Communication-avoiding algorithms minimize movement of data within a memory hierarchy for improving its running-time and energy consumption. These minimize the total of two costs (in terms of time and energy): arithmetic and communication. Communication, in this context refers to moving data, either between levels of memory or between multiple processors over a network. It is much more expensive than arithmetic. == Formal theory == === Two-level memory model === A common computational model in analyzing communication-avoiding algorithms is the two-level memory model: There is one processor and two levels of memory. Level 1 memory is infinitely large. Level 0 memory ("cache") has size M {\displaystyle M} . In the beginning, input resides in level 1. In the end, the output resides in level 1. Processor can only operate on data in cache. The goal is to minimize data transfers between the two levels of memory. === Matrix multiplication === Corollary 6.2: More general results for other numerical linear algebra operations can be found in. The following proof is from. == Motivation == Consider the following running-time model: Measure of computation = Time per FLOP = γ Measure of communication = No. of words of data moved = β ⇒ Total running time = γ·(no. of FLOPs) + β·(no. of words) From the fact that β >> γ as measured in time and energy, communication cost dominates computation cost. Technological trends indicate that the relative cost of communication is increasing on a variety of platforms, from cloud computing to supercomputers to mobile devices. The report also predicts that gap between DRAM access time and FLOPs will increase 100× over coming decade to balance power usage between processors and DRAM. Energy consumption increases by orders of magnitude as we go higher in the memory hierarchy. United States president Barack Obama cited communication-avoiding algorithms in the FY 2012 Department of Energy budget request to Congress: New Algorithm Improves Performance and Accuracy on Extreme-Scale Computing Systems. On modern computer architectures, communication between processors takes longer than the performance of a floating-point arithmetic operation by a given processor. ASCR researchers have developed a new method, derived from commonly used linear algebra methods, to minimize communications between processors and the memory hierarchy, by reformulating the communication patterns specified within the algorithm. This method has been implemented in the TRILINOS framework, a highly-regarded suite of software, which provides functionality for researchers around the world to solve large scale, complex multi-physics problems. == Objectives == Communication-avoiding algorithms are designed with the following objectives: Reorganize algorithms to reduce communication across all memory hierarchies. Attain the lower-bound on communication when possible. The following simple example demonstrates how these are achieved. === Matrix multiplication example === Let A, B and C be square matrices of order n × n. The following naive algorithm implements C = C + A B: for i = 1 to n for j = 1 to n for k = 1 to n C(i,j) = C(i,j) + A(i,k) B(k,j) Arithmetic cost (time-complexity): n2(2n − 1) for sufficiently large n or O(n3). Rewriting this algorithm with communication cost labelled at each step for i = 1 to n {read row i of A into fast memory} - n2 reads for j = 1 to n {read C(i,j) into fast memory} - n2 reads {read column j of B into fast memory} - n3 reads for k = 1 to n C(i,j) = C(i,j) + A(i,k) B(k,j) {write C(i,j) back to slow memory} - n2 writes Fast memory may be defined as the local processor memory (CPU cache) of size M and slow memory may be defined as the DRAM. Communication cost (reads/writes): n3 + 3n2 or O(n3) Since total running time = γ·O(n3) + β·O(n3) and β >> γ the communication cost is dominant. The blocked (tiled) matrix multiplication algorithm reduces this dominant term: ==== Blocked (tiled) matrix multiplication ==== Consider A, B and C to be n/b-by-n/b matrices of b-by-b sub-blocks where b is called the block size; assume three b-by-b blocks fit in fast memory. for i = 1 to n/b for j = 1 to n/b {read block C(i,j) into fast memory} - b2 × (n/b)2 = n2 reads for k = 1 to n/b {read block A(i,k) into fast memory} - b2 × (n/b)3 = n3/b reads {read block B(k,j) into fast memory} - b2 × (n/b)3 = n3/b reads C(i,j) = C(i,j) + A(i,k) B(k,j) - {do a matrix multiply on blocks} {write block C(i,j) back to slow memory} - b2 × (n/b)2 = n2 writes Communication cost: 2n3/b + 2n2 reads/writes << 2n3 arithmetic cost Making b as large possible: 3b2 ≤ M we achieve the following communication lower bound: 31/2n3/M1/2 + 2n2 or Ω (no. of FLOPs / M1/2) == Previous approaches for reducing communication == Most of the approaches investigated in the past to address this problem rely on scheduling or tuning techniques that aim at overlapping communication with computation. However, this approach can lead to an improvement of at most a factor of two. Ghosting is a different technique for reducing communication, in which a processor stores and computes redundantly data from neighboring processors for future computations. Cache-oblivious algorithms represent a different approach introduced in 1999 for fast Fourier transforms, and then extended to graph algorithms, dynamic programming, etc. They were also applied to several operations in linear algebra as dense LU and QR factorizations. The design of architecture specific algorithms is another approach that can be used for reducing the communication in parallel algorithms, and there are many examples in the literature of algorithms that are adapted to a given communication topology.
