AI App That Can Edit Photos

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  XLeratorDB

  XLeratorDB is a suite of database function libraries that enable Microsoft SQL Server to perform a wide range of additional (non-native) business intelligence and ad hoc analytics. The libraries, which are embedded and run centrally on the database, include more than 450 individual functions similar to those found in Microsoft Excel spreadsheets. The individual functions are grouped and sold as six separate libraries based on usage: finance, statistics, math, engineering, unit conversions and strings. WestClinTech, the company that developed XLeratorDB, claims it is "the first commercial function package add-in for Microsoft SQL Server." == Company history == WestClinTech (LLC), founded by software industry veterans Charles Flock and Joe Stampf in 2008, is located in Irvington, New York, United States. Flock was a co-founder of The Frustum Group, developer of the OPICS enterprise banking and trading platform, which was acquired by London-based Misys, PLC in 1996. Stampf joined Frustum in 1994 and with Flock remained active with the company after acquisition, helping to develop successive generations of OPICS now employed by over 150 leading financial institutions worldwide. Following a full year of research, development and testing, WestClinTech introduced and recorded its first commercial sale of XLeratorDB in April 2009. In September 2009, XLeratorDB became available to all Federal agencies through NASA's Strategic Enterprise-Wide Procurement (SEWP-IV) program, a government-wide acquisition contract. == Technology == XLeratorDB uses Microsoft SQL CLR(Common Language Runtime) technology. SQL CLR allows managed code to be hosted by, and run in, the Microsoft SQL Server environment. SQL CLR relies on the creation, deployment and registration of .NET Framework assemblies that are physically stored in managed code dynamic-link libraries (DLL). The assemblies may contain .NET namespaces, classes, functions, and properties. Because managed code compiles to native code prior to execution, functions using SQL CLR can achieve significant performance increases versus the equivalent functions written in T-SQL in some scenarios. XLeratorDB requires Microsoft SQL Server 2005 or SQL Server 2005 Express editions, or later (compatibility mode 90 or higher). The product installs with PERMISSION_SET=SAFE. SAFE mode, the most restrictive permission set, is accessible by all users. Code executed by an assembly with SAFE permissions cannot access external system resources such as files, the network, the internet, environment variables, or the registry. == Functions == In computer science, a function is a portion of code within a larger program which performs a specific task and is relatively independent of the remaining code. As used in database and spreadsheet applications these functions generally represent mathematical formulas widely used across a variety of fields. While this code may be user-generated, it is also embedded as a pre-written sub-routine in applications. These functions are typically identified by common nomenclature which corresponds to their underlying operations: e.g. IRR identifies the function which calculates Internal Rate of Return on a series of periodic cash flows. === Function uses === As subroutines, functions can be integrated and used in a variety of ways, and as part of larger, more complicated applications. Within large enterprise applications they may, for example, play an important role in defining business rules or risk management parameters, while remaining virtually invisible to end users. Within database management systems and spreadsheets, however, these kinds of functions also represent discrete sets of tools; they can be accessed directly and utilized on a stand-alone basis, or in more complex, user-defined configurations. In this context, functions can be used for business intelligence and ad hoc analysis of data in fields such as finance, statistics, engineering, math, etc. === Function types === XLeratorDB uses three kinds of functions to perform analytic operations: scalar, aggregate, and a hybrid form which WestClinTech calls Range Queries. Scalar functions take a single value, perform an operation and return a single value. An example of this type of function is LOG, which returns the logarithm of a number to a specified base. Aggregate functions operate on a series of values but return a single, summarizing value. An example of this type of function is AVG, which returns the average of values in a specified group. In XLeratorDB there are some functions which have characteristics of aggregate functions (operating on multiple series of values) but cannot be processed in SQL CLR using single column inputs, such as AVG does. For example, irregular internal rate of return (XIRR), a financial function, operates on a collection of cash flow values from one column, but must also apply variable period lengths from another column and an initial iterative assumption from a third, in order to return a single, summarizing value. WestClinTech documentation notes that Range Queries specify the data to be included in the result set of the function independently of the WHERE clause associated with the T-SQL statement, by incorporating a SELECT statement into the function as a string argument; the function then traps that SELECT statement, executes it internally and processes the result. Some XLeratorDB functions that employ Range Queries are: NPV, XNPV, IRR, XIRR, MIRR, MULTINOMIAL, and SERIESSUM. Within the application these functions are identified by a "_q" naming convention: e.g. NPV_q, IRR_q, etc. == Analytic functions == === SQL Server functions === Microsoft SQL Server is the #3 selling database management system (DBMS), behind Oracle and IBM. (While versions of SQL Server have been on the market since 1987, XLeratorDB is compatible with only the 2005 edition and later.) Like all major DBMS, SQL Server performs a variety of data mining operations by returning or arraying data in different views (also known as drill-down). In addition, SQL Server uses Transact-SQL (T-SQL) to execute four major classes of pre-defined functions in native mode. Functions operating on the DBMS offer several advantages over client layer applications like Excel: they utilize the most up-to-date data available; they can process far larger quantities of data; and, the data is not subject to exporting and transcription errors. SQL Server 2008 includes a total of 58 functions that perform relatively basic aggregation (12), math (23) and string manipulation (23) operations useful for analytics; it includes no native functions that perform more complex operations directly related to finance, statistics or engineering. === Excel functions === Microsoft Excel, a component of Microsoft Office suite, is one of the most widely used spreadsheet applications on the market today. In addition to its inherent utility as a stand-alone desktop application, Excel overlaps and complements the functionality of DBMS in several ways: storing and arraying data in rows and columns; performing certain basic tasks such as pivot table and aggregating values; and facilitating sharing, importing and exporting of database data. Excel's chief limitation relative to a true database is capacity; Excel 2003 is limited to some 65k rows and 256 columns; Excel 2007 extends this capacity to roughly 1million rows and 16k columns. By comparison, SQL Server is able to manage over 500k terabytes of memory. Excel offers, however, an extensive library of specialized pre-written functions which are useful for performing ad hoc analysis on database data. Excel 2007 includes over 300 of these pre-defined functions, although customized functions can also be created by users, or imported from third party developers as add-ons. Excel functions are grouped by type: === Excel business intelligence functions === Operating on the client computing layer Excel plays an important role as a business intelligence tool because it: performs a wide array of complex analytic functions not native to most DBMS software offers far greater ad hoc reporting and analytic flexibility than most enterprise software provides a medium for sharing and collaborating because of its ubiquity throughout the enterprise Microsoft reinforces this positioning with Business Intelligence documentation that positions Excel in a clearly pivotal role. === XLeratorDB vs. Excel functions === While operating within the database environment, XLeratorDB functions utilize the same naming conventions and input formats, and in most cases, return the same calculation results as Excel functions. XLeratorDB, coupled with SQL Server's native capabilities, compares to Excel's function sets as follows:

