The Lucy–Hook coaddition method is an image processing technique for combining sub-stepped astronomical image data onto a finer grid. The method allows the option of resolution and contrast enhancement or the choice of a conservative, re-convolved, output. Tests with very deep Hubble Space Telescope Wide Field and Planetary Camera 2 (WFPC2) imaging data of excellent quality show that these methods can be very effective and allow fine-scale features to be studied better than on the unprocessed images. The Lucy–Hook coaddition method is an extension of the standard Richardson–Lucy deconvolution iterative restoration method. For many purposes it may be more convenient to combine dithered datasets using the Drizzle method.
Color moments
Color moments are measures that characterise color distribution in an image in the same way that central moments uniquely describe a probability distribution. Color moments are mainly used for color indexing purposes as features in image retrieval applications in order to compare how similar two images are based on color. Usually one image is compared to a database of digital images with pre-computed features in order to find and retrieve a similar Image. Each comparison between images results in a similarity score, and the lower this score is the more identical the two images are supposed to be. == Overview == Color moments are scaling and rotation invariant. It is usually the case that only the first three color moments are used as features in image retrieval applications as most of the color distribution information is contained in the low-order moments. Since color moments encode both shape and color information they are a good feature to use under changing lighting conditions, but they cannot handle occlusion very successfully. Color moments can be computed for any color model. Three color moments are computed per channel (e.g. 9 moments if the color model is RGB and 12 moments if the color model is CMYK). Computing color moments is done in the same way as computing moments of a probability distribution. === Mean === The first color moment can be interpreted as the average color in the image, and it can be calculated by using the following formula E i = ∑ j = 1 N 1 N p i j {\displaystyle E_{i}=\textstyle \sum _{j=1}^{N}{\frac {1}{N}}p_{ij}} where N is the number of pixels in the image and p i j {\displaystyle p_{ij}} is the value of the j-th pixel of the image at the i-th color channel. === Standard Deviation === The second color moment is the standard deviation, which is obtained by taking the square root of the variance of the color distribution. σ i = ( 1 N ∑ j = 1 N ( p i j − E i ) 2 ) {\displaystyle \sigma _{i}={\sqrt {({\frac {1}{N}}\textstyle \sum _{j=1}^{N}(p_{ij}-E_{i})^{2})}}} where E i {\displaystyle E_{i}} is the mean value, or first color moment, for the i-th color channel of the image. === Skewness === The third color moment is the skewness. It measures how asymmetric the color distribution is, and thus it gives information about the shape of the color distribution. Skewness can be computed with the following formula: s i = ( 1 N ∑ j = 1 N ( p i j − E i ) 3 ) 3 σ i {\displaystyle s_{i}={\frac {\sqrt[{3}]{\left({\frac {1}{N}}\textstyle \sum _{j=1}^{N}(p_{ij}-E_{i})^{3}\right)}}{\sigma _{i}}}} === Kurtosis === Kurtosis is the fourth color moment, and, similarly to skewness, it provides information about the shape of the color distribution. More specifically, kurtosis is a measure of how extreme the tails are in comparison to the normal distribution. === Higher-order color moments === Higher-order color moments are usually not part of the color moments feature set in image retrieval tasks as they require more data in order to obtain a good estimate of their value, and also the lower-order moments generally provide enough information. == Applications == Color moments have significant applications in image retrieval. They can be used in order to compare how similar two images are. This is a relatively new approach to color indexing. The greatest advantage of using color moments comes from the fact that there is no need to store the complete color distribution. This greatly speeds up image retrieval since there are less features to compare. In addition, the first three color moments have the same units, which allows for comparison between them. === Color indexing === Color indexing is the main application of color moments. Images can be indexed, and the index will contain the computed color moments. Then, if someone has a particular image and wants to find similar images in the database, the color moments of the image of interest will also be computed. After that the following function will be used in order to compute a similarity score between the image of interest and all the images in the database: d m o m ( H , I ) = ∑ i = 1 r w i 1 | E i 1 − E i 2 | + w i 2 | σ i 1 − σ i 2 | + w i 3 | s i 1 − s i 2 | {\displaystyle d_{mom}(H,I)=\textstyle \sum _{i=1}^{r}w_{i1}|E_{i}^{1}-E_{i}^{2}|+w_{i2}|\sigma _{i}^{1}-\sigma _{i}^{2}|+w_{i3}|s_{i}^{1}-s_{i}^{2}|} where: H and I are the color distributions of the two images that are