Non-negative matrix factorization

3421: 3509:, an application of PCA, using the plot of eigenvalues. A typical choice of the number of components with PCA is based on the "elbow" point, then the existence of the flat plateau is indicating that PCA is not capturing the data efficiently, and at last there exists a sudden drop reflecting the capture of random noise and falls into the regime of overfitting. For sequential NMF, the plot of eigenvalues is approximated by the plot of the fractional residual variance curves, where the curves decreases continuously, and converge to a higher level than PCA, which is the indication of less over-fitting of sequential NMF. 4208:. However, if the noise is non-stationary, the classical denoising algorithms usually have poor performance because the statistical information of the non-stationary noise is difficult to estimate. Schmidt et al. use NMF to do speech denoising under non-stationary noise, which is completely different from classical statistical approaches. The key idea is that clean speech signal can be sparsely represented by a speech dictionary, but non-stationary noise cannot. Similarly, non-stationary noise can also be sparsely represented by a noise dictionary, but speech cannot. 3564: 4287: 8319: 1047:, 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. 4012:

where forward modeling have to be adopted to recover the true flux. Forward modeling is currently optimized for point sources, however not for extended sources, especially for irregularly shaped structures such as circumstellar disks. In this situation, NMF has been an excellent method, being less over-fitting in the sense of the non-negativity and

3113: 2870: 2537:, 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 optimization in these cases may or may not be the same as for standard NMF, but the algorithms need to be rather different. 4249:

data and finding the genes most representative of the clusters. In the analysis of cancer mutations it has been used to identify common patterns of mutations that occur in many cancers and that probably have distinct causes. NMF techniques can identify sources of variation such as cell types, disease

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The data imputation procedure with NMF can be composed of two steps. First, when the NMF components are known, Ren et al. (2020) proved that impact from missing data during data imputation ("target modeling" in their study) is a second order effect. Second, when the NMF components are unknown, the

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in statistics. By first proving that the missing data are ignored in the cost function, then proving that the impact from missing data can be as small as a second order effect, Ren et al. (2020) studied and applied such an approach for the field of astronomy. Their work focuses on two-dimensional

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In direct imaging, to reveal the faint exoplanets and circumstellar disks from bright the surrounding stellar lights, which has a typical contrast from 10⁵ to 10¹⁰, various statistical methods have been adopted, however the light from the exoplanets or circumstellar disks are usually over-fitted,

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in the sense that astrophysical signals are non-negative. NMF has been applied to the spectroscopic observations and the direct imaging observations as a method to study the common properties of astronomical objects and post-process the astronomical observations. The advances in the spectroscopic

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Wahhaj, Zahed; Cieza, Lucas A.; Mawet, Dimitri; Yang, Bin; Canovas, Hector; de Boer, Jozua; Casassus, Simon; Ménard, François; Schreiber, Matthias R.; Liu, Michael C.; Biller, Beth A.; Nielsen, Eric L.; Hayward, Thomas L. (2015). "Improving signal-to-noise in the direct imaging of exoplanets and

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Fractional residual variance (FRV) plots for PCA and sequential NMF; for PCA, the theoretical values are the contribution from the residual eigenvalues. In comparison, the FRV curves for PCA reaches a flat plateau where no signal are captured effectively; while the NMF FRV curves are declining

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Current algorithms are sub-optimal in that they only guarantee finding a local minimum, rather than a global minimum of the cost function. A provably optimal algorithm is unlikely in the near future as the problem has been shown to generalize the k-means clustering problem which is known to be

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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

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The algorithm for NMF denoising goes as follows. Two dictionaries, one for speech and one for noise, need to be trained offline. Once a noisy speech is given, we first calculate the magnitude of the Short-Time-Fourier-Transform. Second, separate it into two parts via NMF, one can be sparsely

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Depending on the way that the NMF components are obtained, the former step above can be either independent or dependent from the latter. In addition, the imputation quality can be increased when the more NMF components are used, see Figure 4 of Ren et al. (2020) for their illustration.

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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

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measurements. This kind of method was firstly introduced in Internet Distance Estimation Service (IDES). Afterwards, as a fully decentralized approach, Phoenix network coordinate system is proposed. It achieves better overall prediction accuracy by introducing the concept of weight.

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Scalability: how to factorize million-by-billion matrices, which are commonplace in Web-scale data mining, e.g., see Distributed Nonnegative Matrix Factorization (DNMF), Scalable Nonnegative Matrix Factorization (ScalableNMF), Distributed Stochastic Singular Value

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Arora, Ge, Halpern, Mimno, Moitra, Sontag, Wu, & Zhu (2013) have given polynomial-time algorithms to learn topic models using NMF. The algorithm assumes that the topic matrix satisfies a separability condition that is often found to hold in these settings.