Stripe, Inc.
Stripe, Inc. is an Irish and American multinational financial services and software as a service (SaaS) company dual-headquartered in South San Francisco, California, United States, and Dublin, Ireland. The company primarily offers payment-processing software and application programming interfaces for e-commerce websites and mobile applications. Stripe is the largest privately owned financial technology company with a valuation of about $159 billion and over $1.9 trillion in payment volume processed in 2025, processing transactions for 5 million businesses in that year. == History == Irish entrepreneur brothers John and Patrick Collison founded Stripe in Palo Alto, California, in 2010, and serve as the company's president and CEO, respectively. In 2011 the company received a $2 million investment, including contributions from Elon Musk, PayPal founder Peter Thiel, Irish entrepreneur Liam Casey, and venture capital firms Sequoia Capital, Andreessen Horowitz, and SV Angel. In March 2013, Stripe made its first acquisition, Kickoff, a chat and task-management application. In 2012 the company moved from Palo Alto to San Francisco. In October 2019, the company announced that it would be moving from the South of Market area to Oyster Point in the neighbouring city of South San Francisco in 2021. In February 2021, Mark Carney, former governor of the Bank of Canada and of the Bank of England, was appointed to the company's board. Carney stepped down from his role with the company in 2025 in order to run for the leadership of the Liberal Party. Stripe acquired accountancy platform Recko in October 2021 whose solution was to be added to Stripe's existing suite of financial tools. In January 2022, Stripe entered a five-year partnership with Ford Motor Company. Through the deal, Stripe would handle transactions for consumer vehicle orders and reservations. That same month, Stripe partnered with Spotify to help the company monetize subscriptions. In April 2022, Twitter announced that it would partner with Stripe, Inc. (digital payments processor) for piloting cryptocurrency pay-outs for limited users in the platform. In April 2022, Stripe announced its strategic partnership with UK-based financial technology company ION. The Wall Street Journal reported in July 2022 that the company's internal share price had fallen, causing its implied valuation to drop from $95 billion to $74 billion. In November 2022, the company announced it intended to initiate layoffs, terminating some 14% of its workforce. Throughout 2022 and 2023, the company announced a number of large enterprise customers, including Airbnb, Amazon, Microsoft, Uber, BMW, Maersk, Zara, Lotus, Alaska Airlines, Le Monde, and Toyota. The company also announced in March 2023 that OpenAI is working with Stripe to commercialize its generative AI technology. In January 2025, Stripe sent layoff notices to nearly 300 workers, primarily affecting roles in Product, Operations and Engineering. The company experienced controversy when the company sent a cartoon picture of a duck to the laid-off employees. Stripe's Chief People Officer Rob McIntosh later apologized for the mistake. After re-enabling cryptocurrency pay-ins in April 2024, starting with USDC, Stripe completed the acquisition of Bridge in February 2025. The acquisition of the two-year-old stablecoin platform company is valued at $1.1 billion. In June 2025, the company acquired Privy, which powers crypto wallets. In September 2025, Stripe announced it was powering Instant Checkout in ChatGPT and released Agentic Commerce Protocol for agentic commerce, which was co-developed with OpenAI. In October 2025, the company opened its second headquarters in Dublin, Ireland. In February 2026, Stripe was valued at $159 billion in a tender offer posted for employees and shareholders. The tender offer was about a 70% increase from Stripe's previous valuation published in February 2025, where it was valued at $91.5 billion. Stripe also announced that its total volume increased to $1.9 trillion USD in 2025, a 34% increase from 2024. == Technology company == === Payment processing === Stripe provides application programming interfaces that web developers can use to integrate payment processing into their websites and mobile applications. The company introduced Stripe Connect in 2012, a multiparty payments solution that lets software developers embed payments natively into their products. In April 2018, Stripe released antifraud tools, branded "Radar", that block fraudulent transactions. The same year, it expanded its services to include a billing product for online businesses, allowing businesses to manage subscription recurring revenue and invoicing. Stripe's point-of-sale service called Terminal was made available to US users on 11 June 2019. Terminal had previously been invitation-only. Terminal is currently available in