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  Word2vec

  Word2vec is a technique in natural language processing for obtaining vector representations of words. These vectors capture information about the meaning of the word based on the surrounding words. The word2vec algorithm estimates these representations by modeling text in a large corpus. Once trained, such a model can detect synonymous words or suggest additional words for a partial sentence. Word2vec was developed by Tomáš Mikolov, Kai Chen, Greg Corrado, Ilya Sutskever and Jeff Dean at Google, and published in 2013. Word2vec represents a word as a high-dimension vector of numbers which capture relationships between words. In particular, words which appear in similar contexts are mapped to vectors which are nearby as measured by cosine similarity. This indicates the level of semantic similarity between the words, so for example the vectors for walk and ran are nearby, as are those for "but" and "however", and "Berlin" and "Germany". == Approach == Word2vec is a group of related models that are used to produce word embeddings. These models are shallow, two-layer neural networks that are trained to reconstruct linguistic contexts of words. Word2vec takes as its input a large corpus of text and produces a mapping of the set of words to a vector space, typically of several hundred dimensions, with each unique word in the corpus being assigned a vector in the space. Word2vec can use either of two model architectures to produce these distributed representations of words: continuous bag of words (CBOW) or continuously sliding skip-gram. In both architectures, word2vec considers both individual words and a sliding context window as it iterates over the corpus. The CBOW can be viewed as a 'fill in the blank' task, where the word embedding represents the way the word influences the relative probabilities of other words in the context window. Words which are semantically similar should influence these probabilities in similar ways, because semantically similar words should be used in similar contexts. The order of context words does not influence prediction (bag of words assumption). In the continuous skip-gram architecture, the model uses the current word to predict the surrounding window of context words. The skip-gram architecture weighs nearby context words more heavily than more distant context words. According to the authors' note, CBOW is faster while skip-gram does a better job for infrequent words. After the model is trained, the learned word embeddings are positioned in the vector space such that words that share common contexts in the corpus — that is, words that are semantically and syntactically similar — are located close to one another in the space. More dissimilar words are located farther from one another in the space. == Mathematical details == This section is based on expositions. A corpus is a sequence of words. Both CBOW and skip-gram are methods to learn one vector per word appearing in the corpus. Let V {\displaystyle V} ("vocabulary") be the set of all words appearing in the corpus C {\displaystyle C} . Our goal is to learn one vector v w ∈ R d {\displaystyle v_{w}\in \mathbb {R} ^{d}} for each word w ∈ V {\displaystyle w\in V} . The idea of skip-gram is that the vector of a word should be close to the vector of each of its neighbors. The idea of CBOW is that the vector-sum of a word's neighbors should be close to the vector of the word. === Continuous bag-of-words (CBOW) === The idea of CBOW is to represent each word with a vector, such that it is possible to predict a word using the sum of the vectors of its neighbors. Specifically, for each word w i {\displaystyle w_{i}} in the corpus, the one-hot encoding of the word is used as the input to the neural network. The output of the neural network is a probability distribution over the dictionary, representing a prediction of individual words in the neighborhood of w i {\displaystyle w_{i}} . The objective of training is to maximize ∑ i ln ⁡ Pr ( w i ∣ w i + j : j ∈ N ) {\displaystyle \sum _{i}\ln \Pr(w_{i}\mid w_{i+j}\colon j\in N)} where N {\displaystyle N} is a set of (non-zero) indices representing the relative locations of nearby words considered to be in w i {\displaystyle w_{i}} 's neighborhood. For example, if we want each word in the corpus to be predicted by every other word in a small span of 4 words. The set of relative indexes of neighbor words will be: N = { − 2 , − 1 , + 1 , + 2 } {\displaystyle N=\{-2,-1,+1,+2\}} , and the objective is to maximize ∑ i ln ⁡ Pr ( w i ∣ w i − 2 , w i − 1 , w i + 1 , w i + 2 ) {\displaystyle \sum _{i}\ln \Pr(w_{i}\mid w_{i-2},w_{i-1},w_{i+1},w_{i+2})} . In standard bag-of-words, a word's context is represented by a word-count (aka a word histogram) of its neighboring words. For example, the "sat" in "the cat sat on the mat" is represented as {"the": 2, "cat": 1, "on": 1}. Note that the last word "mat" is not used to represent "sat", because it is outside the neighborhood N = { − 2 , − 1 , + 1 , + 2 } {\displaystyle N=\{-2,-1,+1,+2\}} . In continuous bag-of-words, the histogram is multiplied by a matrix V {\displaystyle V} to obtain a continuous representation of the word's context. The matrix V {\displaystyle V} is also called a dictionary. Its columns are the word vectors. It has D {\displaystyle D} columns, where D {\displaystyle D} is the size of the dictionary. Let d {\displaystyle d} be the length of each word vector. We have V ∈ R d × D {\displaystyle V\in \mathbb {R} ^{d\times D}} . For example, multiplying the word histogram {"the": 2, "cat": 1, "on": 1} with V {\displaystyle V} , we obtain 2 v the + v cat + v on {\displaystyle 2v_{\text{the}}+v_{\text{cat}}+v_{\text{on}}} . This is then multiplied with another matrix V ′ {\displaystyle V'} of shape R D × d {\displaystyle \mathbb {R} ^{D\times d}} . Each row of it is a word vector v ′ {\displaystyle v'} . This results in a vector of length D {\displaystyle D} , one entry per dictionary entry. Then, apply the softmax to obtain a probability distribution over the dictionary. This system can be visualized as a neural network, similar in spirit to an autoencoder, of architecture linear-linear-softmax, as depicted in the diagram. The system is trained by gradient descent to minimize the cross-entropy loss. In full formula, the cross-entropy loss is: − ∑ i ln ⁡ e v w i ′ ⋅ ( ∑ j ∈ N v w j + i ) ∑ w ′ e v w ′ ′ ⋅ ( ∑ j ∈ N v w j + i ) {\displaystyle -\sum _{i}\ln {\frac {e^{v_{w_{i}}'\cdot (\sum _{j\in N}v_{w_{j+i}})}}{\sum _{w'}e^{v_{w'}'\cdot (\sum _{j\in N}v_{w_{j+i}})}}}} where the outer summation ∑ i {\displaystyle \sum _{i}} is over the words in a corpus, the quantity ∑ j ∈ N v w j + i {\displaystyle \sum _{j\in N}v_{w_{j+i}}} is the sum of a word's neighbors' vectors, etc. Once such a system is trained, we have two trained matrices V , V ′ {\displaystyle V,V'} . Either the column vectors of V {\displaystyle V} or the row vectors of V ′ {\displaystyle V'} can serve as the dictionary. For example, the word "sat" can be represented as either the "sat"-th column of V {\displaystyle V} or the "sat"-th row of V ′ {\displaystyle V'} . It is also possible to simply define V ′ = V ⊤ {\displaystyle V'=V^{\top }} , in which case there would no longer be a choice. === Skip-gram === The idea of skip-gram is to represent each word with a vector, such that it is possible to predict the vectors of its neighbors using the vector of a word. The architecture is still linear-linear-softmax, the same as CBOW, but the input and the output are switched. Specifically, for each word w i {\displaystyle w_{i}} in the corpus, the one-hot encoding of the word is used as the input to the neural network. The output of the neural network is a probability distribution over the dictionary, representing a prediction of individual words in the neighborhood of w i {\displaystyle w_{i}} . The objective of training is to maximize ∑ i ∑ j ∈ N ln ⁡ Pr ( w j + i ∣ w i ) {\displaystyle \sum _{i}\sum _{j\in N}\ln \Pr(w_{j+i}\mid w_{i})} . In full formula, the loss function is − ∑ i ∑ j ∈ N ln ⁡ e v w j + i ′ ⋅ v w i ∑ w ′ e v w ′ ′ ⋅ v w i {\displaystyle -\sum _{i}\sum _{j\in N}\ln {\frac {e^{v_{w_{j+i}}'\cdot v_{w_{i}}}}{\sum _{w'}e^{v_{w'}'\cdot v_{w_{i}}}}}} Same as CBOW, once such a system is trained, we have two trained matrices V , V ′ {\displaystyle V,V'} . Either the column vectors of V {\displaystyle V} or the row vectors of V ′ {\displaystyle V'} can serve as the dictionary. It is also possible to simply define V ′ = V ⊤ {\displaystyle V'=V^{\top }} , in which case there would no longer be a choice. Essentially, skip-gram and CBOW are exactly the same in architecture. They only differ in the objective function during training. == History == During the 1980s, there were some early attempts at using neural networks to represent words and concepts as vectors. In 2010, Tomáš Mikolov (then at Brno University of Technology) with co-authors applied a simple recurrent neural network with a single hidden

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  Time-aware long short-term memory

  Time-aware LSTM (T-LSTM) is a long short-term memory (LSTM) unit capable of handling irregular time intervals in longitudinal patient records. T-LSTM was developed by researchers from Michigan State University, IBM Research, and Cornell University and was first presented in the Knowledge Discovery and Data Mining (KDD) conference. Experiments using real and synthetic data proved that T-LSTM auto-encoder outperformed widely used frameworks including LSTM and MF1-LSTM auto-encoders.

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  Types of artificial neural networks