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in sampled genomes. In human genetic clustering, NMF algorithms provide estimates similar to those of the computer program STRUCTURE, but the algorithms are more efficient computationally and allow analysis of large population genomic data sets.

3450:(PCA) in astronomy. The contribution from the PCA components are ranked by the magnitude of their corresponding eigenvalues; for NMF, its components can be ranked empirically when they are constructed one by one (sequentially), i.e., learn the 2524:

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

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Hassani, Iranmanesh and Mansouri (2019) proposed a feature agglomeration method for term-document matrices which operates using NMF. The algorithm reduces the term-document matrix into a smaller matrix more suitable for text clustering.

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contains cluster membership indicators. This provides a theoretical foundation for using NMF for data clustering. However, k-means does not enforce non-negativity on its centroids, so the closest analogy is in fact with "semi-NMF".

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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

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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

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time in the dense case. Arora, Ge, Halpern, Mimno, Moitra, Sontag, Wu, & Zhu (2013) give a polynomial time algorithm for exact NMF that works for the case where one of the factors W satisfies a separability condition.

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represent data sampled over spatial or temporal dimensions, e.g. time signals, images, or video, features that are equivariant w.r.t. shifts along these dimensions can be learned by Convolutional NMF. In this case,

3713:(SVM). However, SVM and NMF are related at a more intimate level than that of NQP, which allows direct application of the solution algorithms developed for either of the two methods to problems in both domains. 3875: 4212:

represented by the speech dictionary, and the other part can be sparsely represented by the noise dictionary. Third, the part that is represented by the speech dictionary will be the estimated clean speech.

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Stein-O’Brien, Genevieve L.; Arora, Raman; Culhane, Aedin C.; Favorov, Alexander V.; Garmire, Lana X.; Greene, Casey S.; Goff, Loyal A.; Li, Yifeng; Ngom, Aloune; Ochs, Michael F.; Xu, Yanxun (2018-10-01).

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Andrzej Cichocki, Rafal Zdunek, Anh Huy Phan and Shun-ichi Amari: "Nonnegative Matrix and Tensor Factorizations: Applications to Exploratory Multi-way Data Analysis and Blind Source Separation", Wiley,

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To impute missing data in statistics, NMF can take missing data while minimizing its cost function, rather than treating these missing data as zeros. This makes it a mathematically proven method for

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observations by Blanton & Roweis (2007) takes into account of the uncertainties of astronomical observations, which is later improved by Zhu (2016) where missing data are also considered and

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Berry, Michael W.; Browne, Murray; Langville, Amy N.; Paucac, V. Paul; Plemmonsc, Robert J. (15 September 2007). "Algorithms and Applications for Approximate Nonnegative Matrix Factorization".

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Zhang, T.; Fang, B.; Liu, W.; Tang, Y. Y.; He, G.; Wen, J. (2008). "Total variation norm-based nonnegative matrix factorization for identifying discriminant representation of image patterns".

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Cohen and Rothblum 1993 problem: whether a rational matrix always has an NMF of minimal inner dimension whose factors are also rational. Recently, this problem has been answered negatively.

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contains a monomial sub matrix of rank equal to its rank was given by Campbell and Poole in 1981. Kalofolias and Gallopoulos (2012) solved the symmetric counterpart of this problem, where

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Other extensions of NMF include joint factorization of several data matrices and tensors where some factors are shared. Such models are useful for sensor fusion and relational learning.

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of the NMF modeling coefficients, therefore forward modeling can be performed with a few scaling factors, rather than a computationally intensive data re-reduction on generated models.

3823: 915: 3108:{\displaystyle \mathbf {W} _{}^{n+1}\leftarrow \mathbf {W} _{}^{n}{\frac {(\mathbf {V} (\mathbf {H} ^{n+1})^{T})_{}}{(\mathbf {W} ^{n}\mathbf {H} ^{n+1}(\mathbf {H} ^{n+1})^{T})_{}}}} 1725: 3151: 3940: 3911: 1475:

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.

3643: 2283: 2865:{\displaystyle \mathbf {H} _{}^{n+1}\leftarrow \mathbf {H} _{}^{n}{\frac {((\mathbf {W} ^{n})^{T}\mathbf {V} )_{}}{((\mathbf {W} ^{n})^{T}\mathbf {W} ^{n}\mathbf {H} ^{n})_{}}}} 872: 1763: 3425:

continuously, indicating a better ability to capture signal. The FRV curves for NMF also converges to higher levels than PCA, indicating the less-overfitting property of NMF.

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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

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Cédric Févotte; Nancy Bertin & Jean-Louis Durrieu (2009). "Nonnegative Matrix Factorization with the Itakura-Saito Divergence: With Application to Music Analysis".

