a feature set which is not tuned to a specific type of predictive model. This lack of tuning means a feature set from a filter is more general than the set from a wrapper, usually giving lower prediction performance than a wrapper. However the feature set doesn't contain the assumptions of a prediction model, and so is more useful for exposing the relationships between the features. Many filters provide a feature ranking rather than an explicit best feature subset, and the cut off point in the ranking is chosen via cross-validation. Filter methods have also been used as a preprocessing step for wrapper methods, allowing a wrapper to be used on larger problems. One other popular approach is the Recursive Feature Elimination algorithm, commonly used with Support Vector Machines to repeatedly construct a model and remove features with low weights. Embedded methods are a catch-all group of techniques which perform feature selection as part of the model construction process. The exemplar of this approach is the LASSO method for constructing a linear model, which penalizes the regression coefficients with an L1 penalty, shrinking many of them to zero. Any features which have non-zero regression coefficients are 'selected' by the LASSO algorithm. Improvements to the LASSO include Bolasso which bootstraps samples; Elastic net regularization, which combines the L1 penalty of LASSO with the L2 penalty of ridge regression; and FeaLect which scores all the features based on combinatorial analysis of regression coefficients. AEFS further extends LASSO to nonlinear scenario with autoencoders. These approaches tend to be between filters and wrappers in terms of computational complexity. In traditional regression analysis, the most popular form of feature selection is stepwise regression, which is a wrapper technique. It is a greedy algorithm that adds the best feature (or deletes the worst feature) at each round. The main control issue is deciding when to stop the algorithm. In machine learning, this is typically done by cross-validation. In statistics, some criteria are optimized. This leads to the inherent problem of nesting. More robust methods have been explored, such as branch and bound and piecewise linear network. == Subset selection == Subset selection evaluates a subset of features as a group for suitability. Subset selection algorithms can be broken up into wrappers, filters, and embedded methods. Wrappers use a search algorithm to search through the space of possible features and evaluate each subset by running a model on the subset. Wrappers can be computationally expensive and have a risk of over fitting to the model. Filters are similar to wrappers in the search approach, but instead of evaluating against a model, a simpler filter is evaluated. Embedded techniques are embedded in, and specific to, a model. Many popular search approaches use greedy hill climbing, which iteratively evaluates a candidate subset of features, then modifies the subset and evaluates if the new subset is an improvement over the old. Evaluation of the subsets requires a scoring metric that grades a subset of features. Exhaustive search is generally impractical, so at some implementor (or operator) defined stopping point, the subset of features with the highest score discovered up to that point is selected as the satisfactory feature subset. The stopping criterion varies by algorithm; possible criteria include: a subset score exceeds a threshold, a program's maximum allowed run time has been surpassed, etc. Alternative search-based techniques are based on targeted projection pursuit which finds low-dimensional projections of the data that score highly: the features that have the largest projections in the lower-dimensional space are then selected. Search approaches include: Exhaustive Best first Simulated annealing Genetic algorithm Greedy forward selection Greedy backward elimination Particle swarm optimization Targeted projection pursuit Scatter search Variable neighborhood search Two popular filter metrics for classification problems are correlation and mutual information, although neither are true metrics or 'distance measures' in the mathematical sense, since they fail to obey the triangle inequality and thus do not compute any actual 'distance' – they should rather be regarded as 'scores'. These scores are computed between a candidate feature (or set of features) and the desired output category. There are, however, true metrics that are a simple function of the mutual information; see here. Other available filter metrics include: Class separability Error probability Inter-class distance Probabilistic distance Entropy Consistency-based feature selection Correlation-based feature selection == Optimality criteria == The choice of optimality criteria is difficult as there are multiple objectives in a feature selection task. Many common criteria incorporate a measure of accuracy, penalised by the number of features selected. Examples include Akaike information criterion (AIC) and Mallows's Cp, which have a penalty of 2 for each added feature. AIC is based on information theory, and is effectively derived via the maximum entropy principle. Other criteria are Bayesian information criterion (BIC), which uses a penalty of log ⁡ n {\displaystyle {\sqrt {\log {n}}}} for each added feature, minimum description length (MDL) which asymptotically uses log ⁡ n {\displaystyle {\sqrt {\log {n}}}} , Bonferroni / RIC which use 2 log ⁡ p {\displaystyle {\sqrt {2\log {p}}}} , maximum dependency feature selection, and a variety of new criteria that are motivated by false discovery rate (FDR), which use something close to 2 log ⁡ p q {\displaystyle {\sqrt {2\log {\frac {p}{q}}}}} . A maximum entropy rate criterion may also be used to select the most relevant subset of features. == Structure learning == Filter feature selection is a specific case of a more general paradigm called structure learning. Feature selection finds the relevant feature set for a specific target variable whereas structure learning finds the relationships between all the variables, usually by expressing these relationships as a graph. The most common structure learning algorithms