  Types of neural networks (NN) include a family of techniques. The simplest types have static components, including number of units, number of layers, unit weights and topology. Dynamic NNs evolve via learning. Some types allow/require learning to be "supervised" by the operator, while others operate independently. Some types operate purely in hardware, while others are purely software and run on general purpose computers. The main types are: Transformers: these use attention to analyze every token in the input stream against every other token in the stream. That technique has enabled neural networks to reach the general public via chatbots, code generators and many other forms. Convolutional neural networks (CNN): a FNN that uses kernels and regularization to evade problems in prior generations of NNs. They are typically used to analyze visual and other two-dimensional data. Generative adversarial networks set networks (of varying structure) against each other, each trying to push the other(s) to produce better results such as winning a game or to deceive the opponent about the authenticity of an input. == Feedforward == In feedforward neural networks the information moves from the input to output directly in every layer. There can be hidden layers with or without cycles/loops to sequence inputs. Feedforward networks can be constructed with various types of units, such as binary McCulloch–Pitts neurons, the simplest of which is the perceptron. Continuous neurons, frequently with sigmoidal activation, are used in the context of backpropagation. == Group method of data handling == The Group Method of Data Handling (GMDH) features fully automatic structural and parametric model optimization. The node activation functions are Kolmogorov–Gabor polynomials that permit additions and multiplications. It uses a deep multilayer perceptron with eight layers. It is a supervised learning network that grows layer by layer, where each layer is trained by regression analysis. Useless items are detected using a validation set, and pruned through regularization. The size and depth of the resulting network depends on the task. == Autoencoder == An autoencoder, autoassociator or Diabolo network is similar to the multilayer perceptron (MLP) – with an input layer, an output layer and one or more hidden layers connecting them. However, the output layer has the same number of units as the input layer. Its purpose is to reconstruct its own inputs (instead of emitting a target value). Therefore, autoencoders are unsupervised learning models. An autoencoder is used for unsupervised learning of efficient codings, typically for the purpose of dimensionality reduction and for learning generative models of data. == Probabilistic == A probabilistic neural network (PNN) is a four-layer feedforward neural network. The layers are Input, hidden pattern, hidden summation, and output. In the PNN algorithm, the parent probability distribution function (PDF) of each class is approximated by a Parzen window and a non-parametric function. Then, using PDF of each class, the class probability of a new input is estimated and Bayes’ rule is employed to allocate it to the class with the highest posterior probability. It was derived from the Bayesian network and a statistical algorithm called Kernel Fisher discriminant analysis. It is used for classification and pattern recognition. == Time delay == A time delay neural network (TDNN) is a feedforward architecture for sequential data that recognizes features independent of sequence position. In order to achieve time-shift invariance, delays are added to the input so that multiple data points (points in time) are analyzed together. It usually forms part of a larger pattern recognition system. It has been implemented using a perceptron network whose connection weights were trained with back propagation (supervised learning). == Convolutional == A convolutional neural network (CNN, or ConvNet or shift invariant or space invariant) is a class of deep network, composed of one or more convolutional layers with fully connected layers (matching those in typical ANNs) on top. It uses tied weights and pooling layers. In particular, max-pooling. It is often structured via Fukushima's convolutional architecture. They are variations of multilayer perceptrons that use minimal preprocessing. This architecture allows CNNs to take advantage of the 2D structure of input data. Its unit connectivity pattern is inspired by the organization of the visual cortex. Units respond to stimuli in a restricted region of space known as the receptive field. Receptive fields partially overlap, over-covering the entire visual field. Unit response can be approximated mathematically by a convolution operation. CNNs are suitable for processing visual and other two-dimensional data. They have shown superior results in both image and speech applications. They can be trained with standard backpropagation. CNNs are easier to train than other regular, deep, feed-forward neural networks and have many fewer parameters to estimate. Capsule Neural Networks (CapsNet) add structures called capsules to a CNN and reuse output from several capsules to form more stable (with respect to various perturbations) representations. Examples of applications in computer vision include DeepDream and robot navigation. They have wide applications in image and video recognition, recommender systems and natural language processing. == Deep stacking network == A deep stacking network (DSN) (deep convex network) is based on a hierarchy of blocks of simplified neural network modules. It was introduced in 2011 by Deng and Yu. It formulates the learning as a convex optimization problem with a closed-form solution, emphasizing the mechanism's similarity to stacked generalization. Each DSN block is a simple module that is easy to train by itself in a supervised fashion without backpropagation for the entire blocks. Each block consists of a simplified multi-layer perceptron (MLP) with a single hidden layer. The hidden layer h has logistic sigmoidal units, and the output layer has linear units. Connections between these layers are represented by weight matrix U; input-to-hidden-layer connections have weight matrix W. Target vectors t form the columns of matrix T, and the input data vectors x form the columns of matrix X. The matrix of hidden units is H = σ ( W T X ) {\displaystyle {\boldsymbol {H}}=\sigma ({\boldsymbol {W}}^{T}{\boldsymbol {X}})} . Modules are trained in order, so lower-layer weights W are known at each stage. The function performs the element-wise logistic sigmoid operation. Each block estimates the same final label class y, and its estimate is concatenated with original input X to form the expanded input for the next block. Thus, the input to the first block contains the original data only, while downstream blocks' input adds the output of preceding blocks. Then learning the upper-layer weight matrix U given other weights in the network can be formulated as a convex optimization problem: min U T f = ‖ U T H − T ‖ F 2 , {\displaystyle \min _{U^{T}}f=\|{\boldsymbol {U}}^{T}{\boldsymbol {H}}-{\boldsymbol {T}}\|_{F}^{2},} which has a closed-form solution. Unlike other deep architectures, such as DBNs, the goal is not to discover the transformed feature representation. The structure of the hierarchy of this kind of architecture makes parallel learning straightforward, as a batch-mode optimization problem. In purely discriminative tasks, DSNs outperform conventional DBNs. === Tensor deep stacking networks === This architecture is a DSN extension. It offers two important improvements: it uses higher-order information from covariance statistics, and it transforms the non-convex problem of a lower-layer to a convex sub-problem of an upper-layer. TDSNs use covariance statistics in a bilinear mapping from each of two distinct sets of hidden units in the same layer to predictions, via a third-order tensor. While parallelization and scalability are not considered seriously in conventional DNNs, all learning for DSNs and TDSNs is done in batch mode, to allow parallelization. Parallelization allows scaling the design to larger (deeper) architectures and data sets. The basic architecture is suitable for diverse tasks such as classification and regression. == Physics-informed == Such a neural network is designed for the numerical solution of mathematical equations, such as differential, integral, delay, fractional and others. As input parameters, PINN accepts variables (spatial, temporal, and others), transmits them through the network block. At the output, it produces an approximate solution and substitutes it into the mathematical model, considering the initial and boundary conditions. If the solution does not satisfy the required accuracy, one uses the backpropagation and rectify the solution. Besides PINN, other architectures have been developed to produce surrogate models for scientific comput

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  Vector database

  A vector database, vector store or vector search engine is a database that stores and retrieves embeddings of data in vector space. Vector databases typically implement approximate nearest neighbor algorithms so users can search for records semantically similar to a given input, unlike traditional databases which primarily look up records by exact match. Use-cases for vector databases include similarity search, semantic search, multi-modal search, recommendations engines, object detection, and retrieval-augmented generation (RAG). Vector embeddings are mathematical representations of data in a high-dimensional space. In this space, each dimension corresponds to a feature of the data, with the number of dimensions ranging from a few hundred to tens of thousands, depending on the complexity of the data being represented. Each data item is represented by one vector in this space. Words, phrases, or entire documents, as well as images, audio, and other types of data, can all be vectorized. These feature vectors may be computed from the raw data using machine learning methods such as feature extraction algorithms, word embeddings or deep learning networks. The goal is that semantically similar data items receive feature vectors close to each other. Vector retrieval can be combined with metadata filtering or lexical search to support filtered and hybrid retrieval workflows. == Techniques == Common techniques for similarity search on high-dimensional vectors include: Hierarchical Navigable Small World (HNSW) graphs Locality-sensitive hashing (LSH) and sketching Product quantization (PQ) Inverted files These techniques may also be combined in vector search systems. In recent benchmarks, HNSW-based implementations have been among the best performers. Conferences such as the International Conference on Similarity Search and Applications (SISAP) and the Conference on Neural Information Processing Systems (NeurIPS) have hosted competitions on vector search in large databases. == Applications == Vector databases are used in a wide range of machine learning applications including similarity search, semantic search, multi-modal search, recommendations engines, object detection, and retrieval-augmented generation. === Retrieval-augmented generation === An especially common use-case for vector databases is in retrieval-augmented generation (RAG), a method to improve domain-specific responses of large language models. The retrieval component of a RAG can be any search system, but is most often implemented as a vector database. Text documents describing the domain of interest are collected, and for each document or document section, a feature vector (known as an "embedding") is computed, typically using a deep learning network, and stored in a vector database along with a link to the document. Given a user prompt, the feature vector of the prompt is computed, and the database is queried to retrieve the most relevant documents. These are then automatically added into the context window of the large language model, and the large language model proceeds to create a response to the prompt given this context. == Implementations ==

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  Multi-label classification