being compared i is the channel index and r is the total number of channels E i 1 {\displaystyle E_{i}^{1}} and E i 2 {\displaystyle E_{i}^{2}} are the first order moments computed for the image distributions. σ i 1 {\displaystyle \sigma _{i}^{1}} and σ i 2 {\displaystyle \sigma _{i}^{2}} are the second order moments computed for the image distributions. s_i^1 and s_i^2 are the third order moments computed for the image distributions. w i 1 {\displaystyle w_{i1}} , w i 2 {\displaystyle w_{i2}} , and w i 3 {\displaystyle w_{i3}} are weights, specified by the user, for each of the three color moments used. Finally, the images in the database will be ranked according to the computed similarity score with the image of interest, and the database images with the lowest d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} value should be retrieved. "A retrieval based on d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} may produce false positives because the index contains no information about the correlation between the color channels". == Example == A simple and concise example of the use of color moments for image retrieval tasks is illustrated in. Consider having several test images in a database and a "New Image". The goal is to retrieve images from the database that are similar to the "New Image". The first three color moments are used as features. There are several steps in this computation. Image preprocessing (Optional) - The image preprocessing step of the computation process is optional. For example, in this step all images could be modified to be the same size (in terms of pixels). However, since color moments are invariant to scaling, it is not necessary to make all images the same width and height. Computing the features - Use the color moments formulae in order to compute the first three moments for each of the color channels in the image. For example, if the HSV color space is used, this means that for each of the images, 9 features in total will be computed (the first three order moments for the Hue, Saturation, and Value channels). Calculating the similarity score - After computing the color moments the weights for each of the moments in the d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} function should be determined by the user. The weights have to be adjusted each time in accordance with the application or condition and quality of the images. Following that the d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} function is used to calculate a similarity score for the "New Image" and each of the images in the database. Ranking and image retrieval - From the previous step the d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} values were obtained. Now a comparison of these values can be made in order to decide which of the images in the database are more similar to the "New Image", and thus rank the database images accordingly. The smaller the d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} value is the more similar the two color distributions are supposed to be. Finally, some of the top ranked images (the ones with the smallest d m o m ( H , I ) {\displaystyle d_{mom}(H,I)} value) from the database are retrieved.
NOMINATE (scaling method)
NOMINATE (an acronym for nominal three-step estimation) is a multidimensional scaling application developed by US political scientists Keith T. Poole and Howard Rosenthal in the early 1980s to analyze preferential and choice data, such as legislative roll-call voting behavior. In its most well-known application, members of the US Congress are placed on a two-dimensional map, with politicians who are ideologically similar (i.e. who often vote the same) being close together. One of these two dimensions corresponds to the familiar left–right political spectrum (liberal–conservative in the United States). As computing capabilities grew, Poole and Rosenthal developed multiple iterations of their NOMINATE procedure: the original D-NOMINATE method, W-NOMINATE, and most recently DW-NOMINATE (for dynamic, weighted NOMINATE). In 2009, Poole and Rosenthal were the first recipients of the Society for Political Methodology's Best Statistical Software Award for their development of NOMINATE. In 2016, the society awarded Poole its Career Achievement Award, stating that "the modern study of the U.S. Congress would be simply unthinkable without NOMINATE legislative roll call voting scores." == Procedure == The main procedure is an application of multidimensional scaling techniques to political choice data. Though there are important technical differences between these types of NOMINATE scaling procedures, all operate under the same fundamental assumptions. First, that alternative choices can be projected on a basic, low-dimensional (often two-dimensional) Euclidean space. Second, within that space, individuals have utility functions which are bell-shaped (normally distributed), and maximized at their ideal point. Because individuals also have symmetric, single-peaked utility functions which center on their ideal point, ideal points represent individuals' most preferred outcomes. That is, individuals