Sparse principal component analysis (SPCA or sparse PCA) is a technique used in statistical analysis and, in particular, in the analysis of multivariate data sets. It extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by introducing sparsity structures to the input variables. A particular disadvantage of ordinary PCA is that the principal components are usually linear combinations of all input variables. SPCA overcomes this disadvantage by finding components that are linear combinations of just a few input variables (SPCs). This means that some of the coefficients of the linear combinations defining the SPCs, called loadings, are equal to zero. The number of nonzero loadings is called the cardinality of the SPC. == Mathematical formulation == Consider a data matrix, X {\displaystyle X} , where each of the p {\displaystyle p} columns represent an input variable, and each of the n {\displaystyle n} rows represents an independent sample from data population. One assumes each column of X {\displaystyle X} has mean zero, otherwise one can subtract column-wise mean from each element of X {\displaystyle X} . Let Σ = 1 n − 1 X ⊤ X {\displaystyle \Sigma ={\frac {1}{n-1}}X^{\top }X} be the empirical covariance matrix of X {\displaystyle X} , which has dimension p × p {\displaystyle p\times p} . Given an integer k {\displaystyle k} with 1 ≤ k ≤ p {\displaystyle 1\leq k\leq p} , the sparse PCA problem can be formulated as maximizing the variance along a direction represented by vector v ∈ R p {\displaystyle v\in \mathbb {R} ^{p}} while constraining its cardinality: max v T Σ v subject to ‖ v ‖ 2 = 1 ‖ v ‖ 0 ≤ k . {\displaystyle {\begin{aligned}\max \quad &v^{T}\Sigma v\\{\text{subject to}}\quad &\left\Vert v\right\Vert _{2}=1\\&\left\Vert v\right\Vert _{0}\leq k.\end{aligned}}} Eq. 1 The first constraint specifies that v is a unit vector. In the second constraint, ‖ v ‖ 0 {\displaystyle \left\Vert v\right\Vert _{0}} represents the ℓ 0 {\displaystyle \ell _{0}} pseudo-norm of v, which is defined as the number of its non-zero components. So the second constraint specifies that the number of non-zero components in v is less than or equal to k, which is typically an integer that is much smaller than dimension p. The optimal value of Eq. 1 is known as the k-sparse largest eigenvalue. If one takes k=p, the problem reduces to the ordinary PCA, and the optimal value becomes the largest eigenvalue of covariance matrix Σ. After finding the optimal solution v, one deflates Σ to obtain a new matrix Σ 1 = Σ − ( v T Σ v ) v v T , {\displaystyle \Sigma _{1}=\Sigma -(v^{T}\Sigma v)vv^{T},} and iterate this process to obtain further principal components. However, unlike PCA, sparse PCA cannot guarantee that different principal components are orthogonal. In order to achieve orthogonality, additional constraints must be enforced. The following equivalent definition is in matrix form. Let V {\displaystyle V} be a p×p symmetric matrix, one can rewrite the sparse PCA problem as max T r ( Σ V ) subject to T r ( V ) = 1 ‖ V ‖ 0 ≤ k 2 R a n k ( V ) = 1 , V ⪰ 0. {\displaystyle {\begin{aligned}\max \quad &Tr(\Sigma V)\\{\text{subject to}}\quad &Tr(V)=1\\&\Vert V\Vert _{0}\leq k^{2}\\&Rank(V)=1,V\succeq 0.\end{aligned}}} Eq. 2 Tr is the matrix trace, and ‖ V ‖ 0 {\displaystyle \Vert V\Vert _{0}} represents the non-zero elements in matrix V. The last line specifies that V has matrix rank one and is positive semidefinite. The last line means that one has V = v v T {\displaystyle V=vv^{T}} , so Eq. 2 is equivalent to Eq. 1. Moreover, the rank constraint in this formulation is actually redundant, and therefore sparse PCA can be cast as the following mixed-integer semidefinite program max T r ( Σ V ) subject to T r ( V ) = 1 | V i , i | ≤ z i , ∀ i ∈ { 1 , . . . , p } , | V i , j | ≤ 1 2 z i , ∀ i , j ∈ { 1 , . . . , p } : i ≠ j , V ⪰ 0 , z ∈ { 0 , 1 } p , ∑ i z i ≤ k {\displaystyle {\begin{aligned}\max \quad &Tr(\Sigma V)\\{\text{subject to}}\quad &Tr(V)=1\\&\vert V_{i,i}\vert \leq z_{i},\forall i\in \{1,...,p\},\vert V_{i,j}\vert \leq {\frac {1}{2}}z_{i},\forall i,j\in \{1,...,p\}:i\neq j,\\&V\succeq 0,z\in \{0,1\}^{p},\sum _{i}z_{i}\leq k\end{aligned}}} Eq. 3 Because of the cardinality constraint, the maximization problem is hard to solve exactly, especially when dimension p is high. In fact, the sparse PCA problem in Eq. 1 is NP-hard in the strong sense. == Computational considerations == As most sparse problems, variable selection in SPCA is a computationally intractable non-convex NP-hard problem, therefore greedy sub-optimal algorithms are often employed to find solutions. Note also that SPCA introduces hyperparameters