Australia, Canada, France, Germany, Ireland, the Netherlands, New Zealand, Singapore, and the United Kingdom. The service offers physical credit-card readers designed to work with Stripe. On 5 September 2019, Stripe launched a merchant cash-advance scheme called Stripe Capital. The scheme allows Stripe merchants to request an advance on future payments they expect to process through their Stripe merchant account. In June 2021, the company launched Stripe Tax, a service to allow businesses to automatically calculate and collect sales tax, VAT, and GST, initially rolling out to 30 countries and all US states. As of 2025, it has been made available in 102 countries. In May that year, Stripe introduced Payment Links, a no-code product allowing businesses to create a link to a checkout page and begin accepting payments on social platforms or direct channels. In January 2022, Stripe agreed to acquire Terminal manufacturing partner BBPOS, allowing the company to bring the hardware development of Terminal readers in-house. In February, it was announced as Apple's first partner on in-person Tap to Pay, which enables businesses to accept contactless payments using an iPhone and a partner-enabled iOS app. In May, Stripe announced Data Pipeline, a tool for Stripe users who store data with Amazon Redshift or Snowflake Data Cloud. Data Pipeline syncs Stripe data and reports with Amazon Redshift or Snowflake Data Cloud, where they can be queried in combination with other business information. That month, the company also introduced Stripe Financial Connections, enabling businesses to establish direct connections with their customers’ bank accounts to verify accounts for payments and pay-outs, check balances to reduce payment failures, and cut fraud by confirming bank account ownership. In September 2023, Stripe announced that its optimized checkout suite allowed businesses to offer their customers more than 100 payment methods. In May 2025, Stripe announced a new AI foundational model for payments, and introduced stablecoin powered accounts. === Corporate finance === In July 2018, Stripe introduced Stripe Issuing, a product that allows online businesses and platforms to create their own physical and digital credit and debit cards. === Atlas === On 14 February 2016, the company launched the Atlas platform to help start-ups register as US corporations, targeting foreign entrepreneurs. The platform was originally invitation-only. In March 2016, Cuba was added to the list of countries covered under the program. Originally, companies registered using Atlas were set up as Delaware-based C corporations. As of 30 April 2018, the option to be registered as limited liability companies was added. Companies set up using Atlas automatically had a business bank account and Stripe merchant account set up. === Link === In May 2021, Stripe launched Link, a service for saving and auto-filling payment details when paying via Stripe. The service supported payments in over 185 countries and Stripe reported plans to make it available to platform businesses through its API. In September 2025, Patrick Collison announced that Link had surpassed 200 million users. === Other === In 2018, Stripe started a publishing company named Stripe Press to promote ideas that support businesses. In 2019, Stripe began offering loans and credit cards to businesses in the United States. The company stated that loans are approved automatically using machine-learning models, with no human intervention. The following year, the company introduced Stripe Treasury, which provides its platform users APIs to embed financial services, allowing their customers to send, receive, and store funds. In October 2020, Stripe announced Stripe Climate, a service for businesses to fund atmospheric carbon research and capture. In 2022, Stripe started a new subsidiary called Frontier that would direct spending on carbon removal. It announced $925 million in funding from major Silicon Valley companies to fund start up companies performing carbon capture to kick-start the industry. Stripe Identity, launched in Ju
LIBSVM
LIBSVM and LIBLINEAR are two popular open source machine learning libraries, both developed at the National Taiwan University and both written in C++ though with a C API. LIBSVM implements the sequential minimal optimization (SMO) algorithm for kernelized support vector machines (SVMs), supporting classification and regression. LIBLINEAR implements linear SVMs and logistic regression models trained using a coordinate descent algorithm. The SVM learning code from both libraries is often reused in other open source machine learning toolkits, including GATE, KNIME, Orange and scikit-learn. Bindings and ports exist for programming languages such as Java, MATLAB, R, Julia, and Python. It is available in e1071 library in R and scikit-learn in Python. Both libraries are free software released under the 3-clause BSD license.