most desire outcomes closest their ideal point, and will choose/vote probabilistically for the closest outcome. Ideal points can be recovered from observing choices, with individuals exhibiting similar preferences placed more closely than those behaving dissimilarly. It is helpful to compare this procedure to producing maps based on driving distances between cities. For example, Los Angeles is about 1,800 miles from St. Louis; St. Louis is about 1,200 miles from Miami; and Miami is about 2,700 miles from Los Angeles. From this (dis)similarities data, any map of these three cities should place Miami far from Los Angeles, with St. Louis somewhere in between (though a bit closer to Miami than Los Angeles). Just as cities like Los Angeles and San Francisco would be clustered on a map, NOMINATE places ideologically similar legislators (e.g., liberal Senators Barbara Boxer (D-Calif.) and Al Franken (D-Minn.)) closer to each other, and farther from dissimilar legislators (e.g., conservative Senator Tom Coburn (R-Okla.)) based on the degree of agreement between their roll call voting records. At the heart of the NOMINATE procedures (and other multidimensional scaling methods, such as Poole's Optimal Classification method) are algorithms they utilize to arrange individuals and choices in low dimensional (usually two-dimensional) space. Thus, NOMINATE scores provide "maps" of legislatures. Using NOMINATE procedures to study congressional roll call voting behavior from the First Congress to the present-day, Poole and Rosenthal published Congress: A Political-Economic History of Roll Call Voting in 1997 and the revised edition Ideology and Congress in 2007. In 2009, Poole and Rosenthal were named the first recipients of the Society for Political Methodology's Best Statistical Software Award for their development of NOMINATE, a recognition conferred to "individual(s) for developing statistical software that makes a significant research contribution". In 2016, Keith T. Poole was awarded the Society for Political Methodology's Career Achievement Award. The citation for this award reads, in part, "One can say perfectly correctly, and without any hyperbole: the modern study of the U.S. Congress would be simply unthinkable without NOMINATE legislative roll call voting scores. NOMINATE has produced data that entire bodies of our discipline—and many in the press—have relied on to understand the U.S. Congress." == Dimensions == Poole and Rosenthal demonstrate that—despite the many complexities of congressional representation and politics—roll call voting in both the House and the Senate can be organized and explained by no more than two dimensions throughout the sweep of American history. The first dimension (horizontal or x-axis) is the familiar left-right (or liberal-conservative) spectrum on economic matters. The second dimension (vertical or y-axis) picks up attitudes on cross-cutting, salient issues of the day (which include or have included slavery, bimetallism, civil rights, regional, and social/lifestyle issues). Rosenthal and Poole have initially argued that the first dimension refers to socio-economic matters and the second dimension to race-relations. However, the often confusing and residual nature of the second dimension has led to the second dimension being largely ignored by other researchers. For the most part, congressional voting is uni-dimensional, with most of the variation in voting patterns explained by placement along the liberal-conservative first dimension. While the first dimension of the DW-NOMINATE score is able to predict results at 83% accuracy, the addition of the second dimension only increases accuracy to 85%. Furthermore, the second dimension only provided a significant increase in accuracy for Congresses 1-99. As late as the 1990s, the second dimension was able to measure partisan splits in abortion and gun rights issues. However, a 2017 analysis found that since 1987, the votes of the US Congress had best fit a one-dimensional model, suggesting increasing party polarization after 1987. == Interpretation of nominate scores == For illustrative purposes, consider the following plots which use W-NOMINATE scores to scale members of Congress and uses the probabilistic voting model (in which legislators farther from the "cutting line" between "yea" and "nay" outcomes become more likely to vote in the predicted manner) to illustrate some major Congressional votes in the 1990s. Some of these votes, like the House's vote on President Clinton's welfare reform package (the Personal Responsibility and Work Opportunity Act of 1996) are best modeled through the use of the first (economic liberal-conservative) dimension. On the welfare reform vote, nearly all Republicans joined the moderate-conservative bloc of House Democrats in voting for the bill, while opposition was virtually confined to the most liberal Democrats in the House. The errors (those representatives on the "wrong" side of the cutting line which separates predicted "yeas" and predicted "nays") are generally close to the cutting line, which is what we would expect. A legislator directly on the cutting