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It was later shown that some types of NMF are an instance of a more general probabilistic model called "multinomial PCA". When NMF is obtained by minimizing the

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Chistikov, Dmitry; Kiefer, Stefan; Marušić, Ines; Shirmohammadi, Mahsa; Worrell, James (2016-05-22). "Nonnegative Matrix Factorization Requires Irrationality".

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Soummer, Rémi; Pueyo, Laurent; Larkin, James (2012). "Detection and Characterization of Exoplanets and Disks Using Projections on Karhunen-Loève Eigenimages".

3560:, and shows that although the three techniques may be written as factorizations, they implement different constraints and therefore produce different results. 6240:(1999). "The Multilinear Engine: A Table-Driven, Least Squares Program for Solving Multilinear Problems, including the n-Way Parallel Factor Analysis Model". 2382:

is defined on probability distributions). Each divergence leads to a different NMF algorithm, usually minimizing the divergence using iterative update rules.

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C Ding, T Li, MI Jordan, Convex and semi-nonnegative matrix factorizations, IEEE Transactions on Pattern Analysis and Machine Intelligence, 32, 45-55, 2010

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Collective (joint) factorization: factorizing multiple interrelated matrices for multiple-view learning, e.g. multi-view clustering, see CoNMF and MultiNMF

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is constructed with the weights of various terms (typically weighted word frequency information) from a set of documents. This matrix is factored into a

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may affect not only the rate of convergence, but also the overall error at convergence. Some options for initialization include complete randomization,

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Naiyang Guan; Dacheng Tao; Zhigang Luo & Bo Yuan (July 2012). "Online Nonnegative Matrix Factorization With Robust Stochastic Approximation".

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Hassani, Ali; Iranmanesh, Amir; Mansouri, Najme (2019-11-12). "Text Mining using Nonnegative Matrix Factorization and Latent Semantic Analysis".

6178:. Proc. 28th international ACM SIGIR conference on Research and development in information retrieval (SIGIR-05). pp. 601–602. Archived from 3992:

Ren et al. (2018) are able to prove the stability of NMF components when they are constructed sequentially (i.e., one by one), which enables the

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Julian Becker: "Nonnegative Matrix Factorization with Adaptive Elements for Monaural Audio Source Separation: 1 ", Shaker Verlag GmbH, Germany,

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tasks in order to predict novel protein targets and therapeutic indications for approved drugs and to infer pair of synergic anticancer drugs.

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email dataset with 65,033 messages and 91,133 terms into 50 clusters. NMF has also been applied to citations data, with one example clustering

3654: 2073: 6221: 8307: 7877: 7141:"DNA methylation profiling of medulloblastoma allows robust sub-classification and improved outcome prediction using formalin-fixed biopsies" 6417: 5765: 5428: 4569:

Ren, Bin; Pueyo, Laurent; Chen, Christine; Choquet, Elodie; Debes, John H; Duechene, Gaspard; Menard, Francois; Perrin, Marshall D. (2020).

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Hafshejani, Sajad Fathi; Moaberfard, Zahra (November 2022). "Initialization for Nonnegative Matrix Factorization: a Comprehensive Review".

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Naiyang Guan; Dacheng Tao; Zhigang Luo; Bo Yuan (June 2012). "NeNMF: An Optimal Gradient Method for Nonnegative Matrix Factorization".

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Pentti Paatero; Unto Tapper; Pasi Aalto; Markku Kulmala (1991). "Matrix factorization methods for analysing diffusion battery data".

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Andri Mirzal: "Nonnegative Matrix Factorizations for Clustering and LSI: Theory and Programming", LAP LAMBERT Academic Publishing,

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Sitek; Gullberg; Huesman (2002). "Correction for ambiguous solutions in factor analysis using a penalized least squares objective".

6385:. Proceedings of the 26th annual international ACM SIGIR conference on Research and development in information retrieval. New York: 5348: 5321: 4871: 4435:

Blanton, Michael R.; Roweis, Sam (2007). "K-corrections and filter transformations in the ultraviolet, optical, and near infrared".

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NMF extends beyond matrices to tensors of arbitrary order. This extension may be viewed as a non-negative counterpart to, e.g., the

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Andrzej Cichocki, Morten Mrup, et al.: "Advances in Nonnegative Matrix and Tensor Factorization", Hindawi Publishing Corporation,

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is sparse with columns having local non-zero weight windows that are shared across shifts along the spatio-temporal dimensions of

7098:"Sparse non-negative matrix factorizations via alternating non-negativity-constrained least squares for microarray data analysis" 4869:; Unto Tapper; Olli Järvinen (1995). "Source identification of bulk wet deposition in Finland by positive matrix factorization". 4224:

for estimating individual admixture coefficients, detecting genetic clusters of individuals in a population sample or evaluating

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Liu, W.X.; Zheng, N.N. & You, Q.B. (2006). "Nonnegative Matrix Factorization and its applications in pattern recognition".