IDistance
In pattern recognition, iDistance is an indexing and query processing technique for k-nearest neighbor queries on point data in multi-dimensional metric spaces. The kNN query is one of the hardest problems on multi-dimensional data, especially when the dimensionality of the data is high. iDistance is designed to process kNN queries in high-dimensional spaces efficiently and performs extremely well for skewed data distributions, which usually occur in real-life data sets. iDistance employs a two-phase search strategy involving an initial filtering of candidate regions and a subsequent refinement of results, an approach aligned with the Filter and Refine Principle (FRP). This means that the index first prunes the search space to eliminate unlikely candidates, then verifies the true nearest neighbors in a refinement step, following the general FRP paradigm used in database search algorithms. The iDistance index can also be augmented with machine learning models to learn data distributions for improved searching and storage of multi-dimensional data. == Indexing == Building the iDistance index has two steps: A number of reference points in the data space are chosen. There are various ways of choosing reference points. Using cluster centers as reference points is the most efficient way. The data points are partitioned into Voronoi cells based on well-chosen reference points. The distance between a data point and its closest reference point is calculated. This distance plus a scaling value is called the point's iDistance. By this means, points in a multi-dimensional space are mapped to one-dimensional values, and then a B+-tree can be adopted to index the points using the iDistance as the key. The figure on the right shows an example where three reference points (O1, O2, O3) are chosen. The data points are then mapped to a one-dimensional space and indexed in a B+-tree. Various extensions have been proposed to make the selection of reference points for effective query performance, including employing machine learning to learn the identification of reference points. == Query processing == To process a kNN query, the query is mapped to a number of one-dimensional range queries, which can be processed efficiently on a B+-tree. In the above figure, the query Q is mapped to a value in the B+-tree while the kNN search ``sphere" is mapped to a range in the B+-tree. The search sphere expands gradually until the k NNs are found. This corresponds to gradually expanding range searches in the B+-tree. The iDistance technique can be viewed as a way of accelerating the sequential scan. Instead of scanning records from the beginning to the end of the data file, the iDistance starts the scan from spots where the nearest neighbors can be obtained early with a very high probability. == Applications == The iDistance has been used in many applications including Image retrieval Video indexing Similarity search in P2P systems Mobile computing Recommender system == Historical background == The iDistance was first proposed by Cui Yu, Beng Chin Ooi, Kian-Lee Tan and H. V. Jagadish in 2001. Later, together with Rui Zhang, they improved the technique and performed a more comprehensive study on it in 2005.
Probabilistic latent semantic analysis
Probabilistic latent semantic analysis (PLSA), also known as probabilistic latent semantic indexing (PLSI, especially in information retrieval circles) is a statistical technique for the analysis of two-mode and co-occurrence data. In effect, one can derive a low-dimensional representation of the observed variables in terms of their affinity to certain hidden variables, just as in latent semantic analysis, from which PLSA evolved. Compared to standard latent semantic analysis which stems from linear algebra and downsizes the occurrence tables (usually via a singular value decomposition), probabilistic latent semantic analysis is based on a mixture decomposition derived from a latent class model. == Model == Considering observations in the form of co-occurrences ( w , d ) {\displaystyle (w,d)} of words and documents, PLSA models the probability of each co-occurrence as a mixture of conditionally independent multinomial distributions: P ( w , d ) = ∑ c P ( d ) P ( c | d ) P ( w | c ) = P ( d ) ∑ c P ( c | d ) P ( w | c ) {\displaystyle P(w,d)=\sum _{c}P(d)P(c|d)P(w|c)=P(d)\sum _{c}P(c|d)P(w|c)} with c {\displaystyle c} being the words' topic. Note that the number of topics is a hyperparameter that must be chosen in advance and is not estimated from the data. The first formulation is the symmetric formulation, where w {\displaystyle w} and d {\displaystyle d} are both generated from the latent class c {\displaystyle c} in similar ways (using the conditional probabilities P ( d | c ) {\displaystyle P(d|c)} and P ( w | c ) {\displaystyle P(w|c)} ), whereas the second formulation is the asymmetric formulation, where, for each document d {\displaystyle d} , a latent class is chosen conditionally to the document according to