line is indifferent between voting "yea" and "nay" on the measure. All members are shown on the left panel of the plot, while only errors are shown on the right panel: Economic ideology also dominates the Senate vote on the Balanced Budget Amendment of 1995: On other votes, however, a second dimension (which has recently come to represent attitudes on cultural and lifestyle issues) is important. For example, roll call votes on gun control routinely split party coalitions, with socially conservative "blue dog" Democrats joining most Republicans in opposing additional regulation and socially liberal Republicans joining most Democrats in supporting gun control. The addition of the second dimension accounts for these inter-party differences, and the cutting line is more horizontal than vertical (meaning the cleavage is found on the second dimension rather than the first dimension on these votes) This pattern was evident in the 1991 House vote to require waiting periods on handguns: == Political ideology == DW-NOMINATE scores have been used widely to describe the political ideology of political actors, political parties and political institutions. For instance, a score in the first dimension that is close to either pole means that such score is located at one of the extremes in the liberal-conservative scale. So, a score closer to 1 is described as conservative whereas a score closer to −1 can be described as liberal. Finally, a score at zero or close to zero is described as moderate. == Political polarization == Poole and Rosenthal (beginning with their 1984 article "The Polarization of American Politics") have also used NOMINATE data to show that, since the 1970s, party delegations in Congress have become ideologically homogeneous and distant from one another (a phenomenon known as "polarization"). Using DW-NOMINATE scores (which permit direct comparisons between members of different Congress
Algorithmic learning theory
Algorithmic learning theory is a mathematical framework for analyzing machine learning problems and algorithms. Synonyms include formal learning theory and algorithmic inductive inference. Algorithmic learning theory is different from statistical learning theory in that it does not make use of statistical assumptions and analysis. Both algorithmic and statistical learning theory are concerned with machine learning and can thus be viewed as branches of computational learning theory. == Distinguishing characteristics == Unlike statistical learning theory and most statistical theory in general, algorithmic learning theory does not assume that data are random samples, that is, that data points are independent of each other. This makes the theory suitable for domains where observations are (relatively) noise-free but not random, such as language learning and automated scientific discovery. The fundamental concept of algorithmic learning theory is learning in the limit: as the number of data points increases, a learning algorithm should converge to a correct hypothesis on every possible data sequence consistent with the problem space. This is a non-probabilistic version of statistical consistency, which also requires convergence to a correct model in the limit, but allows a learner to fail on data sequences with probability measure 0 . Algorithmic learning theory investigates the learning power of Turing machines. Other frameworks consider a much more restricted class of learning algorithms than Turing machines, for example, learners that compute hypotheses more quickly, for instance in polynomial time. An example of such a framework is probably approximately correct learning . == Learning in the limit == The concept was introduced in E. Mark Gold's seminal paper "Language identification in the limit". The objective of language identification is for a machine running one program to be capable of developing another program by which any given sentence can be tested to determine whether it is "grammatical" or "ungrammatical". The language being learned need not be English or any other natural language - in fact the definition of "grammatical" can be absolutely anything known to the tester. In Gold's learning model, the tester gives the learner an example sentence at each step, and the learner responds with a hypothesis, which is a suggested program to determine grammatical correctness. It is required of the tester that every possible sentence (grammatical or not) appears in the list eventually, but no particular order is required. It is required of the learner that at each step the hypothesis must be correct for all the sentences so far. A particular learner is said to be able to "learn a language in the limit" if there is a certain number of steps beyond which its hypothesis no longer changes. At this point it has indeed learned the language, because every possible sentence appears somewhere in the sequence of inputs (past or future), and the hypothesis is correct for all inputs (past or future), so the hypothesis is correct for every sentence. The learner is not required to be able to tell when it has reached a correct hypothesis, all that is required is that