6447:"Analysis of the emission of very small dust particles from Spitzer spectro-imagery data using blind signal separation methods" 4413: 3982: 5308:. Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining - KDD '11. p. 1064. 3281:{\textstyle {\textstyle {\frac {\mathbf {V} \mathbf {H} ^{\mathsf {T}}}{\mathbf {W} \mathbf {H} \mathbf {H} ^{\mathsf {T}}}}}} 1139: 1107:

investigated the properties of the algorithm and published some simple and useful algorithms for two types of factorizations.

839: 7575:"Reconstruction of 4-D Dynamic SPECT Images From Inconsistent Projections Using a Spline Initialized FADS Algorithm (SIFADS)" 7322: 5829:

Zhu, Guangtun B. (2016-12-19). "Nonnegative Matrix Factorization (NMF) with Heteroscedastic Uncertainties and Missing data".

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Ding, C.; He, X. & Simon, H.D. (2005). "On the equivalence of nonnegative matrix factorization and spectral clustering".

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Jialu Liu; Chi Wang; Jing Gao & Jiawei Han (2013). "Multi-View Clustering via Joint Nonnegative Matrix Factorization".

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Yun Mao; Lawrence Saul & Jonathan M. Smith (2006). "IDES: An Internet Distance Estimation Service for Large Networks".

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Proceedings of the European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases

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and repeatedly using the resulting representation as input to convolutional NMF, deep feature hierarchies can be learned.

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C. Boutsidis & E. Gallopoulos (2008). "SVD based initialization: A head start for nonnegative matrix factorization".

5704:"Algorithms for nonnegative matrix and tensor factorizations: A unified view based on block coordinate descent framework" 5240:"A framework for regularized non-negative matrix factorization, with application to the analysis of gene expression data" 3517:

Exact solutions for the variants of NMF can be expected (in polynomial time) when additional constraints hold for matrix

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Jingu Kim & Haesun Park (2011). "Fast Nonnegative Matrix Factorization: An Active-set-like Method and Comparisons".

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matrices, specifically, it includes mathematical derivation, simulated data imputation, and application to on-sky data.

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Pinoli; Ceddia; Ceri; Masseroli (2021). "Predicting drug synergism by means of non-negative matrix tri-factorization".

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Lafrenière, David; Maroid, Christian; Doyon, René; Barman, Travis (2009). "HST/NICMOS Detection of HR 8799 b in 1998".

4817:"Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values" 1515: 7432:

DiPaola; Bazin; Aubry; Aurengo; Cavailloles; Herry; Kahn (1982). "Handling of dynamic sequences in nuclear medicine".

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Arora, Sanjeev; Ge, Rong; Halpern, Yoni; Mimno, David; Moitra, Ankur; Sontag, David; Wu, Yichen; Zhu, Michael (2013).

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Online: how to update the factorization when new data comes in without recomputing from scratch, e.g., see online CNSC

922: 834: 819: 280: 102: 8091: 5568:"Nonnegative Matrix Factorization Based on Alternating Nonnegativity Constrained Least Squares and Active Set Method" 5517:

Lin, Chih-Jen (2007). "On the Convergence of Multiplicative Update Algorithms for Nonnegative Matrix Factorization".

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NMF, also referred in this field as factor analysis, has been used since the 1980s to analyze sequences of images in

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Ganesh R. Naik(Ed.): "Non-negative Matrix Factorization Techniques: Advances in Theory and Applications", Springer,

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Ngoc-Diep Ho; Paul Van Dooren & Vincent Blondel (2008). "Descent Methods for Nonnegative Matrix Factorization".

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In addition to the optimization step, initialization has a significant effect on NMF. The initial values chosen for

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Alexandrov, Ludmil B.; Nik-Zainal, Serena; Wedge, David C.; Campbell, Peter J.; Stratton, Michael R. (2013-01-31).

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authors proved that the impact from missing data during component construction is a first-to-second order effect.

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Ceddia; Pinoli; Ceri; Masseroli (2020). "Matrix factorization-based technique for drug repurposing predictions".

3209:{\textstyle {\frac {\mathbf {W} ^{\mathsf {T}}\mathbf {V} }{\mathbf {W} ^{\mathsf {T}}\mathbf {W} \mathbf {H} }}} 2565: 936: 542: 310: 180: 7785: 5913:"Detection and Characterization of Exoplanets using Projections on Karhunen Loeve Eigenimages: Forward Modeling" 3788: 2339:

There are different types of non-negative matrix factorizations. The different types arise from using different

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matrix. The features are derived from the contents of the documents, and the feature-document matrix describes

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NMF is also used to analyze spectral data; one such use is in the classification of space objects and debris.

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estimation. That method is commonly used for analyzing and clustering textual data and is also related to the

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is not explicitly imposed, the orthogonality holds to a large extent, and the clustering property holds too.