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  FMLLR

  In signal processing, Feature space Maximum Likelihood Linear Regression (fMLLR) is a global feature transform that are typically applied in a speaker adaptive way, where fMLLR transforms acoustic features to speaker adapted features by a multiplication operation with a transformation matrix. In some literature, fMLLR is also known as the Constrained Maximum Likelihood Linear Regression (cMLLR). == Overview == fMLLR transformations are trained in a maximum likelihood sense on adaptation data. These transformations may be estimated in many ways, but only maximum likelihood (ML) estimation is considered in fMLLR. The fMLLR transformation is trained on a particular set of adaptation data, such that it maximizes the likelihood of that adaptation data given a current model-set. This technique is a widely used approach for speaker adaptation in HMM-based speech recognition. Later research also shows that fMLLR is an excellent acoustic feature for DNN/HMM hybrid speech recognition models. The advantage of fMLLR includes the following: the adaptation process can be performed within a pre-processing phase, and is independent of the ASR training and decoding process. this type of adapted feature can be applied to deep neural networks (DNN) to replace traditionally used mel-spectrogram in end-to-end speech recognition models. fMLLR's speaker adaptation process leads to a significant performance boost for ASR models, hence outperforming other transform or features like MFCCs (Mel-Frequency Cepstral Coefficients) and FBANKs (Filter bank) coefficients. fMLLR features can be efficiently realized with speech toolkits like Kaldi. Major problem and disadvantage of fMLLR: when the amount of adaptation data is limited, the transformation matrices tends to easily overfit the given data. == Computing fMLLR transform == Feature transform of fMLLR can be easily computed with the open source speech tool Kaldi, the Kaldi script uses the standard estimation scheme described in Appendix B of the original paper, in particular the section Appendix B.1 "Direct method over rows". In the Kaldi formulation, fMLLR is an affine feature transform of the form x {\displaystyle x} → A {\displaystyle A} x {\displaystyle x} + b {\displaystyle +b} , which can be written in the form x {\displaystyle x} →W x ^ {\displaystyle {\hat {x}}} , where x ^ {\displaystyle {\hat {x}}} = [ x 1 ] {\displaystyle {\begin{bmatrix}x\\1\end{bmatrix}}} is the acoustic feature x {\displaystyle x} with a 1 appended. Note that this differs from some of the literature where the 1 comes first as x ^ {\displaystyle {\hat {x}}} = [ 1 x ] {\displaystyle {\begin{bmatrix}1\\x\end{bmatrix}}} . The sufficient statistics stored are: K = ∑ t , j , m γ j , m ( t ) Σ j m − 1 μ j m x ( t ) + {\displaystyle K=\sum _{t,j,m}\gamma _{j,m}(t)\textstyle \Sigma _{jm}^{-1}\mu _{jm}x(t)^{+}\displaystyle } where Σ j m − 1 {\displaystyle \textstyle \Sigma _{jm}^{-1}\displaystyle } is the inverse co-variance matrix. And for 0 ≤ i ≤ D {\displaystyle 0\leq i\leq D} where D {\displaystyle D} is the feature dimension: G ( i ) = ∑ t , j , m γ j , m ( t ) ( 1 σ j , m 2 ( i ) ) x ( t ) + x ( t ) + T {\displaystyle G^{(i)}=\sum _{t,j,m}\gamma _{j,m}(t)\left({\frac {1}{\sigma _{j,m}^{2}(i)}}\right)x(t)^{+}x(t)^{+T}\displaystyle } For a thorough review that explains fMLLR and the commonly used estimation techniques, see the original paper "Maximum likelihood linear transformations for HMM-based speech recognition ". Note that the Kaldi script that performs the feature transforms of fMLLR differs with by using a column of the inverse in place of the cofactor row. In other words, the factor of the determinant is ignored, as it does not affect the transform result and can causes potential danger of numerical underflow or overflow. == Comparing with other features or transforms == Experiment result shows that by using the fMLLR feature in speech recognition, constant improvement is gained over other acoustic features on various commonly used benchmark datasets (TIMIT, LibriSpeech, etc). In particular, fMLLR features outperform MFCCs and FBANKs coefficients, which is mainly due to the speaker adaptation process that fMLLR performs. In, phoneme error rate (PER, %) is reported for the test set of TIMIT with various neural architectures: As expected, fMLLR features outperform MFCCs and FBANKs coefficients despite the use of different model architecture. Where MLP (multi-layer perceptron) serves as a simple baseline, on the other hand RNN, LSTM, and GRU are all well known recurrent models. The Li-GRU architecture is based on a single gate and thus saves 33% of the computations over a standard GRU model, Li-GRU thus effectively address the gradient vanishing problem of recurrent models. As a result, the best performance is obtained with the Li-GRU model on fMLLR features. == Extract fMLLR features with Kaldi == fMLLR can be extracted as reported in the s5 recipe of Kaldi. Kaldi scripts can certainly extract fMLLR features on different dataset, below are the basic example steps to extract fMLLR features from the open source speech corpora Librispeech. Note that the instructions below are for the subsets train-clean-100,train-clean-360,dev-clean, and test-clean, but they can be easily extended to support the other sets dev-other, test-other, and train-other-500. These instruction are based on the codes provided in this GitHub repository, which contains Kaldi recipes on the LibriSpeech corpora to execute the fMLLR feature extraction process, replace the files under $KALDI_ROOT/egs/librispeech/s5/ with the files in the repository. Install Kaldi. Install Kaldiio. If running on a single machine, change the following lines in $KALDI_ROOT/egs/librispeech/s5/cmd.sh to replace queue.pl to run.pl: Change the data path in run.sh to your LibriSpeech data path, the directory LibriSpeech/ should be under that path. For example: Install flac with: sudo apt-get install flac Run the Kaldi recipe run.sh for LibriSpeech at least until Stage 13 (included), for simplicity you can use the modified run.sh. Copy exp/tri4b/trans. files into exp/tri4b/decode_tgsmall_train_clean_/ with the following command: Compute the fMLLR features by running the following script, the script can also be downloaded here: Compute alignments using: Apply CMVN and dump the fMLLR features to new .ark files, the script can also be downloaded here: Use the Python script to convert Kaldi generated .ark features to .npy for your own dataloader, an example Python script is provided:

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  Fitness function

  A fitness function is a particular type of objective or cost function that is used to summarize, as a single figure of merit, how close a given candidate solution is to achieving the set aims. It is an important component of evolutionary algorithms (EA), such as genetic programming, evolution strategies or genetic algorithms. An EA is a metaheuristic that reproduces the basic principles of biological evolution as a computer algorithm in order to solve challenging optimization or planning tasks, at least approximately. For this purpose, many candidate solutions are generated, which are evaluated using a fitness function in order to guide the evolutionary development towards the desired goal. Similar quality functions are also used in other metaheuristics, such as ant colony optimization or particle swarm optimization. In the field of EAs, each candidate solution, also called an individual, is commonly represented as a string of numbers (referred to as a chromosome). After each round of testing or simulation the idea is to delete the n worst individuals, and to breed n new ones from the best solutions. Each individual must therefore to be assigned a quality number indicating how close it has come to the overall specification, and this is generated by applying the fitness function to the test or simulation results obtained from that candidate solution. Two main classes of fitness functions exist: one where the fitness function does not change, as in optimizing a fixed function or testing with a fixed set of test cases; and one where the fitness function is mutable, as in niche differentiation or co-evolving the set of test cases. Another way of looking at fitness functions is in terms of a fitness landscape, which shows the fitness for each possible chromosome. In the following, it is assumed that the fitness is determined based on an evaluation that remains unchanged during an optimization run. A fitness function does not necessarily have to be able to calculate an absolute value, as it is sometimes sufficient to compare candidates in order to select the better one. A relative indication of fitness (candidate a is better than b) is sufficient in some cases, such as tournament selection or Pareto optimization. == Requirements of evaluation and fitness function == The quality of the evaluation and calculation of a fitness function is fundamental to the success of an EA optimisation. It implements Darwin's principle of "survival of the fittest". Without fitness-based selection mechanisms for mate selection and offspring acceptance, EA search would be blind and hardly distinguishable from the Monte Carlo method. When setting up a fitness function, one must always be aware that it is about more than just describing the desired target state. Rather, the evolutionary search on the way to the optimum should also be supported as much as possible (see also section on auxiliary objectives), if and insofar as this is not already done by the fitness function alone. If the fitness function is designed badly, the algorithm will either converge on an inappropriate solution, or will have difficulty converging at all. Definition of the fitness function is not straightforward in many cases and often is performed iteratively if the fittest solutions produced by an EA is not what is desired. Interactive genetic algorithms address this difficulty by outsourcing evaluation to external agents which are normally humans. == Computational efficiency == The fitness function should not only closely align with the designer's goal, but also be computationally efficient. Execution speed is crucial, as a typical evolutionary algorithm must be iterated many times in order to produce a usable result for a non-trivial problem. Fitness approximation may be appropriate, especially in the following cases: Fitness computation time of a single solution is extremely high Precise model for fitness computation is missing The fitness function is uncertain or noisy. Alternatively or also in addition to the fitness approximation, the fitness calculations can also be distributed to a parallel computer in order to reduce the execution times. Depending on the population model of the EA used, both the EA itself and the fitness calculations of all offspring of one generation can be executed in parallel. == Multi-objective optimization == Practical applications usually aim at optimizing multiple and at least partially conflicting objectives. Two fundamentally different approaches are often used for this purpose, Pareto optimization and optimization based on fitness calculated using the weighted sum. === Weighted sum and penalty functions === When optimizing with the weighted sum, the single values of the O {\displaystyle O} objectives are first normalized so that they can be compared. This can be done with the help of costs or by specifying target values and determining the current value as the degree of fulfillment. Costs or degrees of fulfillment can then be compared with each other and, if required, can also be mapped to a uniform fitness scale. Without loss of generality, fitness is assumed to represent a value to be maximized. Each objective o i {\displaystyle o_{i}} is assigned a weight w i {\displaystyle w_{i}} in the form of a percentage value so that the overall raw fitness f r a w {\displaystyle f_{raw}} can be calculated as a weighted sum: f r a w = ∑ i = 1 O o i ⋅ w i w i t h ∑ i = 1 O w i = 1 {\displaystyle f_{raw}=\sum _{i=1}^{O}{o_{i}\cdot w_{i}}\quad {\mathsf {with}}\quad \sum _{i=1}^{O}{w_{i}}=1} A violation of R {\displaystyle R} restrictions r j {\displaystyle r_{j}} can be included in the fitness determined in this way in the form of penalty functions. For this purpose, a function p f j ( r j ) {\displaystyle pf_{j}(r_{j})} can be defined for each restriction which returns a value between 0 {\displaystyle 0} and 1 {\displaystyle 1} depending on the degree of violation, with the result being 1 {\displaystyle 1} if there is no violation. The previously determined raw fitness is multiplied by the penalty function(s) and the result is then the final fitness f f i n a l {\displaystyle f_{final}} : f f i n a l = f r a w ⋅ ∏ j = 1 R p f j ( r j ) = ∑ i = 1 O ( o i ⋅ w i ) ⋅ ∏ j = 1 R p f j ( r j ) {\displaystyle f_{final}=f_{raw}\cdot \prod _{j=1}^{R}{pf_{j}(r_{j})}=\sum _{i=1}^{O}{(o_{i}\cdot w_{i})}\cdot \prod _{j=1}^{R}{pf_{j}(r_{j})}} This approach is simple and has the advantage of being able to combine any number of objectives and restrictions. The disadvantage is that different objectives can compensate each other and that the weights have to be defined before the optimization. This means that the compromise lines must be defined before optimization, which is why optimization with the weighted sum is also referred to as the a priori method. In addition, certain solutions may not be obtained, see the section on the comparison of both types of optimization. === Pareto optimization === A solution is called Pareto-optimal if the improvement of one objective is only possible with a deterioration of at least one other objective. The set of all Pareto-optimal solutions, also called Pareto set, represents the set of all optimal compromises between the objectives. The figure below on the right shows an example of the Pareto set of two objectives f 1 {\displaystyle f_{1}} and f 2 {\displaystyle f_{2}} to be maximized. The elements of the set form the Pareto front (green line). From this set, a human decision maker must subsequently select the desired compromise solution. Constraints are included in Pareto optimization in that solutions without constraint violations are per se better than those with violations. If two solutions to be compared each have constraint violations, the respective extent of the violations decides. It was recognized early on that EAs with their simultaneously considered solution set are well suited to finding solutions in one run that cover the Pareto front sufficiently well. They are therefore well suited as a-posteriori methods for multi-objective optimization, in which the final decision is made by a human decision maker after optimization and determination of the Pareto front. Besides the SPEA2, the NSGA-II and NSGA-III have established themselves as standard methods. The advantage of Pareto optimization is that, in contrast to the weighted sum, it provides all alternatives that are equivalent in terms of the objectives as an overall solution. The disadvantage is that a visualization of the alternatives becomes problematic or even impossible from four objectives on. Furthermore, the effort increases exponentially with the number of objectives. If there are more than three or four objectives, some have to be combined using the weighted sum or other aggregation methods. === Comparison of both types of assessment === With the help of the weighted sum, the total Pareto front can be obtained by a suitable choice of weights, provided that it is convex