  In machine learning, multi-label classification or multi-output classification is a variant of the classification problem where multiple nonexclusive labels may be assigned to each instance. Multi-label classification is a generalization of multiclass classification, which is the single-label problem of categorizing instances into precisely one of several (greater than or equal to two) classes. In the multi-label problem the labels are nonexclusive and there is no constraint on how many of the classes the instance can be assigned to. The formulation of multi-label learning was first introduced by Shen et al. in the context of Semantic Scene Classification, and later gained popularity across various areas of machine learning. Formally, multi-label classification is the problem of finding a model that maps inputs x to binary vectors y; that is, it assigns a value of 0 or 1 for each element (label) in y. == Problem transformation methods == Several problem transformation methods exist for multi-label classification, and can be roughly broken down into: === Transformation into binary classification problems === The baseline approach, called the binary relevance method, amounts to independently training one binary classifier for each label. Given an unseen sample, the combined model then predicts all labels for this sample for which the respective classifiers predict a positive result. Although this method of dividing the task into multiple binary tasks may resemble superficially the one-vs.-all (OvA) and one-vs.-rest (OvR) methods for multiclass classification, it is essentially different from both, because a single classifier under binary relevance deals with a single label, without any regard to other labels whatsoever. A classifier chain is an alternative method for transforming a multi-label classification problem into several binary classification problems. It differs from binary relevance in that labels are predicted sequentially, and the output of all previous classifiers (i.e. positive or negative for a particular label) are input as features to subsequent classifiers. Classifier chains have been applied, for instance, in HIV drug resistance prediction. Bayesian network has also been applied to optimally order classifiers in Classifier chains. In case of transforming the problem to multiple binary classifications, the likelihood function reads L = ∏ i = 1 n ( ∏ k ( ∏ j k ( p k , j k ( x i ) δ y i , k , j k ) ) ) {\displaystyle L=\prod _{i=1}^{n}(\prod _{k}(\prod _{j_{k}}(p_{k,j_{k}}(x_{i})^{\delta _{y_{i,k},j_{k}}})))} where index i {\displaystyle i} runs over the samples, index k {\displaystyle k} runs over the labels, j k {\displaystyle j_{k}} indicates the binary outcomes 0 or 1, δ a , b {\displaystyle \delta _{a,b}} indicates the Kronecker delta, y i , k ∈ 0 , 1 {\displaystyle y_{i,k}\in {0,1}} indicates the multiple hot encoded labels of sample i {\displaystyle i} . === Transformation into multi-class classification problem === The label powerset (LP) transformation creates one binary classifier for every label combination present in the training set. For example, if possible labels for an example were A, B, and C, the label powerset representation of this problem is a multi-class classification problem with the classes [0 0 0], [1 0 0], [0 1 0], [0 0 1], [1 1 0], [1 0 1], [0 1 1], and [1 1 1] where for example [1 0 1] denotes an example where labels A and C are present and label B is absent. === Ensemble methods === A set of multi-class classifiers can be used to create a multi-label ensemble classifier. For a given example, each classifier outputs a single class (corresponding to a single label in the multi-label problem). These predictions are then combined by an ensemble method, usually a voting scheme where every class that receives a requisite percentage of votes from individual classifiers (often referred to as the discrimination threshold) is predicted as a present label in the multi-label output. However, more complex ensemble methods exist, such as committee machines. Another variation is the random k-labelsets (RAKEL) algorithm, which uses multiple LP classifiers, each trained on a random subset of the actual labels; label prediction is then carried out by a voting scheme. A set of multi-label classifiers can be used in a similar way to create a multi-label ensemble classifier. In this case, each classifier votes once for each label it predicts rather than for a single label. == Adapted algorithms == Some classification algorithms/models have been adapted to the multi-label task, without requiring problem transformations. Examples of these including for multi-label data are k-nearest neighbors: the ML-kNN algorithm extends the k-NN classifier to multi-label data. decision trees: "Clare" is an adapted C4.5 algorithm for multi-label classification; the modification involves the entropy calculations. MMC, MMDT, and SSC refined MMDT, can classify multi-labeled data based on multi-valued attributes without transforming the attributes into single-values. They are also named multi-valued and multi-labeled decision tree classification methods. kernel methods for vector output neural networks: BP-MLL is an adaptation of the popular back-propagation algorithm for multi-label learning. == Learning paradigms == Based on learning paradigms, the existing multi-label classification techniques can be classified into batch learning and online machine learning. Batch learning algorithms require all the data samples to be available beforehand. It trains the model using the entire training data and then predicts the test sample using the found relationship. The online learning algorithms, on the other hand, incrementally build their models in sequential iterations. In iteration t, an online algorithm receives a sample, xt and predicts its label(s) ŷt using the current model; the algorithm then receives yt, the true label(s) of xt and updates its model based on the sample-label pair: (xt, yt). == Multi-label stream classification == Data streams are possibly infinite sequences of data that continuously and rapidly grow over time. Multi-label stream classification (MLSC) is the version of multi-label classification task that takes place in data streams. It is sometimes also called online multi-label classification. The difficulties of multi-label classification (exponential number of possible label sets, capturing dependencies between labels) are combined with difficulties of data streams (time and memory constraints, addressing infinite stream with finite means, concept drifts). Many MLSC methods resort to ensemble methods in order to increase their predictive performance and deal with concept drifts. Below are the most widely used ensemble methods in the literature: Online Bagging (OzaBagging)-based methods: Observing the probability of having K many of a certain data point in a bootstrap sample is approximately Poisson(1) for big datasets, each incoming data instance in a data stream can be weighted proportional to Poisson(1) distribution to mimic bootstrapping in an online setting. This is called Online Bagging (OzaBagging). Many multi-label methods that use Online Bagging are proposed in the literature, each of which utilizes different problem transformation methods. EBR, ECC, EPS, EBRT, EBMT, ML-Random Rules are examples of such methods. ADWIN Bagging-based methods: Online Bagging methods for MLSC are sometimes combined with explicit concept drift detection mechanisms such as ADWIN (Adaptive Window). ADWIN keeps a variable-sized window to detect changes in the distribution of the data, and improves the ensemble by resetting the components that perform poorly when there is a drift in the incoming data. Generally, the letter 'a' is used as a subscript in the name of such ensembles to indicate the usage of ADWIN change detector. EaBR, EaCC, EaHTPS are examples of such multi-label ensembles. GOOWE-ML-based methods: Interpreting the relevance scores of each component of the ensemble as vectors in the label space and solving a least squares problem at the end of each batch, Geometrically-Optimum Online-Weighted Ensemble for Multi-label Classification (GOOWE-ML) is proposed. The ensemble tries to minimize the distance between the weighted prediction of its components and the ground truth vector for each instance over a batch. Unlike Online Bagging and ADWIN Bagging, GOOWE-ML utilizes a weighted voting scheme where better performing components of the ensemble are given more weight. The GOOWE-ML ensemble grows over time, and the lowest weight component is replaced by a new component when it is full at the end of a batch. GOBR, GOCC, GOPS, GORT are the proposed GOOWE-ML-based multi-label ensembles. Multiple Windows : Here, BR models that use a sliding window are replaced with two windows for each label, one for relevant and one for non-relevant examples. Instances are oversampled or undersampled according to a load factor that is kept

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  Cultural algorithm

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

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  Charge based boundary element fast multipole method

  The charge-based formulation of the boundary element method (BEM) is a dimensionality reduction numerical technique that is used to model quasistatic electromagnetic phenomena in highly complex conducting media (targeting, e.g., the human brain) with a very large (up to approximately 1 billion) number of unknowns. The charge-based BEM solves an integral equation of the potential theory written in terms of the induced surface charge density. This formulation is naturally combined with fast multipole method (FMM) acceleration, and the entire method is known as charge-based BEM-FMM. The combination of BEM and FMM is a common technique in different areas of computational electromagnetics and, in the context of bioelectromagnetism, it provides improvements over the finite element method. == Historical development == Along with more common electric potential-based BEM, the quasistatic charge-based BEM, derived in terms of the single-layer (charge) density, for a single-compartment medium has been known in the potential theory since the beginning of the 20th century. For multi-compartment conducting media, the surface charge density formulation first appeared in discretized form (for faceted interfaces) in the 1964 paper by Gelernter and Swihart. A subsequent continuous form, including time-dependent and dielectric effects, appeared in the 1967 paper by Barnard, Duck, and Lynn. The charge-based BEM has also been formulated for conducting, dielectric, and magnetic media, and used in different applications. In 2009, Greengard et al. successfully applied the charge-based BEM with fast multipole acceleration to molecular electrostatics of dielectrics. A similar approach to realistic modeling of the human brain with multiple conducting compartments was first described by Makarov et al. in 2018. Along with this, the BEM-based multilevel fast multipole method has been widely used in radar and antenna studies at microwave frequencies as well as in acoustics. == Physical background - surface charges in biological media == The charge-based BEM is based on the concept of an impressed (or primary) electric field E i {\displaystyle \mathbf {E} ^{i}} and a secondary electric field E s {\displaystyle \mathbf {E} ^{s}} . The impressed field is usually known a priori or is trivial to find. For the human brain, the impressed electric field can be classified as one of the following: A conservative field E i {\displaystyle \mathbf {E} ^{i}} derived from an impressed density of EEG or MEG current sources in a homogeneous infinite medium with the conductivity σ {\displaystyle \sigma } at the source location; An instantaneous solenoidal field E i {\displaystyle \mathbf {E} ^{i}} of an induction coil obtained from Faraday's law of induction in a homogeneous infinite medium (air), when transcranial magnetic stimulation (TMS) problems are concerned; A surface field E i {\displaystyle \mathbf {E} ^{i}} derived from an impressed surface current density J i = σ E i {\displaystyle \mathbf {J} ^{i}=\sigma \mathbf {E} ^{i}} of current electrodes injecting electric current at a boundary of a compartment with conductivity σ {\displaystyle \sigma } when transcranial direct-current stimulation (tDCS) or deep brain stimulation (DBS) are concerned; A conservative field E i {\displaystyle \mathbf {E} ^{i}} of charges deposited on voltage electrodes for tDCS or DBS. This specific problem requires a coupled treatment since these charges will depend on the environment; In application to multiscale modeling, a field E i {\displaystyle \mathbf {E} ^{i}} obtained from any other macroscopic numerical solution in a small (mesoscale or microscale) spatial domain within the brain. For example, a constant field can be used. When the impressed field is "turned on", free charges located within a conducting volume D immediately begin to redistribute and accumulate at the boundaries (interfaces) of regions of different conductivity in D. A surface charge density ρ ( r ) {\displaystyle \rho (\mathbf {r} )} appears on the conductivity interfaces. This charge density induces a secondary conservative electric field E s {\displaystyle \mathbf {E} ^{s}} following Coulomb's law. One example is a human under a direct current powerline with the known field E i {\displaystyle \mathbf {E} ^{i}} directed down. The superior surface of the human's conducting body will be charged negatively while its inferior portion is charged positively. These surface charges create a secondary electric field that effectively cancels or blocks the primary field everywhere in the body so that no current will flow within the body under DC steady state conditions. Another example is a human head with electrodes attached. At any conductivity interface with a normal vector n {\displaystyle \mathbf {n} } pointing from an "inside" (-) compartment of conductivity σ − {\displaystyle \sigma ^{-}} to an "outside" (+) compartment of conductivity σ + {\displaystyle \sigma ^{+}} , Kirchhoff's current law requires continuity of the normal component of the electric current density. This leads to the interfacial boundary condition in the form for every facet at a triangulated interface. As long as σ ± {\displaystyle \sigma ^{\pm }} are different from each other, the two normal components of the electric field, E ± ⋅ n {\displaystyle \mathbf {E} ^{\pm }\cdot \mathbf {n} } , must also be different. Such a jump across the interface is only possible when a sheet of surface charge exists at that interface. Thus, if an electric current or voltage is applied, the surface charge density follows. The goal of the numerical analysis is to find the unknown surface charge distribution and thus the total electric field E = E i + E s {\displaystyle \mathbf {E} =\mathbf {E} ^{i}+\mathbf {E} ^{s}} (and the total electric potential if required) anywhere in space. == System of equations for surface charges == Below, a derivation is given based on Gauss's law and Coulomb's law. All conductivity interfaces, denoted by S, are discretized into planar triangular facets t m {\displaystyle t_{m}} with centers r m {\displaystyle \mathbf {r} _{m}} . Assume that an m-th facet with the normal vector n m {\displaystyle \mathbf {n} _{m}} and area A m {\displaystyle A_{m}} carries a uniform surface charge density ρ m {\displaystyle \rho _{m}} . If a volumetric tetrahedral mesh were present, the charged facets would belong to tetrahedra with different conductivity values. We first compute the electric field E m + {\displaystyle \mathbf {E} _{m}^{+}} at the point r m + δ n m {\displaystyle \mathbf {r} _{m}+\delta \mathbf {n} _{m}} , for δ → 0 + {\displaystyle \delta \rightarrow 0^{+}} i.e., just outside facet 𝑚 at its center. This field contains three contributions: The continuous impressed electric field E i {\displaystyle \mathbf {E} ^{i}} itself; An electric field of the m-th charged facet itself. Very close to the facet, it can be approximated as the electric field of an infinite sheet of uniform surface charge ρ m {\displaystyle \rho _{m}} . By Gauss's law, it is given by + ρ m / 2 ε 0 ⋅ n m {\displaystyle +\rho _{m}/2\varepsilon _{0}\cdot \mathbf {n} _{m}} where ε 0 {\displaystyle \varepsilon _{0}} is a background electrical permittivity; An electric field generated by all other facets t n {\displaystyle t_{n}} , which we approximate as point charges of charge A n ρ n {\displaystyle A_{n}\rho _{n}} at each center r n {\displaystyle \mathbf {r} _{n}} . A similar treatment holds for the electric field E m − {\displaystyle \mathbf {E} _{m}^{-}} just inside facet 𝑚, but the electric field of the flat sheet of charge changes its sign. Using Coulomb's law to calculate the contribution of facets different from t m {\displaystyle t_{m}} , we find From this equation, we see that the normal component of the electric field indeed undergoes a jump through the charged interface. This is equivalent to a jump relation of the potential theory. As a second step, the two expressions for E m ± {\displaystyle \mathbf {E} _{m}^{\pm }} are substituted into the interfacial boundary condition σ − E m − ⋅ n m = σ + E m + ⋅ n m {\displaystyle \sigma ^{-}\mathbf {E} _{m}^{-}\cdot \mathbf {n} _{m}=\sigma ^{+}\mathbf {E} _{m}^{+}\cdot \mathbf {n} _{m}} , applied to every facet 𝑚. This operation leads to a system of linear equations for unknown charge densities ρ m {\displaystyle \rho _{m}} which solves the problem: where K m = σ − − σ + σ − + σ + {\displaystyle K_{m}={\frac {\sigma ^{-}-\sigma ^{+}}{\sigma ^{-}+\sigma ^{+}}}} is the electric conductivity contrast at the m-th facet. The normalization constant ε 0 {\displaystyle \varepsilon _{0}} will cancel out after the solution is substituted in the expression for E s {\displaystyle \mathbf {E} ^{s}} and becomes redundant. == Application of fast multipole method == For modern characterizations of brain topologies with ever-increasing levels of complexity, the above system of equations for ρ m {\displaystyle \rho _{m}} is very large; it is t