quantifying in what capacity large parameter values are penalized. These might need tuning to achieve satisfactory performance, thereby adding to the total computational cost. == Algorithms for SPCA == Several alternative approaches (of Eq. 1) have been proposed, including a regression framework, a penalized matrix decomposition framework, a convex relaxation/semidefinite programming framework, a generalized power method framework an alternating maximization framework forward-backward greedy search and exact methods using branch-and-bound techniques, a certifiably optimal branch-and-bound approach Bayesian formulation framework. A certifiably optimal mixed-integer semidefinite branch-and-cut approach The methodological and theoretical developments of Sparse PCA as well as its applications in scientific studies are recently reviewed in a survey paper. === Notes on Semidefinite Programming Relaxation === It has been proposed that sparse PCA can be approximated by semidefinite programming (SDP). If one drops the rank constraint and relaxes the cardinality constraint by a 1-norm convex constraint, one gets a semidefinite programming relaxation, which can be solved efficiently in polynomial time: max T r ( Σ V ) subject to T r ( V ) = 1 1 T | V | 1 ≤ k V ⪰ 0. {\displaystyle {\begin{aligned}\max \quad &Tr(\Sigma V)\\{\text{subject to}}\quad &Tr(V)=1\\&\mathbf {1} ^{T}|V|\mathbf {1} \leq k\\&V\succeq 0.\end{aligned}}} Eq. 3 In the second constraint, 1 {\displaystyle \mathbf {1} } is a p×1 vector of ones, and |V| is the matrix whose elements are the absolute values of the elements of V. The optimal solution V {\displaystyle V} to the relaxed problem Eq. 3 is not guaranteed to have rank one. In that case, V {\displaystyle V} can be truncated to retain only the dominant eigenvector. While the semidefinite program does not scale beyond n=300 covariates, it has been shown that a second-order cone relaxation of the semidefinite relaxation is almost as tight and successfully solves problems with n=1000s of covariates == Applications == === Financial Data Analysis === Suppose ordinary PCA is applied to a dataset where each input variable represents a different asset, it may generate principal components that are weighted combination of all the assets. In contrast, sparse PCA would produce principal components that are weighted combination of only a few input assets, so one can easily interpret its meaning. Furthermore, if one uses a trading strategy based on these principal components, fewer assets imply less transaction costs. === Biology === Consider a dataset where each input variable corresponds to a specific gene. Sparse PCA can produce a principal component that involves only a few genes, so researchers can focus on these specific genes for further analysis. === High-dimensional Hypothesis Testing === Contemporary datasets often have the number of input variables ( p {\displaystyle p} ) comparable with or even much larger than the number of samples ( n {\displaystyle n} ). It has been shown that if p / n {\displaystyle p/n} does not converge to zero, the classical PCA is not consistent. In other words, if we let k = p {\displaystyle k=p} in Eq. 1, then the optimal value does not converge to the largest eigenvalue of data population when the sample size n → ∞ {\displaystyle n\rightarrow \infty } , and the optimal solution does not converge to the direction of maximum variance. But sparse PCA can retain consistency even if p ≫ n . {\displaystyle p\gg n.} The k-sparse largest eigenvalue (the optimal value of Eq. 1) can be used to discriminate an isometric model, where every direction has the same variance, from a spiked covariance model in high-dimensional setting. Consider a hypothesis test where the null hypothesis specifies that data X {\displaystyle X} are generated from a multivariate normal distribution with mean 0 and covariance equal to an identity matrix, and the alternative hypothesis specifies that data X {\displaystyle X} is generated from a spiked model with signal strength θ {\displaystyle \theta } : H 0 : X ∼ N ( 0 , I p ) , H 1 : X ∼ N ( 0 , I p + θ v v T ) , {\displaystyle H_{0}:X\sim N(0,I_{p}),\quad H_{1}:X\sim N(0,I_{p}+\theta vv^{T}),} where v ∈ R p {\displaystyle v\in \mathbb {R} ^{p}
Texture filtering