P ( c | d ) {\displaystyle P(c|d)} , and a word is then generated from that class according to P ( w | c ) {\displaystyle P(w|c)} . Although we have used words and documents in this example, the co-occurrence of any couple of discrete variables may be modelled in exactly the same way. So, the number of parameters is equal to c d + w c {\displaystyle cd+wc} . The number of parameters grows linearly with the number of documents. In addition, although PLSA is a generative model of the documents in the collection it is estimated on, it is not a generative model of new documents. Their parameters are learned using the EM algorithm. == Application == PLSA may be used in a discriminative setting, via Fisher kernels. PLSA has applications in information retrieval and filtering, natural language processing, machine learning from text, bioinformatics, and related areas. It is reported that the aspect model used in the probabilistic latent semantic analysis has severe overfitting problems. == Extensions == Hierarchical extensions: Asymmetric: MASHA ("Multinomial ASymmetric Hierarchical Analysis") Symmetric: HPLSA ("Hierarchical Probabilistic Latent Semantic Analysis") Generative models: The following models have been developed to address an often-criticized shortcoming of PLSA, namely that it is not a proper generative model for new documents. Latent Dirichlet allocation – adds a Dirichlet prior on the per-document topic distribution Higher-order data: Although this is rarely discussed in the scientific literature, PLSA extends naturally to higher order data (three modes and higher), i.e. it can model co-occurrences over three or more variables. In the symmetric formulation above, this is done simply by adding conditional probability distributions for these additional variables. This is the probabilistic analogue to non-negative tensor factorisation. == History == This is an example of a latent class model (see references therein), and it is related to non-negative matrix factorization. The present terminology was coined in 1999 by Thomas Hofmann.
Deep learning
In machine learning, deep learning (DL) focuses on utilizing multilayered neural networks to perform tasks such as classification, regression, and representation learning. The field takes inspiration from biological neuroscience and revolves around stacking artificial neurons into layers and "training" them to process data. The adjective "deep" refers to the use of multiple layers (ranging from three to several hundred or thousands) in the network. Methods used can be supervised, semi-supervised or unsupervised. Some common deep learning network architectures include fully connected networks, deep belief networks, recurrent neural networks, convolutional neural networks, generative adversarial networks, transformers, and neural radiance fields. These architectures have been applied to fields including computer vision, speech recognition, natural language processing, machine translation, bioinformatics, drug design, medical image analysis, climate science, material inspection and board game programs, where they have produced results comparable to and in some cases surpassing human expert performance. Early forms of neural networks were inspired by information processing and distributed communication nodes in biological systems, particularly the human brain. However, current neural networks do not intend to model the brain function of organisms, and are generally seen as low-quality models for that purpose. == Overview == Most modern deep learning models are based on multi-layered neural networks such as convolutional neural networks and transformers, although they can also include propositional formulas or latent variables organized layer-wise in deep generative models such as the nodes in deep belief networks and deep Boltzmann machines. Fundamentally, deep learning refers to a class of machine learning algorithms in which a hierarchy of layers is used to transform input data into a progressively more abstract and composite representation. For example, in an image recognition model, the raw input may be an image (represented as a tensor of pixels). The first representational layer may attempt to identify basic shapes such as lines and circles, the second layer may compose and encode arrangements of edges, the third layer may encode a nose and eyes, and the fourth layer may recognize that the image contains a face. Importantly, a deep learning process can learn which features to optimally place at which level on its own. Prior to deep learning, machine learning techniques often involved hand-crafted feature engineering to transform the data into a more suitable representation for a classification algorithm to operate on. In the deep learning approach, features are not hand-crafted and the model discovers useful feature representations from the data automatically. This does not eliminate the need for hand-tuning; for example, varying numbers of layers and layer sizes can provide different