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  Automated attendant

  In telephony, an automated attendant (also auto attendant, auto-attendant, autoattendant, automatic phone menus, AA, or virtual receptionist) allows callers to be automatically transferred to an extension without the intervention of an operator/receptionist. Many AAs will also offer a simple menu system ("for sales, press 1, for service, press 2," etc.). An auto attendant may also allow a caller to reach a live operator by dialing a number, usually "0". Typically the auto attendant is included in a business's phone system such as a PBX, but some services allow businesses to use an AA without such a system. Modern AA services (which now overlap with more complicated interactive voice response or IVR systems) can route calls to mobile phones, VoIP virtual phones, other AAs/IVRs, or other locations using traditional land-line phones or voice message machines. == Feature description == Telephone callers will recognize an automated attendant system as one that greets calls incoming to an organization with a recorded greeting of the form, "Thank you for calling .... If you know your party's extension, you may dial it any time during this message." Callers who have a touch-tone (DTMF) phone can dial an extension number or, in most cases, wait for operator ("attendant") assistance. Since the telephone network does not transmit the DC signals from rotary dial telephones (except for audible clicks), callers who have rotary dial phones have to wait for assistance. On a purely technical level it could be argued that an automated attendant is a very simple kind of IVR however, in the telecom industry the terms IVR and auto attendant are generally considered distinct. An automated attendant serves a very specific purpose (replace live operator and route calls), whereas an IVR can perform all sorts of functions (telephone banking, account inquiries, etc.). An AA will often include a directory which will allow a caller to dial by name in order to find a user on a system. There is no standard format to these directories, and they can use combinations of first name, last name, or both. The following lists common routing steps that are components of an automated attendant: Transfer to extension Transfer to voicemail Play message (i.e., "our address is ...") Go to a sub-menu Repeat choices In addition, an automated attendant would be expected to have values for the following: '0' – where to go when the caller dials '0' Timeout – what to do if the caller does nothing (usually go to the same place as '0') Default mailbox – where to send calls if '0' is not answered (or is not pointing to a live person) == Background == PBXs (private branch exchanges) or PABXs (private automatic branch exchanges) are telephone systems that serve an organization that has many telephone extensions but fewer telephone lines (sometimes called "trunks") that connect that organization to the rest of the global telecommunications network. While persons within an enterprise served by a PBX can call each other by dialing their extension numbers, incoming calls, i.e., calls originating from a telephone not served by the PBX but intended for a party served by the PBX, required assistance from a switchboard operator (also called a "switchboard attendant") or a telephone service called DID ("direct inward dialing"). Direct inward dialing has advantages such as rapid connection to the destination party and disadvantages including cost, lack of identification of the called organization and use of ten-digit telephone numbers. Automated attendants provide, among many other things, a way for an external caller to be directed to an extension or department served by a PBX system without using direct inward dialing or without switchboard attendant assistance. == History == Automated attendants are not part of voicemail systems. Voice messaging (or voicemail or VM) technology has existed since the late 1970s; in the early 1980s companies provided voice-prompting systems that allowed callers to reach (route the call) to an intended party, not necessarily to leave a message. Automated attendant systems are also referred to as automated menu systems and much early work in this field was done by Michael J. Freeman, Ph.D. == Time-based routing == Many auto attendants will have options to allow for time-of-day routing, as well as weekend and holiday routing. The specifics of these features will depend entirely on the particular automated attendant, but typically there would be a normal greeting and routing steps that would take place during normal business hours, and a different greeting and routing for non-business hours.

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  Induction of regular languages