In computer graphics, texture filtering or texture smoothing is the method used to determine the texture color for a texture mapped pixel, using the colors of nearby texels (ie. pixels of the texture). Filtering describes how a texture is applied at many different shapes, size, angles and scales. Depending on the chosen filter algorithm, the result will show varying degrees of blurriness, detail, spatial aliasing, temporal aliasing and blocking. Depending on the circumstances, filtering can be performed in software (such as a software rendering package) or in hardware, eg. with either real time or GPU accelerated rendering circuits, or in a mixture of both. For most common interactive graphical applications, modern texture filtering is performed by dedicated hardware which optimizes memory access through memory cacheing and pre-fetch, and implements a selection of algorithms available to the user and developer. There are two main categories of texture filtering: magnification filtering and minification filtering. Depending on the situation, texture filtering is either a type of reconstruction filter where sparse data is interpolated to fill gaps (magnification), or a type of anti-aliasing (AA) where texture samples exist at a higher frequency than required for the sample frequency needed for texture fill (minification). There are many methods of texture filtering, which make different trade-offs between computational complexity, memory bandwidth and image quality. == The need for filtering == During the texture mapping process for any arbitrary 3D surface, a texture lookup takes place to find out where on the texture each pixel center falls. For texture-mapped polygonal surfaces composed of triangles typical of most surfaces in 3D games and movies, every pixel (or subordinate pixel sample) of that surface will be associated with some triangle(s) and a set of barycentric coordinates, which are used to provide a position within a texture. Such a position may not lie perfectly on the "pixel grid," necessitating some function to account for these cases. In other words, since the textured surface may be at an arbitrary distance and orientation relative to the viewer, one pixel does not usually correspond directly to one texel. Some form of filtering has to be applied to determine the best color for the pixel. Insufficient or incorrect filtering will show up in the image as artifacts (errors in the image), such as 'blockiness', jaggies, or shimmering. There can be different types of correspondence between a pixel and the texel/texels it represents on the screen. These depend on the position of the textured surface relative to the viewer, and different forms of filtering are needed in each case. Given a square texture mapped on to a square surface in the world, at some viewing distance the size of one screen pixel is exactly the same as one texel. Closer than that, the texels are larger than screen pixels, and need to be scaled up appropriately — a process known as texture magnification. Farther away, each texel is smaller than a pixel, and so one pixel covers multiple texels. In this case an appropriate color has to be picked based on the covered texels, via texture minification. Graphics APIs such as OpenGL allow the programmer to set different choices for minification and magnification filters. Note that even in the case where the pixels and texels are exactly the same size, one pixel will not necessarily match up exactly to one texel. It may be misaligned or rotated, and cover parts of up to four neighboring texels. Hence some form of filtering is still required. == Mipmapping == Mipmapping is a standard technique used to save some of the filtering work needed during texture minification. It is also highly beneficial for cache coherency - without it the memory access pattern during sampling from distant textures will exhibit extremely poor locality, adversely affecting performance even if no filtering is performed. During texture magnification, the number of texels that need to be looked up for any pixel is always four or fewer; during minification, however, as the textured polygon moves farther away potentially the entire texture might fall into a single pixel. This would necessitate reading all of its texels and combining their values to correctly determine the pixel color, a prohibitively expensive operation. Mipmapping avoids this by prefiltering the texture and storing it in smaller sizes down to a single pixel. As the textured surface moves farther away, the texture being applied switches to the prefiltered smaller size. Different sizes of the mipmap are referred to as 'levels', with Level 0 being the largest size (used closest to the viewer), and increasing levels used at increasing distances. == Filtering methods == This section lists the most common texture filtering methods, in increasing order of computational cost and image quality. === Nearest-neighbor interpolation === Nearest-neighbor interpolation is the simplest and crudest filtering method — it simply