1681: 6445:; Deville, Y.; Smith, J. D.; Rapacioli, M.; Bernard, J. P.; Thomas, J.; Reach, W.; Abergel, A. (2007-07-01). 8353: 4197: 4025: 2530: 2501: 1071: 1063: 511: 360: 260: 87: 3916: 3887: 8368: 8358: 7855: 7759: 7639: 7579: 7479: 6924: 6870: 6296: 5674: 5584: 5526: 5473: 3710: 3598: 2236: 691: 667: 569: 330: 305: 265: 77: 6446: 5013:"On the equivalence between non-negative matrix factorization and probabilistic latent semantic indexing" 4730: 7524:"Clustering Initiated Factor Analysis (CIFA) Application for Tissue Classification in Dynamic Brain PET" 7145: 6743:

Berry, Michael W.; Browne, Murray (2005). "Email Surveillance Using Non-negative Matrix Factorization".

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is enabled. Their method is then adopted by Ren et al. (2018) to the direct imaging field as one of the

3706: 1015: 645: 467: 419: 275: 190: 62: 4726: 1730: 8100: 8039: 7941: 7751: 7631: 7443: 7050: 6781: 6636: 6580: 6523: 6468: 6325: 6288: 6093: 6057: 5934: 5872: 5666: 5621: 5253: 4924: 4880: 4680: 4592: 4524: 4454: 4359: 4050: 2494: 1032: 1023: 574: 524: 7963: 7860: 7764: 7644: 6929: 6875: 6237: 5478: 4866: 4080: 3398:. However, as in many other data mining applications, a local minimum may still prove to be useful. 1453:

From the treatment of matrix multiplication above it follows that each column in the product matrix

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Current research (since 2010) in nonnegative matrix factorization includes, but is not limited to,

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A particular variant of NMF, namely Non-Negative Matrix Tri-Factorization (NMTF), has been use for

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method, the optimal gradient method, and the block principal pivoting method among several others.

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NMF has an inherent clustering property, i.e., it automatically clusters the columns of input data

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Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining

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Lee and Seung proposed NMF mainly for parts-based decomposition of images. It compares NMF to

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The factorization problem in the squared error version of NMF may be stated as: Given a matrix

8303: 8292: 8281: 8270: 8259: 8248: 8237: 8226: 8214: 8202: 8147: 8008: 7989:

Raul Kompass (2007). "A Generalized Divergence Measure for Nonnegative Matrix Factorization".

7873: 7669:"Distributed Nonnegative Matrix Factorization for Web-Scale Dyadic Data Analysis on MapReduce" 7596: 7555: 7496: 7406: 7363: 7318: 7305:

Ding; Li; Peng; Park (2006). "Orthogonal nonnegative matrix t-factorizations for clustering".

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NMF is applied in scalable Internet distance (round-trip time) prediction. For a network with

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One specific application used hierarchical NMF on a small subset of scientific abstracts from

2483:{\displaystyle F(\mathbf {W} ,\mathbf {H} )=\left\|\mathbf {V} -\mathbf {WH} \right\|_{F}^{2}} 748: 591: 504: 300: 270: 215: 210: 165: 107: 7928:"A receptor model using a specific non-negative transformation technique for ambient aerosol" 6345: 5302:

Fast coordinate descent methods with variable selection for non-negative matrix factorization

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is symmetric and contains a diagonal principal sub matrix of rank r. Their algorithm runs in

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More recently other algorithms have been developed. Some approaches are based on alternating

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model with one layer of observed random variables and one layer of hidden random variables.

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Vamsi K. Potluru; Sergey M. Plis; Morten Morup; Vince D. Calhoun & Terran Lane (2009).

6004: 4134: 3453: 1918: 1498:) where each feature is weighted by the feature's cell value from the document's column in 8086: 4910: 4364: 4246: 4242: 4161: 3949: 1104: 1055: 804: 608: 474: 414: 4273:

dynamic medical imaging. Non-uniqueness of NMF was addressed using sparsity constraints.

3133:

Note that the updates are done on an element by element basis not matrix multiplication.

2378:) and an extension of the Kullback–Leibler divergence to positive matrices (the original 2022:-th cluster. This centroid's representation can be significantly enhanced by convex NMF. 8324: 8104: 8043: 7945: 7755: 7635: 7447: 7054: 6584: 6527: 6292: 6097: 6061: 5938: 5876: 5670: 5625: 5456: 5257: 4998: 4928: 4884: 4684: 4596: 4528: 4458: 4300:

Please help update this article to reflect recent events or newly available information.