  In computational learning theory, induction of regular languages refers to the task of learning a formal description (e.g. grammar) of a regular language from a given set of example strings. Although E. Mark Gold has shown that not every regular language can be learned this way (see language identification in the limit), approaches have been investigated for a variety of subclasses. They are sketched in this article. For learning of more general grammars, see Grammar induction. == Definitions == A regular language is defined as a (finite or infinite) set of strings that can be described by one of the mathematical formalisms called "finite automaton", "regular grammar", or "regular expression", all of which have the same expressive power. Since the latter formalism leads to shortest notations, it shall be introduced and used here. Given a set Σ of symbols (a.k.a. alphabet), a regular expression can be any of ∅ (denoting the empty set of strings), ε (denoting the singleton set containing just the empty string), a (where a is any character in Σ; denoting the singleton set just containing the single-character string a), r + s (where r and s are, in turn, simpler regular expressions; denoting their set's union) r ⋅ s (denoting the set of all possible concatenations of strings from r's and s's set), r + (denoting the set of n-fold repetitions of strings from r's set, for any n ≥ 1), or r (similarly denoting the set of n-fold repetitions, but also including the empty string, seen as 0-fold repetition). For example, using Σ = {0,1}, the regular expression (0+1+ε)⋅(0+1) denotes the set of all binary numbers with one or two digits (leading zero allowed), while 1⋅(0+1)⋅0 denotes the (infinite) set of all even binary numbers (no leading zeroes). Given a set of strings (also called "positive examples"), the task of regular language induction is to come up with a regular expression that denotes a set containing all of them. As an example, given {1, 10, 100}, a "natural" description could be the regular expression 1⋅0, corresponding to the informal characterization "a 1 followed by arbitrarily many (maybe even none) 0's". However, (0+1) and 1+(1⋅0)+(1⋅0⋅0) is another regular expression, denoting the largest (assuming Σ = {0,1}) and the smallest set containing the given strings, and called the trivial overgeneralization and undergeneralization, respectively. Some approaches work in an extended setting where also a set of "negative example" strings is given; then, a regular expression is to be found that generates all of the positive, but none of the negative examples. == Lattice of automata == Dupont et al. have shown that the set of all structurally complete finite automata generating a given input set of example strings forms a lattice, with the trivial undergeneralized and the trivial overgeneralized automaton as bottom and top element, respectively. Each member of this lattice can be obtained by factoring the undergeneralized automaton by an appropriate equivalence relation. For the above example string set {1, 10, 100}, the picture shows at its bottom the undergeneralized automaton Aa,b,c,d in grey, consisting of states a, b, c, and d. On the state set {a,b,c,d}, a total of 15 equivalence relations exist, forming a lattice. Mapping each equivalence E to the corresponding quotient automaton language L(Aa,b,c,d / E) obtains the partially ordered set shown in the picture. Each node's language is denoted by a regular expression. The language may be recognized by quotient automata w.r.t. different equivalence relations, all of which are shown below the node. An arrow between two nodes indicates that the lower node's language is a proper subset of the higher node's. If both positive and negative example strings are given, Dupont et al. build the lattice from the positive examples, and then investigate the separation border between automata that generate some negative example and such that do not. Most interesting are those automata immediately below the border. In the picture, separation borders are shown for the negative example strings 11 (green), 1001 (blue), 101 (cyan), and 0 (red). Coste and Nicolas present an own search method within the lattice, which they relate to Mitchell's version space paradigm. To find the separation border, they use a graph coloring algorithm on the state inequality relation induced by the negative examples. Later, they investigate several ordering relations on the set of all possible state fusions. Kudo and Shimbo use the representation by automaton factorizations to give a unique framework for the following approaches (sketched below): k-reversible languages and the "tail clustering" follow-up approach, Successor automata and the predecessor-successor method, and pumping-based approaches (framework-integration challenged by Luzeaux, however). Each of these approaches is shown to correspond to a particular kind of equivalence relations used for factorization. == Approaches == === k-reversible languages === Angluin considers so-called "k-reversible" regular automata, that is, deterministic automata in which each state can be reached from at most one state by following a transition chain of length k. Formally, if Σ, Q, and δ denote the input alphabet, the state set, and the transition function of an automaton A, respectively, then A is called k-reversible if: ∀a0, ..., ak ∈ Σ ∀s1, s2 ∈ Q: δ(s1, a0...ak) = δ(s2, a0...ak) ⇒ s1 = s2, where δ means the homomorphic extension of δ to arbitrary words. Angluin gives a cubic algorithm for learning of the smallest k-reversible language from a given set of input words; for k = 0, the algorithm has even almost linear complexity. The required state uniqueness after k + 1 given symbols forces unifying automaton states, thus leading to a proper generalization different from the trivial undergeneralized automaton. This algorithm has been used to learn simple parts of English syntax; later, an incremental version has been provided. Another approach based on k-reversible automata is the tail clustering method. === Successor automata === From a given set of input strings, Vernadat and Richetin build a so-called successor automaton, consisting of one state for each distinct character and a transition between each two adjacent characters' states. For example, the singleton input set {aabbaabb} leads to an automaton corresponding to the regular expression (a+⋅b+). An extension of this approach is the predecessor-successor method which generalizes each character repetition immediately to a Kleene + and then includes for each character the set of its possible predecessors in its state. Successor automata can learn exactly the class of local languages. Since each regular language is the homomorphic image of a local language, grammars from the former class can be learned by lifting, if an appropriate (depending on the intended application) homomorphism is provided. In particular, there is such a homomorphism for the class of languages learnable by the predecessor-successor method. The learnability of local languages can be reduced to that of k-reversible languages. === Early approaches === Chomsky and Miller (1957) used the pumping lemma: they guess a part v of an input string uvw and try to build a corresponding cycle into the automaton to be learned; using membership queries they ask, for appropriate k, which of the strings uw, uvvw, uvvvw, ..., uvkw also belongs to the language to be learned, thereby refining the structure of their automaton. In 1959, Solomonoff generalized this approach to context-free languages, which also obey a pumping lemma. === Cover automata === Câmpeanu et al. learn a finite automaton as a compact representation of a large finite language. Given such a language F, they search a so-called cover automaton A such that its language L(A) covers F in the following sense: L(A) ∩ Σ≤ l = F, where l is the length of the longest string in F, and Σ≤ l denotes the set of all strings not longer than l. If such a cover automaton exists, F is uniquely determined by A and l. For example, F = {ad, read, reread } has l = 6 and a cover automaton corresponding to the regular expression (r⋅e)⋅a⋅d. For two strings x and y, Câmpeanu et al. define x ~ y if xz ∈ F ⇔ yz ∈ F for all strings z of a length such that both xz and yz are not longer than l. Based on this relation, whose lack of transitivity causes considerable technical problems, they give an O(n4) algorithm to construct from F a cover automaton A of minimal state count. Moreover, for union, intersection, and difference of two finite languages they provide corresponding operations on their cover automata. Păun et al. improve the time complexity to O(n2). === Residual automata === For a set S of strings and a string u, the Brzozowski derivative u−1S is defined as the set of all rest-strings obtainable from a string in S by cutting off its prefix u (if possible), formally: u−1S = {v ∈ Σ: uv ∈ S}, cf. picture. Denis et al. define a

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  Mathematics of neural networks in machine learning

  An artificial neural network (ANN) or neural network combines biological principles with advanced statistics to solve problems in domains such as pattern recognition and game-play. ANNs adopt the basic model of neuron analogues connected to each other in a variety of ways. == Structure == === Neuron === A neuron with label j {\displaystyle j} receiving an input p j ( t ) {\displaystyle p_{j}(t)} from predecessor neurons consists of the following components: an activation a j ( t ) {\displaystyle a_{j}(t)} , the neuron's state, depending on a discrete time parameter, an optional threshold θ j {\displaystyle \theta _{j}} , which stays fixed unless changed by learning, an activation function f {\displaystyle f} that computes the new activation at a given time t + 1 {\displaystyle t+1} from a j ( t ) {\displaystyle a_{j}(t)} , θ j {\displaystyle \theta _{j}} and the net input p j ( t ) {\displaystyle p_{j}(t)} giving rise to the relation a j ( t + 1 ) = f ( a j ( t ) , p j ( t ) , θ j ) , {\displaystyle a_{j}(t+1)=f(a_{j}(t),p_{j}(t),\theta _{j}),} and an output function f out {\displaystyle f_{\text{out}}} computing the output from the activation o j ( t ) = f out ( a j ( t ) ) . {\displaystyle o_{j}(t)=f_{\text{out}}(a_{j}(t)).} Often the output function is simply the identity function. An input neuron has no predecessor but serves as input interface for the whole network. Similarly an output neuron has no successor and thus serves as output interface of the whole network. === Propagation function === The propagation function computes the input p j ( t ) {\displaystyle p_{j}(t)} to the neuron j {\displaystyle j} from the outputs o i ( t ) {\displaystyle o_{i}(t)} and typically has the form p j ( t ) = ∑ i o i ( t ) w i j . {\displaystyle p_{j}(t)=\sum _{i}o_{i}(t)w_{ij}.} === Bias === A bias term can be added, changing the form to the following: p j ( t ) = ∑ i o i ( t ) w i j + w 0 j , {\displaystyle p_{j}(t)=\sum _{i}o_{i}(t)w_{ij}+w_{0j},} where w 0 j {\displaystyle w_{0j}} is a bias. == Neural networks as functions == Neural network models can be viewed as defining a function that takes an input (observation) and produces an output (decision) f : X → Y {\displaystyle \textstyle f:X\rightarrow Y} or a distribution over X {\displaystyle \textstyle X} or both X {\displaystyle \textstyle X} and Y {\displaystyle \textstyle Y} . Sometimes models are intimately associated with a particular learning rule. A common use of the phrase "ANN model" is really the definition of a class of such functions (where members of the class are obtained by varying parameters, connection weights, or specifics of the architecture such as the number of neurons, number of layers or their connectivity). Mathematically, a neuron's network function f ( x ) {\displaystyle \textstyle f(x)} is defined as a composition of other functions g i ( x ) {\displaystyle \textstyle g_{i}(x)} , that can further be decomposed into other functions. This can be conveniently represented as a network structure, with arrows depicting the dependencies between functions. A widely used type of composition is the nonlinear weighted sum, where f ( x ) = K ( ∑ i w i g i ( x ) ) {\displaystyle \textstyle f(x)=K\left(\sum _{i}w_{i}g_{i}(x)\right)} , where K {\displaystyle \textstyle K} (commonly referred to as the activation function) is some predefined function, such as the hyperbolic tangent, sigmoid function, softmax function, or rectifier function. The important characteristic of the activation function is that it provides a smooth transition as input values change, i.e. a small change in input produces a small change in output. The following refers to a collection of functions g i {\displaystyle \textstyle g_{i}} as a vector g = ( g 1 , g 2 , … , g n ) {\displaystyle \textstyle g=(g_{1},g_{2},\ldots ,g_{n})} . This figure depicts such a decomposition of f {\displaystyle \textstyle f} , with dependencies between variables indicated by arrows. These can be interpreted in two ways. The first view is the functional view: the input x {\displaystyle \textstyle x} is transformed into a 3-dimensional vector h {\displaystyle \textstyle h} , which is then transformed into a 2-dimensional vector g {\displaystyle \textstyle g} , which is finally transformed into f {\displaystyle \textstyle f} . This view is most commonly encountered in the context of optimization. The second view is the probabilistic view: the random variable F = f ( G ) {\displaystyle \textstyle F=f(G)} depends upon the random variable G = g ( H ) {\displaystyle \textstyle G=g(H)} , which depends upon H = h ( X ) {\displaystyle \textstyle H=h(X)} , which depends upon the random variable X {\displaystyle \textstyle X} . This view is most commonly encountered in the context of graphical models. The two views are largely equivalent. In either case, for this particular architecture, the components of individual layers are independent of each other (e.g., the components of g {\displaystyle \textstyle g} are independent of each other given their input h {\displaystyle \textstyle h} ). This naturally enables a degree of parallelism in the implementation. Networks such as the previous one are commonly called feedforward, because their graph is a directed acyclic graph. Networks with cycles are commonly called recurrent. Such networks are commonly depicted in the manner shown at the top of the figure, where f {\displaystyle \textstyle f} is shown as dependent upon itself. However, an implied temporal dependence is not shown. == Backpropagation == Backpropagation training algorithms fall into three categories: steepest descent (with variable learning rate and momentum, resilient backpropagation); quasi-Newton (Broyden–Fletcher–Goldfarb–Shanno, one step secant); Levenberg–Marquardt and conjugate gradient (Fletcher–Reeves update, Polak–Ribiére update, Powell–Beale restart, scaled conjugate gradient). === Algorithm === Let N {\displaystyle N} be a network with e {\displaystyle e} connections, m {\displaystyle m} inputs and n {\displaystyle n} outputs. Below, x 1 , x 2 , … {\displaystyle x_{1},x_{2},\dots } denote vectors in R m {\displaystyle \mathbb {R} ^{m}} , y 1 , y 2 , … {\displaystyle y_{1},y_{2},\dots } vectors in R n {\displaystyle \mathbb {R} ^{n}} , and w 0 , w 1 , w 2 , … {\displaystyle w_{0},w_{1},w_{2},\ldots } vectors in R e {\displaystyle \mathbb {R} ^{e}} . These are called inputs, outputs and weights, respectively. The network corresponds to a function y = f N ( w , x ) {\displaystyle y=f_{N}(w,x)} which, given a weight w {\displaystyle w} , maps an input x {\displaystyle x} to an output y {\displaystyle y} . In supervised learning, a sequence of training examples ( x 1 , y 1 ) , … , ( x p , y p ) {\displaystyle (x_{1},y_{1}),\dots ,(x_{p},y_{p})} produces a sequence of weights w 0 , w 1 , … , w p {\displaystyle w_{0},w_{1},\dots ,w_{p}} starting from some initial weight w 0 {\displaystyle w_{0}} , usually chosen at random. These weights are computed in turn: first compute w i {\displaystyle w_{i}} using only ( x i , y i , w i − 1 ) {\displaystyle (x_{i},y_{i},w_{i-1})} for i = 1 , … , p {\displaystyle i=1,\dots ,p} . The output of the algorithm is then w p {\displaystyle w_{p}} , giving a new function x ↦ f N ( w p , x ) {\displaystyle x\mapsto f_{N}(w_{p},x)} . The computation is the same in each step, hence only the case i = 1 {\displaystyle i=1} is described. w 1 {\displaystyle w_{1}} is calculated from ( x 1 , y 1 , w 0 ) {\displaystyle (x_{1},y_{1},w_{0})} by considering a variable weight w {\displaystyle w} and applying gradient descent to the function w ↦ E ( f N ( w , x 1 ) , y 1 ) {\displaystyle w\mapsto E(f_{N}(w,x_{1}),y_{1})} to find a local minimum, starting at w = w 0 {\displaystyle w=w_{0}} . This makes w 1 {\displaystyle w_{1}} the minimizing weight found by gradient descent. == Learning pseudocode == To implement the algorithm above, explicit formulas are required for the gradient of the function w ↦ E ( f N ( w , x ) , y ) {\displaystyle w\mapsto E(f_{N}(w,x),y)} where the function is E ( y , y ′ ) = | y − y ′ | 2 {\displaystyle E(y,y')=|y-y'|^{2}} . The learning algorithm can be divided into two phases: propagation and weight update. === Propagation === Propagation involves the following steps: Propagation forward through the network to generate the output value(s) Calculation of the cost (error term) Propagation of the output activations back through the network using the training pattern target to generate the deltas (the difference between the targeted and actual output values) of all output and hidden neurons. === Weight update === For each weight: Multiply the weight's output delta and input activation to find the gradient of the weight. Subtract the ratio (percentage) of the weight's gradient from the weight. The learning rate is the ratio (percentage) that influences the speed and quality of learning. The greater the ratio, the faster the neuron trains, but the lower the ratio, the more accurat