uses the color of the texel closest to the pixel center for the pixel color. While simple, this results in a large number of artifacts - texture 'blockiness' during magnification, and aliasing and shimmering during minification. This method is fast during magnification but during minification the stride through memory becomes arbitrarily large and it can often be less efficient than MIP-mapping due to the lack of spatially coherent texture access and cache-line reuse. === Nearest-neighbor with mipmapping === This method still uses nearest neighbor interpolation, but adds mipmapping — first the nearest mipmap level is chosen according to distance, then the nearest texel center is sampled to get the pixel color. This reduces the aliasing and shimmering significantly during minification but does not eliminate it entirely. In doing so it improves texture memory access and cache-line reuse through avoiding arbitrarily large access strides through texture memory during rasterization. This does not help with blockiness during magnification as each magnified texel will still appear as a large rectangle. === Linear mipmap filtering === Less commonly used, OpenGL and other APIs support nearest-neighbor sampling from individual mipmaps whilst linearly interpolating the two nearest mipmaps relevant to the sample. === Bilinear filtering === In Bilinear filtering, the four nearest texels to the pixel center are sampled (at the closest mipmap level), and their colors are combined by weighted average according to distance. This removes the 'blockiness' seen during magnification, as there is now a smooth gradient of color change from one texel to the next, instead of an abrupt jump as the pixel center crosses the texel boundary. Bilinear filtering for magnification filtering is common. When used for minification it is often used with mipmapping; though it can be used without, it would suffer the same aliasing and shimmering problems as nearest-neighbor filtering when minified too much. For modest minification ratios, however, it can be used as an inexpensive hardware accelerated weighted texture supersample. The Nintendo 64 used an unusual version of bilinear filtering where only three pixels are used known as 3-point texture filtering, instead of four due to hardware optimization concerns. This introduces a noticeable "triangulation bias" in some textures. === Trilinear filtering === Trilinear filtering is a remedy to a common artifact seen in mipmapped bilinearly filtered images: an abrupt and very noticeable change in quality at boundaries where the renderer switches from one mipmap level to the next. Trilinear filtering solves this by doing a texture lookup and bilinear filtering on the two closest mipmap levels (one higher and one lower quality), and then linearly interpolating the results. This results in a smooth degradation of texture quality as distance from the viewer increases, rather than a series of sudden drops. Of course, closer than Level 0 there is only one mipmap level available, and the algorithm reverts to bilinear filtering. === Anisotropic filtering === Anisotropic filtering is the highest quality filtering available in current consumer 3D graphics cards. Simpler, "isotropic" techniques use only square mipmaps which are then interpolated using bi– or trilinear filtering. (Isotropic means same in all directions, and hence is used to describe a system in which all the maps are squares rather than rectangles or other quadrilaterals.) When a surface is at a high angle relative to the camera, the fill area for a texture will not be approximately square. Consider the common case of a floor in a game: the fill area is far wider than it is tall. In this case, none of the square maps are a good fit. The result is blurriness and/or shimmering, depending on how the fit is chosen. Anisotropic filtering corrects this by sampling the texture as a non-square shape. The goal is
Defining length
In the field of genetic algorithms, a schema (plural: schemata) is a template that represents a subset of potential solutions. These templates use fixed symbols (e.g., `0` or `1`) for specific positions and a wildcard or "don't care" symbol (often `#` or ``) for others. The defining length of a schema, denoted as L(H), measures the distance between the outermost fixed positions in the template. According to the Schema theorem, a schema with a shorter defining length is less likely to be disrupted by the genetic operator of crossover. As a result, short schemata are considered more robust and are more likely to be propagated to the next generation. In genetic programming, where solutions are often represented