degrees of abstraction. The word "deep" in "deep learning" refers to the number of layers through which the data is transformed. More precisely, deep learning systems have a substantial credit assignment path (CAP) depth. The CAP is the chain of transformations from input to output. CAPs describe potentially causal connections between input and output. For a feedforward neural network, the depth of the CAPs is that of the network and is the number of hidden layers plus one (as the output layer is also parameterized). For recurrent neural networks, in which a signal may propagate through a layer more than once, the CAP depth is potentially unlimited. No universally agreed-upon threshold of depth divides shallow learning from deep learning, but most researchers agree that deep learning involves CAP depth higher than two. CAP of depth two has been shown to be a universal approximator in the sense that it can emulate any function. Beyond that, more layers do not add to the function approximator ability of the network. Deep models (CAP > two) are able to extract better features than shallow models and hence, extra layers help in learning the features effectively. Deep learning architectures can be constructed with a greedy layer-by-layer method. Deep learning helps to disentangle these abstractions and pick out which features improve performance. Deep learning algorithms can be applied to unsupervised learning tasks. This is an important benefit because unlabeled data is more abundant than labeled data. Examples of deep structures that can be trained in an unsupervised manner are deep belief networks. The term deep learning was introduced to the machine learning community by Rina Dechter in 1986, and to artificial neural networks by Igor Aizenberg and colleagues in 2000, in the context of Boolean threshold neurons. The etymology of the term is more complicated. == Interpretations == Deep neural networks are generally interpreted in terms of the universal approximation theorem or probabilistic inference. The classic universal approximation theorem concerns the capacity of feedforward neural networks with a single hidden layer of finite size to approximate continuous functions. In 1989, the first proof was published by George Cybenko for sigmoid activation functions and was generalised to feed-forward multi-layer architectures in 1991 by Kurt Hornik. Recent work also showed that universal approximation also holds for non-bounded activation functions such as Kunihiko Fukushima's rectified linear unit. The universal approximation theorem for deep neural networks concerns the capacity of networks with bounded width but the depth is allowed to grow. Lu et al. proved that if the width of a deep neural network with ReLU activation is strictly larger than the input dimension, then the network can approximate any Lebesgue integrable function; if the width is smaller or equal to the input dimension, then a deep neural network is not a universal approximator. The probabilistic interpretation derives from the field of machine learning. It features inference, as well as the optimization concepts of training and testing, related to fitting and generalization, respectively. More specifically, the probabilistic interpretation considers the activation nonlinearity as a cumulative distribution function. The probabilistic interpretation led to the introduction of dropout as regularizer in neural networks. The probabilistic interpretation was introduced by researchers including Hopfield, Widrow and Narendra and popularized in surveys such as the one by Bishop. == History == === Before 1980 === There are two types of artificial neural network (ANN): feedforward neural network (FNN) or multilayer perceptron (MLP) and recurrent neural networks (RNN). RNNs have cycles in their connectivity structure, whereas FNNs do not. In the 1920s, Wilhelm Lenz and Ernst Ising created the Ising model which is essentially a non-learning RNN architecture consisting of neuron-like threshold elements. In 1972, Shun'ichi Amari made this architecture adaptive. His learning RNN was republished by John Hopfield in 1982. Other early recurrent neural networks were published by Kaoru Nakano in 1971. Already in 1948, Alan Turing produced work on "Intelligent Machinery" that was not published in his lifetime, containing "ideas related to artificial evolution and learning RNNs". Frank Rosenblatt (1958) proposed the perceptron, an MLP with 3 layers: an input layer, a hidden layer with randomized weights that did not learn, and an output layer. He later published a 1962 book that also introduced variants and computer experiments, including a version with four-layer perceptrons "with adaptive preterminal networks" where the last two layers have learned weights (here he credits H. D. Block and B. W. Knight). The book cites an earlier network by R. D. Joseph (1960) "functionally equivalent to a variation of" this four-layer system (the book mentions Joseph over 30 times). Should Joseph therefore