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  Scale-invariant feature operator

  In the fields of computer vision and image analysis, the scale-invariant feature operator (or SFOP) is an algorithm to detect local features in images. The algorithm was published by Förstner et al. in 2009. == Algorithm == The scale-invariant feature operator (SFOP) is based on two theoretical concepts: spiral model feature operator Desired properties of keypoint detectors: Invariance and repeatability for object recognition Accuracy to support camera calibration Interpretability: Especially corners and circles, should be part of the detected keypoints (see figure). As few control parameters as possible with clear semantics Complementarity to known detectors scale-invariant corner/circle detector. == Theory == === Maximize the weight === Maximize the weight w {\displaystyle w} = 1/variance of a point p {\displaystyle p} w ( p , α , τ , σ ) = ( N ( σ ) − 2 ) λ m i n ( M ( p , α , τ , σ ) ) Ω ( p , α , τ , σ ) {\displaystyle w(\mathbf {p} ,\alpha ,\tau ,\sigma )=\left(N(\sigma )-2\right){\frac {\lambda _{min}(M(\mathbf {p} ,\alpha ,\tau ,\sigma ))}{\Omega (\mathbf {p} ,\alpha ,\tau ,\sigma )}}} comprising: 1. the image model Ω ( p , α , τ , σ ) = ∑ n = 1 N ( σ ) [ ( q n − p ) T R α ∇ T g ( q n ) ] 2 G σ ( q n − p ) = N ( σ ) t r { R α ∇ τ ∇ τ T R α T ∗ p p T G σ ( p ) } {\displaystyle {\begin{aligned}\Omega (\mathbf {p} ,\alpha ,\tau ,\sigma )&=\sum _{n=1}^{N(\sigma )}[(\mathbf {q} _{n}-\mathbf {p} )^{T}\mathbf {R} _{\alpha }\mathbf {\nabla } _{T}g(\mathbf {q} _{n})]^{2}G_{\sigma }(\mathbf {q} _{n}-\mathbf {p} )\\&=N(\sigma )\mathbf {tr} \left\{R_{\alpha }\mathbf {\nabla } _{\tau }\mathbf {\nabla } _{\tau }^{T}R_{\alpha }^{T}\mathbf {p} \mathbf {p} ^{T}G_{\sigma }(\mathbf {p} )\right\}\end{aligned}}} 2. the smaller eigenvalue of the structure tensor M ( p , α , τ , σ ) ⏟ structure tensor = G σ ( p ) ⏟ weighted summation ∗ ( R σ ∇ τ ∇ τ T R σ T ) ⏟ squared rotated gradients {\displaystyle \underbrace {M(\mathbf {p} ,\alpha ,\tau ,\sigma )} _{\text{structure tensor}}=\underbrace {G_{\sigma }(\mathbf {p} )} _{\text{weighted summation}}\underbrace {(R_{\sigma }\nabla _{\tau }\nabla _{\tau }^{T}R_{\sigma }^{T})} _{\text{squared rotated gradients}}} === Reduce the search space === Reduce the 5-dimensional search space by linking the differentiation scale τ {\displaystyle \tau } to the integration scale τ = σ / 3 {\displaystyle \tau =\sigma /3} solving for the optimal α ^ {\displaystyle {\hat {\alpha }}} using the model Ω ( α ) = a − b cos ⁡ ( 2 α − 2 α 0 ) {\displaystyle \Omega (\alpha )=a-b\cos(2\alpha -2\alpha _{0})} and determining the parameters from three angles, e. g. Ω ( 0 ∘ ) , Ω ( 60 ∘ ) , Ω ( 120 ∘ ) → a , b , α 0 → α ^ {\displaystyle \Omega (0^{\circ }),\Omega (60^{\circ }),\Omega (120^{\circ })\quad \rightarrow \quad a,b,\alpha _{0}\quad \rightarrow \quad {\hat {\alpha }}} pre-selection possible: α = 0 ∘ → junctions , α = 90 ∘ → circular features {\displaystyle \alpha =0^{\circ }\,\rightarrow \,{\mbox{junctions}},\quad \alpha =90^{\circ }\,\rightarrow \,{\mbox{circular features}}} === Filter potential keypoints === non-maxima suppression over scale, space and angle thresholding the isotropy λ 2 ( M ) {\displaystyle \lambda _{2(M)}} :eigenvalues characterize the shape of the keypoint, smallest eigenvalue has to be larger than threshold T λ {\displaystyle T_{\lambda }} derived from noise variance V ( n ) {\displaystyle V(n)} and significance level S {\displaystyle S} : T λ ( V ( n ) , τ , σ , S ) = N ( σ ) 16 π τ 4 V ( n ) χ 2 , S 2 {\displaystyle T_{\lambda }(V(n),\tau ,\sigma ,S)={\frac {N(\sigma )}{16\pi \tau ^{4}}}V(n)\chi _{2,S}^{2}} == Algorithm == == Results == === Interpretability of SFOP keypoints ===