it be true. Gold showed that any language which is defined by a Turing machine program can be learned in the limit by another Turing-complete machine using enumeration. This is done by the learner testing all possible Turing machine programs in turn until one is found which is correct so far - this forms the hypothesis for the current step. Eventually, the correct program will be reached, after which the hypothesis will never change again (but note that the learner does not know that it won't need to change). Gold also showed that if the learner is given only positive examples (that is, only grammatical sentences appear in the input, not ungrammatical sentences), then the language can only be guaranteed to be learned in the limit if there are only a finite number of possible sentences in the language (this is possible if, for example, sentences are known to be of limited length). Language identification in the limit is a highly abstract model. It does not allow for limits of runtime or computer memory which can occur in practice, and the enumeration method may fail if there are errors in the input. However the framework is very powerful, because if these strict conditions are maintained, it allows the learning of any program known to be computable. This is because a Turing machine program can be written to mimic any program in any conventional programming language. See Church-Turing thesis. == Other identification criteria == Learning theorists have investigated other learning criteria, such as the following. Efficiency: minimizing the number of data points required before convergence to a correct hypothesis. Mind Changes: minimizing the number of hypothesis changes that occur before convergence. Mind change bounds are closely related to mistake bounds that are studied in statistical learning theory. Kevin Kelly has suggested that minimizing mind changes is closely related to choosing maximally simple hypotheses in the sense of Occam’s Razor. == Annual conference == Since 1990, there is an International Conference on Algorithmic Learning Theory (ALT), called Workshop in its first years (1990–1997). Between 1992 and 2016, proceedings were published in the LNCS series. Starting from 2017, they are published by the Proceedings of Machine Learning Research. The 34th conference will be held in Singapore in Feb 2023. The topics of the conference cover all of theoretical machine learning, including statistical and computational learning theory, online learning, active learning, reinforcement learning, and deep learning.
NSynth
NSynth (a portmanteau of "Neural Synthesis") is a WaveNet-based autoencoder for synthesizing audio, outlined in a paper in April 2017. == Overview == The model generates sounds through a neural network based synthesis, employing a WaveNet-style autoencoder to learn its own temporal embeddings from four different sounds. Google then released an open source hardware interface for the algorithm called NSynth Super, used by notable musicians such as Grimes and YACHT to generate experimental music using artificial intelligence. The research and development of the algorithm was part of a collaboration between Google Brain, Magenta and DeepMind. == Technology == === Dataset === The NSynth dataset is composed of 305,979 one-shot instrumental notes featuring a unique pitch, timbre, and envelope, sampled from 1,006 instruments from commercial sample libraries. For each instrument the dataset contains four-second 16 kHz audio snippets by ranging over every pitch of a standard MIDI piano, as well as five different velocities. The dataset is made available under a Creative Commons Attribution 4.0 International (CC BY 4.0) license. === Machine learning model === A spectral autoencoder model and a WaveNet autoencoder model are publicly available on GitHub. The baseline model uses a spectrogram with fft_size 1024 and hop_size 256, MSE loss on the magnitudes, and the Griffin-Lim algorithm for reconstruction. The WaveNet model trains on mu-law encoded waveform chunks of size 6144. It learns embeddings with 16 dimensions that are downsampled by 512 in time. == NSynth Super == In 2018 Google released a hardware interface for the NSynth algorithm, called NSynth Super, designed to provide an accessible physical interface to the algorithm for musicians to use in their artistic production. Design files, source code and internal components are released under an open source Apache License 2.0, enabling hobbyists and musicians to freely build and use the instrument. At the core of the NSynth Super there is a Raspberry Pi, extended with a custom printed circuit board to accommodate the interface elements. == Influence == Despite not being publicly available as a commercial product, NSynth Super has been used by notable artists, including Grimes and YACHT. Grimes reported using the instrument in her 2020 studio album Miss Anthropocene. YACHT announced an extensive use of NSynth Super in their album Chain Tripping. Claire L. Evans compared the potential influence of the instrument to the Roland TR-808. The NSynth Super design was honored with a D&AD Yellow Pencil award in 2018.