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  Low-rank approximation

  In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank. The problem is used for mathematical modeling and data compression. The rank constraint is related to a constraint on the complexity of a model that fits the data. In applications, often there are other constraints on the approximating matrix apart from the rank constraint, e.g., non-negativity and Hankel structure. Low-rank approximation is closely related to numerous other techniques, including principal component analysis, factor analysis, total least squares, latent semantic analysis, orthogonal regression, and dynamic mode decomposition. == Definition == Given structure specification S : R n p → R m × n {\displaystyle {\mathcal {S}}:\mathbb {R} ^{n_{p}}\to \mathbb {R} ^{m\times n}} , vector of structure parameters p ∈ R n p {\displaystyle p\in \mathbb {R} ^{n_{p}}} , norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} , and desired rank r {\displaystyle r} , minimize over p ^ ‖ p − p ^ ‖ subject to rank ⁡ ( S ( p ^ ) ) ≤ r . {\displaystyle {\text{minimize}}\quad {\text{over }}{\widehat {p}}\quad \|p-{\widehat {p}}\|\quad {\text{subject to}}\quad \operatorname {rank} {\big (}{\mathcal {S}}({\widehat {p}}){\big )}\leq r.} == Applications == Linear system identification, in which case the approximating matrix is Hankel structured. Machine learning, in which case the approximating matrix is nonlinearly structured. Recommender systems, in which cases the data matrix has missing values and the approximation is categorical. Distance matrix completion, in which case there is a positive definiteness constraint. Natural language processing, in which case the approximation is nonnegative. Computer algebra, in which case the approximation is Sylvester structured. Matrix product states, in which case the approximation is usually rescaled to have fixed Frobenius norm. == Basic low-rank approximation problem == The unstructured problem with fit measured by the Frobenius norm, i.e., minimize over D ^ ‖ D − D ^ ‖ F subject to rank ⁡ ( D ^ ) ≤ r {\displaystyle {\text{minimize}}\quad {\text{over }}{\widehat {D}}\quad \|D-{\widehat {D}}\|_{\text{F}}\quad {\text{subject to}}\quad \operatorname {rank} {\big (}{\widehat {D}}{\big )}\leq r} has an analytic solution in terms of the singular value decomposition of the data matrix. The result is referred to as the matrix approximation lemma or Eckart–Young–Mirsky theorem. This problem was originally solved by Erhard Schmidt in the infinite dimensional context of integral operators (although his methods easily generalize to arbitrary compact operators on Hilbert spaces) and later rediscovered by C. Eckart and G. Young. L. Mirsky generalized the result to arbitrary unitarily invariant norms. Let D = U Σ V ⊤ ∈ R m × n , m ≥ n {\displaystyle D=U\Sigma V^{\top }\in \mathbb {R} ^{m\times n},\quad m\geq n} be the singular value decomposition of D {\displaystyle D} , where Σ =: diag ⁡ ( σ 1 , … , σ r ) {\displaystyle \Sigma =:\operatorname {diag} (\sigma _{1},\ldots ,\sigma _{r})} , where r ≤ min { m , n } = n {\displaystyle r\leq \min\{m,n\}=n} , is the m × n {\displaystyle m\times n} rectangular diagonal matrix with r {\displaystyle r} non-zero singular values σ 1 ≥ … ≥ σ r > σ r + 1 = … = σ n = 0 {\displaystyle \sigma _{1}\geq \ldots \geq \sigma _{r}>\sigma _{r+1}=\ldots =\sigma _{n}=0} . For a given k ∈ { 1 , … , r } {\displaystyle k\in \{1,\dots ,r\}} , partition U {\displaystyle U} , Σ {\displaystyle \Sigma } , and V {\displaystyle V} as follows: U =: [ U 1 U 2 ] , Σ =: [ Σ 1 0 0 Σ 2 ] , and V =: [ V 1 V 2 ] , {\displaystyle U=:{\begin{bmatrix}U_{1}&U_{2}\end{bmatrix}},\quad \Sigma =:{\begin{bmatrix}\Sigma _{1}&0\\0&\Sigma _{2}\end{bmatrix}},\quad {\text{and}}\quad V=:{\begin{bmatrix}V_{1}&V_{2}\end{bmatrix}},} where U 1 {\displaystyle U_{1}} is m × k {\displaystyle m\times k} , Σ 1 {\displaystyle \Sigma _{1}} is k × k {\displaystyle k\times k} , and V 1 {\displaystyle V_{1}} is n × k {\displaystyle n\times k} . Then the rank k {\displaystyle k} matrix D ^ ∗ := U 1 Σ 1 V 1 ⊤ , {\displaystyle {\widehat {D}}^{}:=U_{1}\Sigma _{1}V_{1}^{\top },} obtained from the truncated singular value decomposition is such that ‖ D − D ^ ∗ ‖ F = min rank ⁡ ( D ^ ) ≤ k ‖ D − D ^ ‖ F = σ k + 1 2 + ⋯ + σ r 2 . {\displaystyle \|D-{\widehat {D}}^{}\|_{\text{F}}=\min _{\operatorname {rank} ({\widehat {D}})\leq k}\|D-{\widehat {D}}\|_{\text{F}}={\sqrt {\sigma _{k+1}^{2}+\cdots +\sigma _{r}^{2}}}.} The minimizer D ^ ∗ {\displaystyle {\widehat {D}}^{}} is unique if and only if σ k > σ k + 1 {\displaystyle \sigma _{k}>\sigma _{k+1}} . == Proof of Eckart–Young–Mirsky theorem (for spectral norm) == Let A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} be a real (possibly rectangular) matrix with m ≤ n {\displaystyle m\leq n} . Suppose that A = U Σ V ⊤ {\displaystyle A=U\Sigma V^{\top }} is the singular value decomposition of A {\displaystyle A} . Recall that U {\displaystyle U} and V {\displaystyle V} are orthogonal matrices, and Σ {\displaystyle \Sigma } is an m × n {\displaystyle m\times n} diagonal matrix with entries ( σ 1 , σ 2 , ⋯ , σ m ) {\displaystyle (\sigma _{1},\sigma _{2},\cdots ,\sigma _{m})} such that σ 1 ≥ σ 2 ≥ ⋯ ≥ σ m ≥ 0 {\displaystyle \sigma _{1}\geq \sigma _{2}\geq \cdots \geq \sigma _{m}\geq 0} . We claim that the best rank- k {\displaystyle k} approximation to A {\displaystyle A} in the spectral norm, denoted by ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} , is given by A k := ∑ i = 1 k σ i u i v i ⊤ {\displaystyle A_{k}:=\sum _{i=1}^{k}\sigma _{i}u_{i}v_{i}^{\top }} where u i {\displaystyle u_{i}} and v i {\displaystyle v_{i}} denote the i {\displaystyle i} th column of U {\displaystyle U} and V {\displaystyle V} , respectively. First, note that we have ‖ A − A k ‖ 2 = ‖ ∑ i = 1 n σ i u i v i ⊤ − ∑ i = 1 k σ i u i v i ⊤ ‖ 2 = ‖ ∑ i = k + 1 n σ i u i v i ⊤ ‖ 2 = σ k + 1 {\displaystyle \|A-A_{k}\|_{2}=\left\|\sum _{i=1}^{\color {red}{n}}\sigma _{i}u_{i}v_{i}^{\top }-\sum _{i=1}^{\color {red}{k}}\sigma _{i}u_{i}v_{i}^{\top }\right\|_{2}=\left\|\sum _{i=\color {red}{k+1}}^{n}\sigma _{i}u_{i}v_{i}^{\top }\right\|_{2}=\sigma _{k+1}} Therefore, we need to show that if B k = X Y ⊤ {\displaystyle B_{k}=XY^{\top }} where X {\displaystyle X} and Y {\displaystyle Y} have k {\displaystyle k} columns then ‖ A − A k ‖ 2 = σ k + 1 ≤ ‖ A − B k ‖ 2 {\displaystyle \|A-A_{k}\|_{2}=\sigma _{k+1}\leq \|A-B_{k}\|_{2}} . Since Y {\displaystyle Y} has k {\displaystyle k} columns, then there must be a nontrivial linear combination of the first k + 1 {\displaystyle k+1} columns of V {\displaystyle V} , i.e., w = γ 1 v 1 + ⋯ + γ k + 1 v k + 1 , {\displaystyle w=\gamma _{1}v_{1}+\cdots +\gamma _{k+1}v_{k+1},} such that Y ⊤ w = 0 {\displaystyle Y^{\top }w=0} . Without loss of generality, we can scale w {\displaystyle w} so that ‖ w ‖ 2 = 1 {\displaystyle \|w\|_{2}=1} or (equivalently) γ 1 2 + ⋯ + γ k + 1 2 = 1 {\displaystyle \gamma _{1}^{2}+\cdots +\gamma _{k+1}^{2}=1} . Therefore, ‖ A − B k ‖ 2 2 ≥ ‖ ( A − B k ) w ‖ 2 2 = ‖ A w ‖ 2 2 = γ 1 2 σ 1 2 + ⋯ + γ k + 1 2 σ k + 1 2 ≥ σ k + 1 2 . {\displaystyle \|A-B_{k}\|_{2}^{2}\geq \|(A-B_{k})w\|_{2}^{2}=\|Aw\|_{2}^{2}=\gamma _{1}^{2}\sigma _{1}^{2}+\cdots +\gamma _{k+1}^{2}\sigma _{k+1}^{2}\geq \sigma _{k+1}^{2}.} The result follows by taking the square root of both sides of the above inequality. == Proof of Eckart–Young–Mirsky theorem (for Frobenius norm) == Let A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} be a real (possibly rectangular) matrix with m ≤ n {\displaystyle m\leq n} . Suppose that A = U Σ V ⊤ {\displaystyle A=U\Sigma V^{\top }} is the singular value decomposition of A {\displaystyle A} . We claim that the best rank k {\displaystyle k} approximation to A {\displaystyle A} in the Frobenius norm, denoted by ‖ ⋅ ‖ F {\displaystyle \|\cdot \|_{F}} , is given by A k = ∑ i = 1 k σ i u i v i ⊤ {\displaystyle A_{k}=\sum _{i=1}^{k}\sigma _{i}u_{i}v_{i}^{\top }} where u i {\displaystyle u_{i}} and v i {\displaystyle v_{i}} denote the i {\displaystyle i} th column of U {\displaystyle U} and V {\displaystyle V} , respectively. First, note that we have ‖ A − A k ‖ F 2 = ‖ ∑ i = k + 1 n σ i u i v i ⊤ ‖ F 2 = ∑ i = k + 1 n σ i 2 {\displaystyle \|A-A_{k}\|_{F}^{2}=\left\|\sum _{i=k+1}^{n}\sigma _{i}u_{i}v_{i}^{\top }\right\|_{F}^{2}=\sum _{i=k+1}^{n}\sigma _{i}^{2}} Therefore, we need to show that if B k = X Y ⊤ {\displaystyle B_{k}=XY^{\top }} where X {\displaystyle X} and Y {\displaystyle Y} have k {\displaystyle k} columns then ‖ A − A k ‖ F 2 = ∑ i = k + 1 n σ i 2 ≤ ‖ A − B k ‖ F 2 . {\displaystyle \|A-A_{k}\|_{F}^{2}=\sum _{i=k+1}^{n}\sigma _{i}^{2}\leq \|A-B_{k}\|_{F}^{2}.} By the triangle inequality with the spectral norm