be considered the originator of proper adaptive multilayer perceptrons with learning hidden units? Unfortunately, the learning algorithm was not a functional one, and fell into oblivion. The first working deep learning algorithm was the Group method of data handling, a method to train arbitrarily deep neural networks, published by Alexey Ivakhnenko and Lapa in 1965. They regarded it as a form of polynomial regression, or a generalization of Rosenblatt's perceptron to handle more complex, nonlinear, and hierarchical relationships. A 1971 paper described a deep network with eight layers trained by this method, which is based on layer by layer training through regression analysis. Superfluous hidden units are pruned using a separate validation set. Since the activation functions of the nodes are Kolmogorov-Gabor polynomials, these were also the first deep networks with multiplicative units or "gates". The first deep learning multilayer perceptron trained by stochastic gradient descent was published in 1967 by Shun'ichi
Linear discriminant analysis
Linear discriminant analysis (LDA), normal discriminant analysis (NDA), canonical variates analysis (CVA), or discriminant function analysis is a generalization of Fisher's linear discriminant, a method used in statistics and other fields, to find a linear combination of features that characterizes or separates two or more classes of objects or events. The resulting combination may be used as a linear classifier, or, more commonly, for dimensionality reduction before later classification. LDA is closely related to analysis of variance (ANOVA) and regression analysis, which also attempt to express one dependent variable as a linear combination of other features or measurements. However, ANOVA uses categorical independent variables and a continuous dependent variable, whereas discriminant analysis has continuous independent variables and a categorical dependent variable (i.e. the class label). Logistic regression and probit regression are more similar to LDA than ANOVA is, as they also explain a categorical variable by the values of continuous independent variables. These other methods are preferable in applications where it is not reasonable to assume that the independent variables have a normal distribution, which is a fundamental assumption of the LDA method. LDA is also closely related to principal component analysis (PCA) and factor analysis in that they both look for linear combinations of variables which best explain the data. LDA explicitly attempts to model the difference between the classes of data. PCA, in contrast, does not take into account any difference in class, and factor analysis builds the feature combinations based on similarities rather than differences. Discriminant analysis is also different from factor analysis in that it is not an interdependence technique: a distinction between independent variables and dependent variables (also called criterion variables) must be made. LDA works when the measurements made on independent variables for each observation are continuous quantities. When dealing with categorical independent variables, the equivalent technique is discriminant correspondence analysis. Discriminant analysis is used when groups are known a priori (unlike in cluster analysis). Each case must have a score on one or more quantitative predictor measures, and a score on a group measure. In simple terms, discriminant function analysis is classification - the act of distributing things into groups, classes or categories of the same type. == History == The original dichotomous discriminant analysis was developed by Sir Ronald Fisher in 1936. It is different from an ANOVA or MANOVA, which is used to predict one (ANOVA) or multiple (MANOVA) continuous dependent variables by one or more independent categorical variables. Discriminant function analysis is useful in determining whether a set of variables is effective in predicting category membership. == LDA for two classes == Consider a set of observations x → {\displaystyle {\vec {x}}} (also called features, attributes, variables or measurements) for each sample of an object or event with known class y {\displaystyle y} . This set of samples is called the training set in a supervised learning context. The classification problem is then to find a good predictor for the class y {\displaystyle y} of any sample of the same distribution (not necessarily from the training set) given only an observation x → {\displaystyle {\vec {x}}} . LDA approaches the problem by assuming that the conditional probability density functions p ( x → | y = 0 ) {\displaystyle p({\vec {x}}|y=0)} and p ( x → | y = 1 ) {\displaystyle p({\vec {x}}|y=1)} are both the normal distribution with mean and covariance parameters ( μ → 0 , Σ 0 ) {\displaystyle \left({\vec {\mu }}_{0},\Sigma _{0}\right)} and ( μ → 1 , Σ 1 ) {\displaystyle \left({\vec {\mu }}_{1},\Sigma _{1}\right)} , respectively. Under this assumption, the Bayes-optimal solution is to predict points