VistaCreate
VistaCreate (formerly Crello) is an online graphic design platform for non-designers, launched in 2016. As of 2022, it has more than 10 million users in 192 countries. == Overview == VistaCreate (then known as Crello) was launched in 2016 as a part of Depositphotos. In 2019, the product hit a milestone of 1 million registered users and also launched mobile apps. In 2020, the library of templates and objects became free. A music library and a background remover tool were added to the platform. In May 2021, Moufflons Basketball, in collaboration with VistaCreate, organized a poster design competition in support of gender equality in sports. In October 2021, Vistaprint acquired Crello and its parent company, Depositphotos, for a total price of $85 million. After the acquisition, Crello was rebranded to VistaCreate. Along with Vistaprint and 99designs, it became part of the new Vista parent brand. After Russia started a full-scale war on the territory of Ukraine in February 2022, VistaCreate suspended all business in Russia and Belarus. VistaCreate's team and Depositphotos gathered collections of images and templates dedicated to the war in Ukraine.
Non-negative matrix factorization
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms or muscular activity, non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically. NMF finds applications in such fields as astronomy, computer vision, document clustering, missing data imputation, chemometrics, audio signal processing, recommender systems, and bioinformatics. == History == In chemometrics non-negative matrix factorization has a long history under the name "self modeling curve resolution". In this framework the vectors in the right matrix are continuous curves rather than discrete vectors. Also early work on non-negative matrix factorizations was performed by a Finnish group of researchers in the 1990s under the name positive matrix factorization. It became more widely known as non-negative matrix factorization after Lee and Seung investigated the properties of the algorithm and published some simple and useful algorithms for two types of factorizations. == Background == Let matrix V be the product of the matrices W and H, V = W H . {\displaystyle \mathbf {V} =\mathbf {W} \mathbf {H} \,.} Matrix multiplication can be implemented as computing the column vectors of V as linear combinations of the column vectors in W using coefficients supplied by columns of H. That is, each column of V can be computed as follows: v i = W h i , {\displaystyle \mathbf {v} _{i}=\mathbf {W} \mathbf {h} _{i}\,,} where vi is the i-th column vector of the product matrix V and hi is the i-th column vector of the matrix H. When multiplying matrices, the dimensions of the factor matrices may be significantly lower than those of the product matrix and it is this property that forms the basis of NMF. NMF generates factors with significantly reduced dimensions compared to the original matrix. For example, if V is an m × n matrix, W is an m × p matrix, and H is a p × n matrix then p can be significantly less than both m and n. Here is an example based on a text-mining application: Let the input matrix (the matrix to be factored) be V with 10000 rows and 500 columns where words are in rows and documents are in columns. That is, we have 500 documents indexed by 10000 words. It follows that a column vector v in V represents a document. Assume we ask the algorithm to find 10 features in order to generate a features matrix W with 10000 rows and 10 columns and a coefficients matrix H with 10 rows and 500 columns. The product of W and H is a matrix with 10000 rows and 500 columns, the same shape as the input matrix V and, if the factorization worked, it is a reasonable approximation to the input matrix V. From the treatment of matrix multiplication above it follows that each column in the product matrix WH is a linear combination of the 10 column vectors in the features matrix W with coefficients supplied by the coefficients matrix H. This last point is the basis of NMF because we can consider each original document in our example as being built from a small set of hidden features. NMF generates these features. It is useful to think of each feature (column vector) in the features matrix W as a document archetype comprising a set of words where each word's cell value defines the word's rank in the feature: The higher a word's cell value the higher the word's rank in the feature. A column in the coefficients matrix H represents an original document with a cell value defining the document's rank for a feature. We can now reconstruct a document (column vector) from our input matrix by a linear combination of our features (column vectors in W) where each feature is weighted by the feature's cell value from the document's column in H. == Clustering property == NMF has an inherent clustering property, i.e., it automatically clusters the columns of input data V = ( v 1 , … , v n ) {\displaystyle \mathbf {V} =(v_{1},\dots ,v_{n})} . More specifically, the approximation of V {\displaystyle \mathbf {V} } by V ≃ W H {\displaystyle \mathbf {V} \simeq \mathbf {W} \mathbf {H} } is achieved by finding W {\displaystyle W} and H {\displaystyle H} that minimize the error function (using the Frobenius norm) ‖ V − W H ‖ F , {\displaystyle \left\|V-WH\right\|_{F},} subject to W ≥ 0 , H ≥ 0. {\displaystyle W\geq 0,H\geq 0.