8197: 8168: 7550: 7523: 7282: 7249: 7224: 7191: 7167: 7140: 7073: 7036: 7012: 6985: 6535: 5567: 5276: 5239: 5035: 4703: 4668: 4238: 4205: 4200:. There are many algorithms for denoising if the noise is stationary. For example, the 4114: 3722: 3485: 3417:, k-means clustering, and more advanced strategies based on these and other paradigms. 3289: 2632: 2598:

has been a popular method due to the simplicity of implementation. This algorithm is:

2375: 2005: 1985: 1965: 1945: 1840: 1655: 1635: 1444:

and, if the factorization worked, it is a reasonable approximation to the input matrix

1079: 1019: 824: 355: 92: 7981: 6310: 6207: 5947: 5912: 5884: 4785: 4639:. Proc. ACM SIGKDD Int'l Conf. on Knowledge discovery and data mining. pp. 69–77. 8347: 7954: 7927: 7846: 7668: 7418: 7375: 6692: 6602: 6593: 6558: 6203: 5988: 5971: 5956: 5703: 4892: 4735: 4614: 4201: 2513: 2340: 2196:

by significantly less data, then one has to infer some latent structure in the data.

743: 672: 554: 285: 170: 8159: 8059: 7781: 7608: 7463: 7116: 7097: 6938: 6892: 6766: 6708: 6656: 6427: 6402:

2004 IEEE International Joint Conference on Neural Networks (IEEE Cat. No.04CH37541)

5892: 5734: 5338: 4474: 4419:(Report). Max Planck Institute for Biological Cybernetics. Technical Report No. 193. 2325:

is called a nonnegative rank factorization (NRF). The problem of finding the NRF of

8120: 8020: 7508: 6946: 6671: 6543: 6368:. Proceedings of the 2009 SIAM Conference on Data Mining (SDM). pp. 1218–1229. 6121: 5641: 5548: 5503: 5438: 5393: 5164: 4952: 4546: 4065: 3959:

More control over the non-uniqueness of NMF is obtained with sparsity constraints.

2374:

Two simple divergence functions studied by Lee and Seung are the squared error (or

1091: 1067: 7332: 6648: 6210:". International Conference on Computer Vision (ICCV) Beijing, China, Oct., 2005. 5340:

Online Discussion Participation Prediction Using Non-negative Matrix Factorization

5300: 5106:

Thomas, L.B. (1974). "Problem 73-14, Rank factorization of nonnegative matrices".

4321:

Algorithmic: searching for global minima of the factors and factor initialization.

1830:, then the above minimization is mathematically equivalent to the minimization of 7653: 7207: 7063: 6950: 6481: 5487: 5266: 5193: 4693: 8143: 7887: 7003: 6967: 6672:"Mining the posterior cingulate: segregation between memory and pain components" 5563: 4999:"On the Equivalence of Nonnegative Matrix Factorization and Spectral Clustering" 4046: 4000:

property is used to separate the stellar light and the light scattered from the

3953: 3395: 2526: 1673: 1438:

is a matrix with 10000 rows and 500 columns, the same shape as the input matrix

549: 43: 8247:

Yong Xiang: "Blind Source Separation: Dependent Component Analysis", Springer,

8004: 7869: 6845: 6409: 6380: 5804: 5757: 5420: 5408: 5377: 5133:

Vavasis, S.A. (2009). "On the complexity of nonnegative matrix factorization".

5031: 4605: 4570: 4537: 4502: 8051: 7402: 7359: 7265: 7158: 6908:"Phoenix: A Weight-based Network Coordinate System Using Matrix Factorization" 6758: 6679: 6023: 5725: 5077: 4913:(1999). "Learning the parts of objects by non-negative matrix factorization". 4250:

subtypes, population stratification, tissue composition, and tumor clonality.

4001: 3387: 698: 394: 320: 8112: 7773: 7592: 7455: 7273: 7215: 6884: 6490: 5812: 5633: 5233: 5231: 4840: 4793: 4637:

Large-scale matrix factorization with distributed stochastic gradient descent

4503:"Non-negative Matrix Factorization: Robust Extraction of Extended Structures" 1478:

It is useful to think of each feature (column vector) in the features matrix

7966:(1997). "Least squares formulation of robust non-negative factor analysis". 7667:

Chao Liu; Hung-chih Yang; Jinliang Fan; Li-Wei He & Yi-Min Wang (2010).

7574: 7314: 6787: 5540: 5313: 4979: 3997: 3993: 3523:. A polynomial time algorithm for solving nonnegative rank factorization if 1178:

Matrix multiplication can be implemented as computing the column vectors of

1051: 1011: 857: 638: 8280:

Jen-Tzung Chien: "Source Separation and Machine Learning", Academic Press,

8206: 8151: 8012: 7600: 7559: 7500: 7410: 7367: 7291: 7233: 7176: 7125: 7082: 7021: 6700: 6150: 6113: 5495: 5385: 5285: 4944: 4832: 4712: 3505:

The contribution of the sequential NMF components can be compared with the

2172:

they become easier to store and manipulate. Another reason for factorizing

8187: 7541: 7192:"Deciphering signatures of mutational processes operative in human cancer" 6223:

Exponential Family Harmoniums with an Application to Information Retrieval

5413:

Proceedings of the International Joint Conference on Neural Networks, 2003

4848: 4801: 2151:. The elements of the residual matrix can either be negative or positive. 967:

Illustration of approximate non-negative matrix factorization: the matrix

6463: 5244: 4653:

TopicMF: Simultaneously Exploiting Ratings and Reviews for Recommendation

4449: 4013: 6277:

Max Welling & Markus Weber (2001). "Positive Tensor Factorization".