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  GNU Octave

  GNU Octave is a scientific programming language for scientific computing and numerical computation. Among other things, Octave can be used to solve linear and nonlinear problems numerically and to perform other numerical experiments using a language that is mostly compatible with MATLAB. It may also be used as a batch-oriented language. As part of the GNU Project, it is free software under the terms of the GNU General Public License. == History == The project was conceived around 1988. At first it was intended to be a companion to a chemical reactor design course. Full development was started by John W. Eaton in 1992. The first alpha release dates back to 4 January 1993 and on 17 February 1994 version 1.0 was released. Version 9.2.0 was released on 7 June 2024. The program is named after Octave Levenspiel, a former professor of the principal author. Levenspiel was known for his ability to perform quick back-of-the-envelope calculations. == Development history == == Developments == In addition to use on desktops for personal scientific computing, Octave is used in academia and industry. For example, Octave was used on a massive parallel computer at Pittsburgh Supercomputing Center to find vulnerabilities related to guessing social security numbers. Acceleration with OpenCL or CUDA is also possible with use of GPUs. == Technical details == Octave is written in C++ using the C++ standard library. Octave uses an interpreter to execute the Octave scripting language. Octave is extensible using dynamically loadable modules. Octave interpreter has an OpenGL-based graphics engine to create plots, graphs and charts and to save or print them. Alternatively, gnuplot can be used for the same purpose. Octave includes a graphical user interface (GUI) in addition to the traditional command-line interface (CLI); see #User interfaces for details. == Octave, the language == The Octave language is an interpreted programming language. It is a structured programming language (similar to C) and supports many common C standard library functions, and also certain UNIX system calls and functions. However, it does not support passing arguments by reference although function arguments are copy-on-write to avoid unnecessary duplication. Octave programs consist of a list of function calls or a script. The syntax is matrix-based and provides various functions for matrix operations. It supports various data structures and allows object-oriented programming. Its syntax is very similar to MATLAB, and careful programming of a script will allow it to run on both Octave and MATLAB. Because Octave is made available under the GNU General Public License, it may be freely changed, copied and used. The program runs on Microsoft Windows and most Unix and Unix-like operating systems, including Linux, Android, and macOS. == Notable features == === Command and variable name completion === Typing a TAB character on the command line causes Octave to attempt to complete variable, function, and file names (similar to Bash's tab completion). Octave uses the text before the cursor as the initial portion of the name to complete. === Command history === When running interactively, Octave saves the commands typed in an internal buffer so that they can be recalled and edited. === Data structures === Octave includes a limited amount of support for organizing data in structures. In this example, we see a structure x with elements a, b, and c, (an integer, an array, and a string, respectively): === Short-circuit Boolean operators === Octave's && and || logical operators are evaluated in a short-circuit fashion (like the corresponding operators in the C language), in contrast to the element-by-element operators & and |. === Increment and decrement operators === Octave includes the C-like increment and decrement operators ++ and -- in both their prefix and postfix forms. Octave also does augmented assignment, e.g. x += 5. === Unwind-protect === Octave supports a limited form of exception handling modelled after the unwind_protect of Lisp. The general form of an unwind_protect block looks like this: As a general rule, GNU Octave recognizes as termination of a given block either the keyword end (which is compatible with the MATLAB language) or a more specific keyword endblock or, in some cases, end_block. As a consequence, an unwind_protect block can be terminated either with the keyword end_unwind_protect as in the example, or with the more portable keyword end. The cleanup part of the block is always executed. In case an exception is raised by the body part, cleanup is executed immediately before propagating the exception outside the block unwind_protect. GNU Octave also supports another form of exception handling (compatible with the MATLAB language): This latter form differs from an unwind_protect block in two ways. First, exception_handling is only executed when an exception is raised by body. Second, after the execution of exception_handling the exception is not propagated outside the block (unless a rethrow( lasterror ) statement is explicitly inserted within the exception_handling code). === Variable-length argument lists === Octave has a mechanism for handling functions that take an unspecified number of arguments without explicit upper limit. To specify a list of zero or more arguments, use the special argument varargin as the last (or only) argument in the list. varargin is a cell array containing all the input arguments. === Variable-length return lists === A function can be set up to return any number of values by using the special return value varargout. For example: === C++ integration === It is also possible to execute Octave code directly in a C++ program. For example, here is a code snippet for calling rand([10,1]): C and C++ code can be integrated into GNU Octave by creating oct files, or using the MATLAB compatible MEX files. == MATLAB compatibility == Octave has been built with MATLAB compatibility in mind, and shares many features with MATLAB: % Script: myscript.m a = 5; b = a 2 % Function: myfunc.m function result = myfunc(x) result = x^2 + 3; end Matrices as fundamental data type. Built-in support for complex numbers. Powerful built-in math functions and extensive function libraries. Extensibility in the form of user-defined functions. Octave treats incompatibility with MATLAB as a bug; therefore, it could be considered a software clone, which does not infringe software copyright as per Lotus v. Borland court case. MATLAB scripts from the MathWorks' FileExchange repository in principle are compatible with Octave. However, while they are often provided and uploaded by users under an Octave compatible and proper open source BSD license, the FileExchange Terms of use prohibit any usage beside MathWorks' proprietary MATLAB. === Syntax compatibility === There are a few purposeful, albeit minor, syntax additions Archived 2012-04-26 at the Wayback Machine: Comment lines can be prefixed with the # character as well as the % character; Various C-based operators ++, --, +=, =, /= are supported; Elements can be referenced without creating a new variable by cascaded indexing, e.g. [1:10](3); Strings can be defined with the double-quote " character as well as the single-quote ' character; When the variable type is single (a single-precision floating-point number), Octave calculates the "mean" in the single-domain (MATLAB in double-domain) which is faster but gives less accurate results; Blocks can also be terminated with more specific Control structure keywords, i.e., endif, endfor, endwhile, etc.; Functions can be defined within scripts and at the Octave prompt; Presence of a do-until loop (similar to do-while in C). === Function compatibility === Many, but not all, of the numerous MATLAB functions are available in GNU Octave, some of them accessible through packages in Octave Forge. The functions available as part of either core Octave or Forge packages are listed online Archived 2024-03-14 at the Wayback Machine. A list of unavailable functions is included in the Octave function __unimplemented.m__. Unimplemented functions are also listed under many Octave Forge packages in the Octave Wiki. When an unimplemented function is called the following error message is shown: == User interfaces == Octave comes with an official graphical user interface (GUI) and an integrated development environment (IDE) based on Qt. It has been available since Octave 3.8, and has become the default interface (over the command-line interface) with the release of Octave 4.0. It was well-received by an EDN contributor, who wrote "[Octave] now has a very workable GUI" in reviewing the then-new GUI in 2014. Several 3rd-party graphical front-ends have also been developed, like ToolboX for coding education. == GUI applications == With Octave code, the user can create GUI applications. See GUI Development (GNU Octave (version 7.1.0)). Below are some examples: Button, edit control, checkboxTextboxListbox wit