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  Color management

  Color management is the process of ensuring consistent and accurate colors across various devices, such as monitors, printers, and cameras. It involves the use of color profiles, which are standardized descriptions of how colors should be displayed or reproduced. Color management is necessary because different devices have different color capabilities and characteristics. For example, a monitor may display colors differently than a printer can reproduce them. Without color management, the same image may appear differently on different devices, leading to inconsistencies and inaccuracies. To achieve color management, a color profile is created for each device involved in the color workflow. This profile describes the device's color capabilities and characteristics, such as its color gamut (range of colors it can display or reproduce) and color temperature. These profiles are then used to translate colors between devices, ensuring consistent and accurate color reproduction. Color management is particularly important in industries such as graphic design, photography, and printing, where accurate color representation is crucial. It helps to maintain color consistency throughout the entire workflow, from capturing an image to displaying or printing it. Parts of color management are implemented in the operating system (OS), helper libraries, the application, and devices. The type of color profile that is typically used is called an ICC profile. A cross-platform view of color management is the use of an ICC-compatible color management system. The International Color Consortium (ICC) is an industry consortium that has defined: an open standard for a Color Matching Module (CMM) at the OS level color profiles for: devices, including DeviceLink profiles that transform one device profile (color space) to another device profile without passing through an intermediate color space, such as LAB, more accurately preserving color working spaces, the color spaces in which color data is meant to be manipulated There are other approaches to color management besides using ICC profiles. This is partly due to history and partly because of other needs than the ICC standard covers. The film and broadcasting industries make use of some of the same concepts, but they frequently rely on more limited boutique solutions. The film industry, for instance, often uses 3D LUTs (lookup table) to represent a complete color transformation for a specific RGB encoding. At the consumer level, system wide color management is available in most of Apple's products (macOS, iOS, iPadOS, watchOS). Microsoft Windows lacks system wide color management and virtually all applications do not employ color management. Windows' media player API is not color space aware, and if applications want to color manage videos manually, they have to incur significant performance and power consumption penalties. Android supports system wide color management, but most devices ship with color management disabled. == Overview == Characterize. Every color-managed device requires a personalized table, or "color profile," which characterizes the color response of that particular device. Standardize. Each color profile describes these colors relative to a standardized set of reference colors (the "Profile Connection Space"). Translate. Color-managed software then uses these standardized profiles to translate color from one device to another. This is usually performed by a color management module (CMM). == Hardware == === Characterization === To describe the behavior of various output devices, they must be compared (measured) in relation to a standard color space. Often a step called linearization is performed first, to undo the effect of gamma correction that was done to get the most out of limited 8-bit color paths. Instruments used for measuring device colors include colorimeters and spectrophotometers. As an intermediate result, the device gamut is described in the form of scattered measurement data. The transformation of the scattered measurement data into a more regular form, usable by the application, is called profiling. Profiling is a complex process involving mathematics, intense computation, judgment, testing, and iteration. After the profiling is finished, an idealized color description of the device is created. This description is called a profile. === Calibration === Calibration is like characterization, except that it can include the adjustment of the device, as opposed to just the measurement of the device. Color management is sometimes sidestepped by calibrating devices to a common standard color space such as sRGB; when such calibration is done well enough, no color translations are needed to get all devices to handle colors consistently. This avoidance of the complexity of color management was one of the goals in the development of sRGB. == Color profiles == === Embedding === Image formats themselves (such as TIFF, JPEG, PNG, EPS, PDF, and SVG) may contain embedded color profiles but are not required to do so by the image format. The International Color Consortium standard was created to bring various developers and manufacturers together. The ICC standard permits the exchange of output device characteristics and color spaces in the form of metadata. This allows the embedding of color profiles into images as well as storing them in a database or a profile directory. === Working spaces === Working spaces, such as sRGB, Adobe RGB or ProPhoto are color spaces that facilitate good results while editing. For instance, pixels with equal values of R,G,B should appear neutral. Using a large (gamut) working space will lead to posterization, while using a small working space will lead to clipping. This trade-off is a consideration for the critical image editor. == Color transformation == Color transformation, or color space conversion, is the transformation of the representation of a color from one color space to another. This calculation is required whenever data is exchanged inside a color-managed chain and carried out by a Color Matching Module. Transforming profiled color information to different output devices is achieved by referencing the profile data into a standard color space. It makes it easier to convert colors from one device to a selected standard color space and from that to the colors of another device. By ensuring that the reference color space covers the many possible colors that humans can see, this concept allows one to exchange colors between many different color output devices. Color transformations can be represented by two profiles (source profile and target profile) or by a devicelink profile. In this process there are approximations involved which make sure that the image keeps its important color qualities and also gives an opportunity to control on how the colors are being changed. === Profile connection space === In the terminology of the International Color Consortium, a translation between two color spaces can go through a profile connection space (PCS): Color Space 1 → PCS (CIELAB or CIEXYZ) → Color space 2; conversions into and out of the PCS are each specified by a profile. === Gamut mapping === In nearly every translation process, we have to deal with the fact that the color gamut of different devices vary in range which makes an accurate reproduction impossible. They therefore need some rearrangement near the borders of the gamut. Some colors must be shifted to the inside of the gamut, as they otherwise cannot be represented on the output device and would simply be clipped. This so-called gamut mismatch occurs for example, when we translate from the RGB color space with a wider gamut into the CMYK color space with a narrower gamut range. In this example, the dark highly saturated purplish-blue color of a typical computer monitor's "blue" primary is impossible to print on paper with a typical CMYK printer. The nearest approximation within the printer's gamut will be much less saturated. Conversely, an inkjet printer's "cyan" primary, a saturated mid-brightness blue, is outside the gamut of a typical computer monitor. The color management system can utilize various methods to achieve desired results and give experienced users control of the gamut mapping behavior. ==== Rendering intent ==== When the gamut of source color space exceeds that of the destination, saturated colors are liable to become clipped (inaccurately represented), or more formally burned. The color management module can deal with this problem in several ways. The ICC specification includes four different rendering intents, listed below. Before the actual rendering intent is carried out, one can temporarily simulate the rendering by soft proofing. It is a useful tool as it predicts the outcome of the colors and is available as an application in many color management systems: Absolute colorimetric Absolute colorimetry and relative colorimetry actually use the same table but differ in the adjust

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  Multimodal learning

  Multimodal learning is a type of deep learning that integrates and processes multiple types of data, referred to as modalities, such as text, audio, images, or video. This integration allows for a more holistic understanding of complex data, improving model performance in tasks like visual question answering, cross-modal retrieval, text-to-image generation, aesthetic ranking, and image captioning. Multimodal learning was proposed in 2011 at the beginning of the deep learning period. Large multimodal models, such as Google Gemini and GPT-4o, have become increasingly popular since 2023, enabling increased versatility and a broader understanding of real-world phenomena. == Motivation == Data usually comes with different modalities which carry different information. For example, it is very common to caption an image to convey the information not presented in the image itself. Similarly, sometimes it is more straightforward to use an image to describe information which may not be obvious from text. As a result, if different words appear in similar images, then these words likely describe the same thing. Conversely, if a word is used to describe seemingly dissimilar images, then these images may represent the same object. Thus, in cases dealing with multi-modal data, it is important to use a model which is able to jointly represent the information such that the model can capture the combined information from different modalities. == Multimodal transformers == Models such as CLIP (Contrastive Language–Image Pretraining) learn joint representations of images and text by optimizing contrastive objectives, allowing the model to match images with their corresponding textual descriptions. == Multimodal deep Boltzmann machines == A Boltzmann machine is a type of stochastic neural network invented by Geoffrey Hinton and Terry Sejnowski in 1985. Boltzmann machines can be seen as the stochastic, generative counterpart of Hopfield nets. They are named after the Boltzmann distribution in statistical mechanics. The units in Boltzmann machines are divided into two groups: visible units and hidden units. Each unit is like a neuron with a binary output that represents whether it is activated or not. General Boltzmann machines allow connection between any units. However, learning is impractical using general Boltzmann Machines because the computational time is exponential to the size of the machine. A more efficient architecture is called restricted Boltzmann machine where connection is only allowed between hidden unit and visible unit, which is described in the next section. Multimodal deep Boltzmann machines can process and learn from different types of information, such as images and text, simultaneously. This can notably be done by having a separate deep Boltzmann machine for each modality, for example one for images and one for text, joined at an additional top hidden layer. == Applications == Multimodal machine learning has numerous applications across various domains: Cross-modal retrieval: cross-modal retrieval allows users to search for data across different modalities (e.g., retrieving images based on text descriptions), improving multimedia search engines and content recommendation systems. Classification and missing data retrieval: multimodal Deep Boltzmann Machines outperform traditional models like support vector machines and latent Dirichlet allocation in classification tasks and can predict missing data in multimodal datasets, such as images and text. Healthcare diagnostics: multimodal models integrate medical imaging, genomic data, and patient records to improve diagnostic accuracy and early disease detection, especially in cancer screening. Content generation: models like DALL·E generate images from textual descriptions, benefiting creative industries, while cross-modal retrieval enables dynamic multimedia searches. Robotics and human-computer interaction: multimodal learning improves interaction in robotics and AI by integrating sensory inputs like speech, vision, and touch, aiding autonomous systems and human-computer interaction. Emotion recognition: combining visual, audio, and text data, multimodal systems enhance sentiment analysis and emotion recognition, applied in customer service, social media, and marketing.

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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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