as trees, the defining length is the number of links in the minimum tree fragment that includes all the non-wildcard symbols within a schema H. == Example == The defining length is calculated by subtracting the position of the first fixed symbol from the position of the last one. Using 1-based indexing for a string of length 5: The schema `1##0#` has its first fixed symbol (`1`) at position 1 and its last fixed symbol (`0`) at position 4. Its defining length is 4 − 1 = 3. The schema `00##0` has its first fixed symbol at position 1 and its last at position 5. Its defining length is 5 − 1 = 4. The schema `##0##` has only one fixed symbol at position 3. The first and last fixed positions are the same, so its defining length is 3 − 3 = 0.
Generalized canonical correlation
In statistics, the generalized canonical correlation analysis (gCCA), is a way of making sense of cross-correlation matrices between the sets of random variables when there are more than two sets. While a conventional CCA generalizes principal component analysis (PCA) to two sets of random variables, a gCCA generalizes PCA to more than two sets of random variables. The canonical variables represent those common factors that can be found by a large PCA of all of the transformed random variables after each set underwent its own PCA. == Applications == The Helmert-Wolf blocking (HWB) method of estimating linear regression parameters can find an optimal solution only if all cross-correlations between the data blocks are zero. They can always be made to vanish by introducing a new regression parameter for each common factor. The gCCA method can be used for finding those harmful common factors that create cross-correlation between the blocks. However, no optimal HWB solution exists if the random variables do not contain enough information on all of the new regression parameters.
Latent space
A latent space, also known as a latent feature space or embedding space, is an embedding of a set of items within a manifold in which items resembling each other are positioned closer to one another. Position within the latent space can be viewed as being defined by a set of latent variables that emerge from the resemblances between the objects. In most cases, the dimensionality of the latent space is chosen to be lower than the dimensionality of the feature space from which the data points are drawn, making the construction of a latent space an example of dimensionality reduction, which can also be viewed as a form of data compression. Latent spaces are usually fit via machine learning, and they can then be used as feature spaces in machine learning models, including classifiers and other supervised predictors. The interpretation of latent spaces in machine learning models is an ongoing area of research, but achieving clear interpretations remains challenging. The black-box nature of these models often makes the latent space unintuitive, while its high-dimensional, complex, and nonlinear characteristics further complicate the task of understanding it. Analysis of the latent space geometry of diffusion models reveals a fractal structure of phase transitions in the latent space, characterized by abrupt changes in the Fisher information metric. Some visualization techniques have been developed to connect the latent space to the visual world, but there is often not a direct connection between the latent space interpretation and the model itself. Such techniques include t-distributed stochastic neighbor embedding (t-SNE), where the latent space is mapped to two dimensions for visualization. Latent space distances lack physical units, so the interpretation of these distances may depend on the application. == Embedding models == Several embedding models have been developed to perform this transformation to create latent space embeddings given a set of data items and a similarity function. These models learn the embeddings by leveraging statistical techniques and machine learning algorithms. Here are some commonly used embedding models: Word2Vec: Word2Vec is a popular embedding model used in natural language processing (NLP). It learns word embeddings by training a neural network on a large corpus of text. Word2Vec captures semantic and syntactic relationships between words, allowing for meaningful computations like word analogies. GloVe: GloVe (Global Vectors for Word Representation) is another widely used embedding model for NLP. It combines global statistical information from a corpus with local context information to learn word embeddings. GloVe embeddings are known for capturing both semantic and relational similarities between words. Siamese