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  Seq2seq

  Seq2seq is a family of machine learning approaches used for natural language processing. Originally developed by Lê Viết Quốc, a Vietnamese computer scientist and a machine learning pioneer at Google Brain, this framework has become foundational in many modern AI systems. Applications include language translation, image captioning, conversational models, speech recognition, and text summarization. Seq2seq uses sequence transformation: it turns one sequence into another sequence. == History == One naturally wonders if the problem of translation could conceivably be treated as a problem in cryptography. When I look at an article in Russian, I say: 'This is really written in English, but it has been coded in some strange symbols. I will now proceed to decode. seq2seq is an approach to machine translation (or more generally, sequence transduction) with roots in information theory, where communication is understood as an encode-transmit-decode process, and machine translation can be studied as a special case of communication. This viewpoint was elaborated, for example, in the noisy channel model of machine translation. In practice, seq2seq maps an input sequence into a real-numerical vector by using a neural network (the encoder), and then maps it back to an output sequence using another neural network (the decoder). The idea of encoder-decoder sequence transduction had been developed in the early 2010s. The papers most commonly cited as the originators that produced seq2seq are two papers from 2014. In the seq2seq as proposed by them, both the encoder and the decoder were LSTMs. This had the "bottleneck" problem, since the encoding vector has a fixed size, so for long input sequences, information would tend to be lost, as they are difficult to fit into the fixed-length encoding vector. The attention mechanism, proposed in 2014, resolved the bottleneck problem. They called their model RNNsearch, as it "emulates searching through a source sentence during decoding a translation". A problem with seq2seq models at this point was that recurrent neural networks are difficult to parallelize. The 2017 publication of Transformers resolved the problem by replacing the encoding RNN with self-attention Transformer blocks ("encoder blocks"), and the decoding RNN with cross-attention causally-masked Transformer blocks ("decoder blocks"). === Priority dispute === One of the papers cited as the originator for seq2seq is (Sutskever et al 2014), published at Google Brain while they were on Google's machine translation project. The research allowed Google to overhaul Google Translate into Google Neural Machine Translation in 2016. Tomáš Mikolov claims to have developed the idea (before joining Google Brain) of using a "neural language model on pairs of sentences... and then [generating] translation after seeing the first sentence"—which he equates with seq2seq machine translation, and to have mentioned the idea to Ilya Sutskever and Quoc Le (while at Google Brain), who failed to acknowledge him in their paper. Mikolov had worked on RNNLM (using RNN for language modelling) for his PhD thesis, and is more notable for developing word2vec. == Architecture == The main reference for this section is. === Encoder === The encoder is responsible for processing the input sequence and capturing its essential information, which is stored as the hidden state of the network and, in a model with attention mechanism, a context vector. The context vector is the weighted sum of the input hidden states and is generated for every time instance in the output sequences. === Decoder === The decoder takes the context vector and hidden states from the encoder and generates the final output sequence. The decoder operates in an autoregressive manner, producing one element of the output sequence at a time. At each step, it considers the previously generated elements, the context vector, and the input sequence information to make predictions for the next element in the output sequence. Specifically, in a model with attention mechanism, the context vector and the hidden state are concatenated together to form an attention hidden vector, which is used as an input for the decoder. The seq2seq method developed in the early 2010s uses two neural networks: an encoder network converts an input sentence into numerical vectors, and a decoder network converts those vectors to sentences in the target language. The Attention mechanism was grafted onto this structure in 2014 and is shown below. Later it was refined into the encoder-decoder Transformer architecture of 2017. === Training vs prediction === There is a subtle difference between training and prediction. During training time, both the input and the output sequences are known. During prediction time, only the input sequence is known, and the output sequence must be decoded by the network itself. Specifically, consider an input sequence x 1 : n {\displaystyle x_{1:n}} and output sequence y 1 : m {\displaystyle y_{1:m}} . The encoder would process the input x 1 : n {\displaystyle x_{1:n}} step by step. After that, the decoder would take the output from the encoder, as well as the as input, and produce a prediction y ^ 1 {\displaystyle {\hat {y}}_{1}} . Now, the question is: what should be input to the decoder in the next step? A standard method for training is "teacher forcing". In teacher forcing, no matter what is output by the decoder, the next input to the decoder is always the reference. That is, even if y ^ 1 ≠ y 1 {\displaystyle {\hat {y}}_{1}\neq y_{1}} , the next input to the decoder is still y 1 {\displaystyle y_{1}} , and so on. During prediction time, the "teacher" y 1 : m {\displaystyle y_{1:m}} would be unavailable. Therefore, the input to the decoder must be y ^ 1 {\displaystyle {\hat {y}}_{1}} , then y ^ 2 {\displaystyle {\hat {y}}_{2}} , and so on. It is found that if a model is trained purely by teacher forcing, its performance would degrade during prediction time, since generation based on the model's own output is different from generation based on the teacher's output. This is called exposure bias or a train/test distribution shift. A 2015 paper recommends that, during training, randomly switch between teacher forcing and no teacher forcing. === Attention for seq2seq === The attention mechanism is an enhancement introduced by Bahdanau et al. in 2014 to address limitations in the basic Seq2Seq architecture where a longer input sequence results in the hidden state output of the encoder becoming irrelevant for the decoder. It enables the model to selectively focus on different parts of the input sequence during the decoding process. At each decoder step, an alignment model calculates the attention score using the current decoder state and all of the attention hidden vectors as input. An alignment model is another neural network model that is trained jointly with the seq2seq model used to calculate how well an input, represented by the hidden state, matches with the previous output, represented by attention hidden state. A softmax function is then applied to the attention score to get the attention weight. In some models, the encoder states are directly fed into an activation function, removing the need for alignment model. An activation function receives one decoder state and one encoder state and returns a scalar value of their relevance. Consider the seq2seq language English-to-French translation task. To be concrete, let us consider the translation of "the zone of international control ", which should translate to "la zone de contrôle international ". Here, we use the special token as a control character to delimit the end of input for both the encoder and the decoder. An input sequence of text x 0 , x 1 , … {\displaystyle x_{0},x_{1},\dots } is processed by a neural network (which can be an LSTM, a Transformer encoder, or some other network) into a sequence of real-valued vectors h 0 , h 1 , … {\displaystyle h_{0},h_{1},\dots } , where h {\displaystyle h} stands for "hidden vector". After the encoder has finished processing, the decoder starts operating over the hidden vectors, to produce an output sequence y 0 , y 1 , … {\displaystyle y_{0},y_{1},\dots } , autoregressively. That is, it always takes as input both the hidden vectors produced by the encoder, and what the decoder itself has produced before, to produce the next output word: ( h 0 , h 1 , … {\displaystyle h_{0},h_{1},\dots } , "") → "la" ( h 0 , h 1 , … {\displaystyle h_{0},h_{1},\dots } , " la") → "la zone" ( h 0 , h 1 , … {\displaystyle h_{0},h_{1},\dots } , " la zone") → "la zone de" ... ( h 0 , h 1 , … {\displaystyle h_{0},h_{1},\dots } , " la zone de contrôle international") → "la zone de contrôle international " Here, we use the special token as a control character to delimit the start of input for the decoder. The decoding terminates as soon as "" appears in the decoder output. ==