} , If we furthermore impose an orthogonality constraint on H {\displaystyle \mathbf {H} } , i.e. H H T = I {\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} , then the above minimization is mathematically equivalent to the minimization of K-means clustering. Furthermore, the computed H {\displaystyle H} gives the cluster membership, i.e., if H k j > H i j {\displaystyle \mathbf {H} _{kj}>\mathbf {H} _{ij}} for all i ≠ k, this suggests that the input data v j {\displaystyle v_{j}} belongs to k {\displaystyle k} -th cluster. The computed W {\displaystyle W} gives the cluster centroids, i.e., the k {\displaystyle k} -th column gives the cluster centroid of k {\displaystyle k} -th cluster. This centroid's representation can be significantly enhanced by convex NMF. When the orthogonality constraint H H T = I {\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} is not explicitly imposed, the orthogonality holds to a large extent, and the clustering property holds too. When the error function to be used is Kullback–Leibler divergence, NMF is identical to the probabilistic latent semantic analysis (PLSA), a popular document clustering method. == Types == === Approximate non-negative matrix factorization === Usually the number of columns of W and the number of rows of H in NMF are selected so the product WH will become an approximation to V. The full decomposition of V then amounts to the two non-negative matrices W and H as well as a residual U, such that: V = WH + U. The elements of the residual matrix can either be negative or positive. When W and H are smaller than V they become easier to store and manipulate. Another reason for factorizing V into smaller matrices W and H, is that if one's goal is to approximately represent the elements of V by significantly less data, then one has to infer some latent structure in the data. === Convex non-negative matrix factorization === In standard NMF, matrix factor W ∈ R+m × k， i.e., W can be anything in that space. Convex NMF restricts the columns of W to convex combinations of the input data vectors ( v 1 , … , v n ) {\displaystyle (v_{1},\dots ,v_{n})} . This greatly improves the quality of data representation of W. Furthermore, the resulting matrix factor H becomes more sparse and orthogonal. === Nonnegative rank factorization === In case the nonnegative rank of V is equal to its actual rank, V = WH is called a nonnegative rank factorization (NRF). The problem of finding the NRF of V, if it exists, is known to be NP-hard. === Different cost functions and regularizations === There are different types of non-negative matrix factorizations. The different types arise from using different cost functions for measuring the divergence between V and WH and possibly by regularization of the W and/or H matrices. Two simple divergence functions studied by Lee and Seung are the squared error (or Frobenius norm) and an extension of the Kullback–Leibler divergence to positive matrices (the original Kullback–Leibler divergence is defined on probability distributions). Each divergence leads to a different NMF algorithm, usually minimizing the divergence using iterative update rules. The factorization problem in the squared error version of NMF may be stated as: Given a matrix V {\displaystyle \mathbf {V} } find nonnegative matrices W and H that minimize the function F ( W , H ) = ‖ V − W H ‖ F 2 {\displaystyle F(\mathbf {W} ,\mathbf {H} )=\left\|\mathbf {V} -\mathbf {WH} \right\|_{F}^{2}} Another type of NMF for images is based on the total variation norm. When L1 regularization (akin to Lasso) is added to NMF with the mean squared error cost function, the resulting problem may be called non-negative sparse coding due to the similarity to the sparse coding problem, although it may also still be referred to as NMF. === Online NMF === Many standard NMF algorithms analyze all the data together; i.e., the whole matrix is available from the start. This may be unsatisfactory in applications where there are too many data to fit into memory or where the data are provided in streaming fashion. One such use is for collaborative filtering in recommendation systems, where there may be many users and many items to recommend, and it would be inefficient to recalculate everything when one user or one item is added to the system. The cost function for o