Networks: Siamese networks are a type of neural network architecture commonly used for similarity-based embedding. They consist of two identical subnetworks that process two input samples and produce their respective embeddings. Siamese networks are often used for tasks like image similarity, recommendation systems, and face recognition. Variational Autoencoders (VAEs): VAEs are generative models that simultaneously learn to encode and decode data. The latent space in VAEs acts as an embedding space. By training VAEs on high-dimensional data, such as images or audio, the model learns to encode the data into a compact latent representation. VAEs are known for their ability to generate new data samples from the learned latent space. == Multimodality == Multimodality refers to the integration and analysis of multiple modes or types of data within a single model or framework. Embedding multimodal data involves capturing relationships and interactions between different data types, such as images, text, audio, and structured data. Multimodal embedding models aim to learn joint representations that fuse information from multiple modalities, allowing for cross-modal analysis and tasks. These models enable applications like image captioning, visual question answering, and multimodal sentiment analysis. To embed multimodal data, specialized architectures such as deep multimodal networks or multimodal transformers are employed. These architectures combine different types of neural network modules to process and integrate information from various modalities. The resulting embeddings capture the complex relationships between different data types, facilitating multimodal analysis and understanding. == Applications == Embedding latent space and multimodal embedding models have found numerous applications across various domains: Information retrieval: Embedding techniques enable efficient similarity search and recommendation systems by representing data points in a compact space. Natural language processing: Word embeddings have revolutionized NLP tasks like sentiment analysis, machine translation, and document classification. Computer vision: Image and video embeddings enable tasks like object recognition, image retrieval, and video summarization. Recommendation systems: Embeddings help capture user preferences and item characteristics, enabling personalized recommendations. Healthcare: Embedding techniques have been applied to electronic health records, medical imaging, and genomic data for disease prediction, diagnosis, and treatment. Social systems: Embedding techniques can be used to learn latent representations of social systems such as internal migration systems, academic citation networks, and world trade networks.
Token maxxing
Token Maxxing or Token Maxing is a metric used in an attempt to track productivity in the workplace especially for those using Artificial Intelligence (AI) based services. AI services charge for each token which represent units of effort expended by an AI service to solve a problem. Some believe that token consumption equates to productivity and thus can be used as a metric to monitor an employee's work. Supporters believe that higher token usage indicates higher productivity and higher utilization of powerful AI services. This also suggests that those not consuming enough tokens may be less productive and underutilizing powerful AI services. This belief might lead to an environment that incentivizes higher token usage to predict increased productivity. Critics of token maxxing as a metric claim that prudent workers will maximize any metric that management wants increased to gain a workplace advantage. For example: Engineers in the tech industries pressed to consume as many tokens as possible might run several AI agents in tandem, enter longer input prompts, or automate their tasks to maximize their token consumption. To management, this higher token usage may indicate potential productivity, but in reality may cause additional token costs, worker burnout, or actually create more bloated code of lower quality. Another claim is AI service companies potentially benefit from such an emphasis on token consumption and actively encourage the trend. Some developers have publicly advocated the practice. Developer Sigrid Jin, who said he used 50 billion tokens in a single year, has argued that maximizing token consumption is the best way to understand the value of AI, advising others to spend as much on AI usage as they pay in rent to obtain a return on investment. == See Also == Goodhart's law Perverse incentive Jevons Paradox
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