6046:. Proceedings of the 30th International Conference on Machine Learning. 4414:

Sparse nonnegative matrix approximation: new formulations and algorithms

4395:"Generalized Nonnegative Matrix Approximations with Bregman Divergences" 3668:

NMF with the least-squares objective is equivalent to a relaxed form of

1459:

is a linear combination of the 10 column vectors in the features matrix

7735:

Dong Wang; Ravichander Vipperla; Nick Evans; Thomas Fang Zheng (2013).

6263: 4756: 4087:

based on the outbound scientific citations in English Knowledge (XXG).

3696: 1405:

Assume we ask the algorithm to find 10 features in order to generate a

633: 7492: 5688: 5598: 5156: 5064:

Berman, A.; R.J. Plemmons (1974). "Inverses of nonnegative matrices".

4571:"Using Data Imputation for Signal Separation in High Contrast Imaging" 2190:, is that if one's goal is to approximately represent the elements of 7737:"Online Non-Negative Convolutive Pattern Learning for Speech Signals" 5337:

Fung, Yik-Hing; Li, Chun-Hung; Cheung, William K. (2 November 2007).

4501:

Ren, Bin; Pueyo, Laurent; Zhu, Guangtun B.; Duchêne, Gaspard (2018).

4354: 4072: 3370:

may be the same or different, as some NMF variants regularize one of

384: 7848:

Proceedings of the 2013 SIAM International Conference on Data Mining

6327:

Fast Nonnegative Tensor Factorization with an Active-set-like Method

6255: 6077:"Learning the parts of objects by non-negative matrix factorization" 5119: 4748: 2223:

can be anything in that space. Convex NMF restricts the columns of

7907: 7691:"Scalable Nonnegative Matrix Factorization with Block-wise Updates" 6986:"Fast and efficient estimation of individual ancestry coefficients" 6817: 6631: 5929: 5835: 5795: 5220: 4587: 4519: 4466: 3870:{\displaystyle \mathbf {\tilde {H}} =\mathbf {B} ^{-1}\mathbf {H} } 3653:, it is in fact equivalent to another instance of multinomial PCA, 8073: 6984:

Frichot E, Mathieu F, Trouillon T, Bouchard G, Francois O (2014).

6796: 6575: 6518: 6105: 6052: 5867: 5147: 4936: 4266: 4076: 3725:

can be used to transform the two factorization matrices by, e.g.,

3562: 3550:

Learning the parts of objects by non-negative matrix factorization

628: 623: 350: 7391:

IEEE/ACM Transactions on Computational Biology and Bioinformatics

6723: 6043:

A practical algorithm for topic modeling with provable guarantees

5457:"Projected Gradient Methods for Nonnegative Matrix Factorization" 3952:. In this simple case it will just correspond to a scaling and a 8169:"Bayesian Inference for Nonnegative Matrix Factorisation Models" 5214:. Proc. IEEE Workshop on Neural Networks for Signal Processing. 1251:{\displaystyle \mathbf {v} _{i}=\mathbf {W} \mathbf {h} _{i}\,,} 7818:

Proceedings of the 23rd International World Wide Web Conference

7676:

Proceedings of the 19th International World Wide Web Conference

7250:"Enter the Matrix: Factorization Uncovers Knowledge from Omics" 6670:

Nielsen, Finn Årup; Balslev, Daniela; Hansen, Lars Kai (2005).

5415:. Vol. 4. Portland, Oregon USA: IEEE. pp. 2758–2763. 6559:"PYNPOINT: an image processing package for finding exoplanets" 6382:

Document clustering based on non-negative matrix factorization

4280: 2285:. This greatly improves the quality of data representation of 963: 7809:

Xiangnan He; Min-Yen Kan; Peichu Xie & Xiao Chen (2014).

7711: 3775:{\displaystyle \mathbf {WH} =\mathbf {WBB} ^{-1}\mathbf {H} } 2407:

find nonnegative matrices W and H that minimize the function

7689:

Jiangtao Yin; Lixin Gao & Zhongfei (Mark) Zhang (2014).

6366:

Efficient Multiplicative updates for Support Vector Machines

6344:

Kenan Yilmaz; A. Taylan Cemgil & Umut Simsekli (2011).