as being from the second class if the log of the likelihood ratios is bigger than some threshold T, so that: 1 2 ( x → − μ → 0 ) T Σ 0 − 1 ( x → − μ → 0 ) + 1 2 ln | Σ 0 | − 1 2 ( x → − μ → 1 ) T Σ 1 − 1 ( x → − μ → 1 ) − 1 2 ln | Σ 1 | > T {\displaystyle {\frac {1}{2}}({\vec {x}}-{\vec {\mu }}_{0})^{\mathrm {T} }\Sigma _{0}^{-1}({\vec {x}}-{\vec {\mu }}_{0})+{\frac {1}{2}}\ln |\Sigma _{0}|-{\frac {1}{2}}({\vec {x}}-{\vec {\mu }}_{1})^{\mathrm {T} }\Sigma _{1}^{-1}({\vec {x}}-{\vec {\mu }}_{1})-{\frac {1}{2}}\ln |\Sigma _{1}|\ >\ T} Without any further assumptions, the resulting classifier is referred to as quadratic discriminant analysis (QDA). LDA instead makes the additional simplifying homoscedasticity assumption (i.e. that the class covariances are identical, so Σ 0 = Σ 1 = Σ {\displaystyle \Sigma _{0}=\Sigma _{1}=\Sigma } ) and that the covariances have full rank. In this case, several terms cancel: x → T Σ 0 − 1 x → = x → T Σ 1 − 1 x → {\displaystyle {\vec {x}}^{\mathrm {T} }\Sigma _{0}^{-1}{\vec {x}}={\vec {x}}^{\mathrm {T} }\Sigma _{1}^{-1}{\vec {x}}} x → T Σ i − 1 μ → i = μ → i T Σ i − 1 x → {\displaystyle {\vec {x}}^{\mathrm {T} }{\Sigma _{i}}^{-1}{\vec {\mu }}_{i}={{\vec {\mu }}_{i}}^{\mathrm {T} }{\Sigma _{i}}^{-1}{\vec {x}}} because both sides are scalar and transpose to each other ( Σ i {\displaystyle \Sigma _{i}} is Hermitian) and the above decision criterion becomes a threshold on the dot product w → T x → > c {\displaystyle {\vec {w}}^{\mathrm {T} }{\vec {x}}>c} for some threshold constant c, where w → = Σ − 1 ( μ → 1 − μ → 0 ) {\displaystyle {\vec {w}}=\Sigma ^{-1}({\vec {\mu }}_{1}-{\vec {\mu }}_{0})} c = 1 2 w → T ( μ → 1 + μ → 0 ) {\displaystyle c={\frac {1}{2}}\,{\vec {w}}^{\mathrm {T} }({\vec {\mu }}_{1}+{\vec {\mu }}_{0})} This means that the criterion of an input x → {\displaystyle {\vec {x}}} being in a class y {\displaystyle y} is purely a function of this linear combination of the known observations. It is often useful to see this conclusion in geometrical terms: the criterion of an input x → {\displaystyle {\vec {x}}} being in a class y {\displaystyle y} is purely a function of projection of multidimensional-space point x → {\displaystyle {\vec {x}}} onto vector w → {\displaystyle {\vec {w}}} (thus, we only consider its direction). In other words, the observation belongs to y {\displaystyle y} if corresponding x → {\displaystyle {\vec {x}}} is located on a certain side of a hyperplane perpendicular to w → {\displaystyle {\vec {w}}} . The location of the plane is defined by the threshold c {\displaystyle c} . == Assumptions == The assumptions of discriminant analysis are the same as those for MANOVA. The analysis is quite sensitive to outliers and the size of the smallest group must be larger than the number of predictor variables. Multivariate normality: Independent variables are normal for each level of the grouping variable. Homogeneity of variance/covariance (homoscedasticity): Variances among group variables are the same across levels of predictors. Can be tested with Box's M statistic. It has been suggested, however, that linear discriminant analysis be used when covariances are equal, and that quadratic discriminant analysis may be used when covariances are not equal. Independence: Participants are assumed to be randomly sampled, and a participant's score on one variable is assumed to be independent of scores on that variable for all other participants. It has been suggested that discriminant analysis is relatively robust to slight violations of these assumptions, and it has also been shown that discriminant analysis may still be reliable when using dichotomous variables (where multivariate normality is often violated). == Discriminant functions == Discriminant analysis works by creating one or more linear combinations of predictors, creating a new latent variable for each function. These functions are called discriminant functions. The number of functions possible is either N g − 1 {\displaystyle N_{g}-1} where N g {\displaystyle N_{g}} = number of groups, or p {\displaystyle p} (the number of predictors), whichever is smaller. The first function created maximizes the differences between groups on that function. The second function maximizes differences on that function, but also must not be correlated with the previous function. This continues with subsequent functions with the requirement that the new function not be correlated with any of the previous functions. Given group j {\displaystyle j} , with R j {\displaystyle \mathbb {R} _{j}} sets of sample space, there is a discriminant rule such that if x ∈ R j {\displaystyle x\in \mathbb {R} _{j}} , then x ∈ j {\displaystyle x\in j} . Discriminant analysis then, finds “good” regions of R j {\displaystyle \mathbb {R} _{j}} to minimize classification error, therefore leading to a high percent correct classified in the classification table. Each function is given a discriminant score to determine how well it predicts group placement. Structure Corr