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  Memtransistor

  The memtransistor (a blend word from Memory Transfer Resistor) is an experimental multi-terminal passive electronic component that might be used in the construction of artificial neural networks. It is a combination of the memristor and transistor technology. This technology is different from the 1T-1R approach since the devices are merged into one single entity. Multiple memristors can be embedded with a single transistor, enabling it to more accurately model a neuron with its multiple synaptic connections. A neural network produced from these would provide hardware-based artificial intelligence with a good foundation. == Applications == These types of devices would allow for a synapse model that could realise a learning rule, by which the synaptic efficacy is altered by voltages applied to the terminals of the device. An example of such a learning rule is spike-timing-dependant-plasticty by which the weight of the synapse, in this case the conductivity, could be modulated based on the timing of pre and post synaptic spikes arriving at each terminal. The advantage of this approach over two terminal memristive devices is that read and write protocols have the possibility to occur simultaneously and distinctly.

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  Joseph Nechvatal

  Joseph Nechvatal (born January 15, 1951) is an American post-conceptual digital artist and art theoretician who creates computer-assisted paintings and computer animations, often using custom computer viruses. == Life and work == Joseph Nechvatal was born in Chicago. He studied fine art and philosophy at Southern Illinois University Carbondale, Cornell University, and Columbia University. He earned a Doctor of Philosophy in Philosophy of Art and Technology at the Planetary Collegium at University of Wales, Newport and has taught art theory and art history at the School of Visual Arts. He has had many solo exhibitions and is one of five artists that art historian Patrick Frank examines in his 2024 book Art of the 1980s: As If the Digital Mattered. His work in the late 1970s and early 1980s chiefly consisted of postminimal gray palimpsest-like drawings that were often photo-mechanically enlarged. Beginning in 1979 he became associated with the artist group Colab, organized the Public Arts International/Free Speech series, and helped established the non-profit group ABC No Rio. In 1983 he co-founded the avant-garde electronic art music audio project Tellus Audio Cassette Magazine. In 1984, Nechvatal began work on an opera called XS: The Opera Opus (1984-6) with the no wave musical composer Rhys Chatham. He began using computers and robotics to make post-conceptual paintings in 1986 and later, in his signature work, began to employ self-created computer viruses. From 1991 to 1993, he was artist-in-residence at the Louis Pasteur Atelier in Arbois, France and at the Saline Royale/Ledoux Foundation's computer lab. There he worked on The Computer Virus Project, his first artistic experiment with computer viruses and computer virus animation. He exhibited computer-robotic paintings at Documenta 8 in 1987. In 2002 he extended his experimentation into viral artificial life through a collaboration with the programmer Stephane Sikora of music2eye in a work called the Computer Virus Project II. Nechvatal has also created a noise music work called viral symphOny, a collaborative sound symphony created by using his computer virus software at the Institute for Electronic Arts at Alfred University. In 2021 Pentiments released Nechvatal's retrospective audio cassette called Selected Sound Works (1981-2021) and in 2022 his The Viral Tempest, a double vinyl LP of new audio work. In 2025, he joined the roster of artists/musicians at Table of the Elements with two CD/book releases: Selected Sound Works (1981-2021) and The Marriage of Orlando and Artaud, Even. From 1999 to 2013, Nechvatal taught art theories of immersive virtual reality and the viractual at the School of Visual Arts in New York City (SVA). A book of his collected essays entitled Towards an Immersive Intelligence: Essays on the Work of Art in the Age of Computer Technology and Virtual Reality (1993–2006) was published by Edgewise Press in 2009. Also in 2009, his virtual reality art theory and art history book Immersive Ideals / Critical Distances was published. In 2011, his book Immersion Into Noise was published by Open Humanities Press in conjunction with the University of Michigan Library's Scholarly Publishing Office. Nechvatal has also published three books with Punctum Books: Minóy (noise music—ed.—2014), Destroyer of Naivetés (poetry—2015), and Styling Sagaciousness (poetry—2022). In 2023 his art theory cybersex farce novella venus©~Ñ~vibrator, even was published by Orbis Tertius Press The Joseph Nechvatal archive is housed at The Fales Library Downtown Collection at the NYU Special Collections Library in New York City. === Viractualism === Viractualism is an art theory concept developed by Nechvatal in 1999 from Ph.D. research Nechvatal conducted at the Planetary Collegium at University of Wales, Newport. There he developed his concept of the viractual, which strives to create an interface between the actual and the virtual.

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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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  Manifold hypothesis

  The manifold hypothesis posits that many high-dimensional data sets that occur in the real world actually lie along low-dimensional latent manifolds inside that high-dimensional space. As a consequence of the manifold hypothesis, many data sets that appear to initially require many variables to describe, can actually be described by a comparatively small number of variables, linked to the local coordinate system of the underlying manifold. It is suggested that this principle underpins the effectiveness of machine learning algorithms in describing high-dimensional data sets by considering a few common features. The manifold hypothesis is related to the effectiveness of nonlinear dimensionality reduction techniques in machine learning. Many techniques of dimensional reduction make the assumption that data lies along a low-dimensional submanifold, such as manifold sculpting, manifold alignment, and manifold regularization. The major implications of this hypothesis is that Machine learning models only have to fit relatively simple, low-dimensional, highly structured subspaces within their potential input space (latent manifolds). Within one of these manifolds, it's always possible to interpolate between two inputs, that is to say, morph one into another via a continuous path along which all points fall on the manifold. The ability to interpolate between samples is the key to generalization in deep learning. == The information geometry of statistical manifolds == An empirically-motivated approach to the manifold hypothesis focuses on its correspondence with an effective theory for manifold learning under the assumption that robust machine learning requires encoding the dataset of interest using methods for data compression. This perspective gradually emerged using the tools of information geometry thanks to the coordinated effort of scientists working on the efficient coding hypothesis, predictive coding and variational Bayesian methods. The argument for reasoning about the information geometry on the latent space of distributions rests upon the existence and uniqueness of the Fisher information metric. In this general setting, we are trying to find a stochastic embedding of a statistical manifold. From the perspective of dynamical systems, in the big data regime this manifold generally exhibits certain properties such as homeostasis: We can sample large amounts of data from the underlying generative process. Machine Learning experiments are reproducible, so the statistics of the generating process exhibit stationarity. In a sense made precise by theoretical neuroscientists working on the free energy principle, the statistical manifold in question possesses a Markov blanket.

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  Sum of absolute differences

  In digital image processing, the sum of absolute differences (SAD) is a measure of the similarity between image blocks. It is calculated by taking the absolute difference between each pixel in the original block and the corresponding pixel in the block being used for comparison. These differences are summed to create a simple metric of block similarity, the L1 norm of the difference image or Manhattan distance between two image blocks. The sum of absolute differences may be used for a variety of purposes, such as object recognition, the generation of disparity maps for stereo images, and motion estimation for video compression. == Example == This example uses the sum of absolute differences to identify which part of a search image is most similar to a template image. In this example, the template image is 3 by 3 pixels in size, while the search image is 3 by 5 pixels in size. Each pixel is represented by a single integer from 0 to 9. Template Search image 2 5 5 2 7 5 8 6 4 0 7 1 7 4 2 7 7 5 9 8 4 6 8 5 There are exactly three unique locations within the search image where the template may fit: the left side of the image, the center of the image, and the right side of the image. To calculate the SAD values, the absolute value of the difference between each corresponding pair of pixels is used: the difference between 2 and 2 is 0, 4 and 1 is 3, 7 and 8 is 1, and so forth. Calculating the values of the absolute differences for each pixel, for the three possible template locations, gives the following: Left Center Right 0 2 0 5 0 3 3 3 1 3 7 3 3 4 5 0 2 0 1 1 3 3 1 1 1 3 4 For each of these three image patches, the 9 absolute differences are added together, giving SAD values of 20, 25, and 17, respectively. From these SAD values, it could be asserted that the right side of the search image is the most similar to the template image, because it has the lowest sum of absolute differences as compared to the other two locations. == Comparison to other metrics == === Object recognition === The sum of absolute differences provides a simple way to automate the searching for objects inside an image, but may be unreliable due to the effects of contextual factors such as changes in lighting, color, viewing direction, size, or shape. The SAD may be used in conjunction with other object recognition methods, such as edge detection, to improve the reliability of results. === Video compression === SAD is an extremely fast metric due to its simplicity; it is effectively the simplest possible metric that takes into account every pixel in a block. Therefore, it is very effective for a wide motion search of many different blocks. SAD is also easily parallelizable since it analyzes each pixel separately, making it easily implementable with such instructions as ARM NEON or x86 SSE2. For example, SSE has packed sum of absolute differences instruction (PSADBW) specifically for this purpose. Once candidate blocks are found, the final refinement of the motion estimation process is often done with other slower but more accurate metrics, which better take into account human perception. These include the sum of absolute transformed differences (SATD), the sum of squared differences (SSD), and rate–distortion optimization.

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  Natarajan dimension

  In the theory of Probably Approximately Correct Machine Learning, the Natarajan dimension characterizes the complexity of learning a set of functions, generalizing from the Vapnik–Chervonenkis dimension for boolean functions to multi-class functions. Originally introduced as the Generalized Dimension by Natarajan, it was subsequently renamed the Natarajan Dimension by Haussler and Long. == Definition == Let H {\displaystyle H} be a set of functions from a set X {\displaystyle X} to a set Y {\displaystyle Y} . H {\displaystyle H} shatters a set C ⊂ X {\displaystyle C\subset X} if there exist two functions f 0 , f 1 ∈ H {\displaystyle f_{0},f_{1}\in H} such that For every x ∈ C , f 0 ( x ) ≠ f 1 ( x ) {\displaystyle x\in C,f_{0}(x)\neq f_{1}(x)} . For every B ⊂ C {\displaystyle B\subset C} , there exists a function h ∈ H {\displaystyle h\in H} such that for all x ∈ B , h ( x ) = f 0 ( x ) {\displaystyle x\in B,h(x)=f_{0}(x)} and for all x ∈ C − B , h ( x ) = f 1 ( x ) {\displaystyle x\in C-B,h(x)=f_{1}(x)} . The Natarajan dimension of H is the maximal cardinality of a set shattered by H {\displaystyle H} . It is easy to see that if | Y | = 2 {\displaystyle |Y|=2} , the Natarajan dimension collapses to the Vapnik–Chervonenkis dimension. Shalev-Shwartz and Ben-David present comprehensive material on multi-class learning and the Natarajan dimension, including uniform convergence and learnability. Recently, Cohen et al showed that the Natarajan dimension is the dominant term governing agnostic multi-class PAC learnability.

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