5001:. Proc. SIAM Int'l Conf. Data Mining, pp. 606-610. May 2005 2516:

problem, although it may also still be referred to as NMF.

1625:{\displaystyle \mathbf {V} \simeq \mathbf {W} \mathbf {H} } 8302:

Nicolas Gillis: "Nonnegative Matrix Factorization", SIAM,

6972:

Machine Learning for Signal Processing, IEEE Workshop on

5366:

IEEE Transactions on Neural Networks and Learning Systems

8291:

Shoji Makino(Ed.): "Audio Source Separation", Springer,

6400:

Eggert, J.; Korner, E. (2004). "Sparse coding and NMF".

6208:

A Unifying Approach to Hard and Probabilistic Clustering

5343:. Wi-Iatw '07. IEEE Computer Society. pp. 284–287. 4158:

end-to-end links can be predicted after conducting only

3881:

they form another parametrization of the factorization.

1768:

If we furthermore impose an orthogonality constraint on

1380:

Here is an example based on a text-mining application:

1168:{\displaystyle \mathbf {V} =\mathbf {W} \mathbf {H} \,.} 916:

List of datasets in computer vision and image processing

8089:(2008). "Nonnegative Matrix and Tensor Factorization". 3567:

NMF as a probabilistic graphical model: visible units (

3358:

is found analogously. The procedures used to solve for

7811:"Comment-based Multi-View Clustering of Web 2.0 Items" 4131:

hosts, with the help of NMF, the distances of all the

3224: 3222: 3154: 1900:{\displaystyle \mathbf {H} _{kj}>\mathbf {H} _{ij}} 1465:

with coefficients supplied by the coefficients matrix

6968:

Wind noise reduction using non-negative sparse coding

6783:

Clustering of scientific citations in Knowledge (XXG)

6005:"Computing symmetric nonnegative rank factorizations" 4399:

Advances in Neural Information Processing Systems 18

4164: 4137: 4117: 3919: 3890: 3831: 3791: 3734: 3601: 3488: 3456: 3320:{\displaystyle \mathbf {V} =\mathbf {W} \mathbf {H} } 3298: 2882: 2657: 2635: 2416: 2391: 2239: 2031: 2008: 1988: 1968: 1948: 1921: 1863: 1843: 1796: 1774: 1733: 1684: 1658: 1638: 1603: 1581: 1518: 1211: 1142: 6906:

Yang Chen; Xiao Wang; Cong Shi; et al. (2011).

4964: 4962: 4196:

Speech denoising has been a long lasting problem in

1384:

Let the input matrix (the matrix to be factored) be

985:, which, when multiplied, approximately reconstruct 6966:

Schmidt, M.N., J. Larsen, and F.T. Hsiao. (2007). "

6916:

IEEE Transactions on Network and Service Management

5783:

International Journal of Data Science and Analytics

4412:Tandon, Rashish; Sra, Suvrit (September 13, 2010). 3346:found by a non-negative least squares solver, then 6746:Computational and Mathematical Organization Theory 4179: 4150: 4123: 3934: 3905: 3869: 3817: 3774: 3721:The factorization is not unique: A matrix and its 3637: 3494: 3474: 3319: 3280: 3208: 3107: 2864: 2641: 2482: 2399: 2277: 2057: 2014: 1994: 1974: 1954: 1934: 1899: 1849: 1822: 1782: 1757: 1719: 1664: 1644: 1624: 1589: 1564: 1250: 1167: 7522:Boutchko; Mitra; Baker; Jagust; Gullberg (2015). 7348:IEEE Journal of Biomedical and Health Informatics 6563:Monthly Notices of the Royal Astronomical Society 6243:Journal of Computational and Graphical Statistics 5093:Nonnegative matrices in the Mathematical Sciences 4430: 4428: 4426: 3926: 3897: 3838: 3799: 1565:{\displaystyle \mathbf {V} =(v_{1},\dots ,v_{n})} 6862:IEEE Journal on Selected Areas in Communications 6152:Variational Extensions to EM and Multinomial PCA 5702:Jingu Kim; Yunlong He & Haesun Park (2013). 5576:SIAM Journal on Matrix Analysis and Applications 4972:Algorithms for Non-negative Matrix Factorization 4075:. Another research group clustered parts of the 1184:as linear combinations of the column vectors in 7969:Chemometrics and Intelligent Laboratory Systems 4969:Daniel D. Lee & H. Sebastian Seung (2001). 3434:The sequential construction of NMF components ( 2493:Another type of NMF for images is based on the 6172:Relation between PLSA and NMF and Implications 4993: 4991: 4989: 2119:then amounts to the two non-negative matrices 2076:(PLSA), a popular document clustering method. 2058:{\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} 1823:{\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} 911:List of datasets for machine-learning research 7529:Journal of Cerebral Blood Flow and Metabolism 6075:Lee, Daniel D.; Sebastian, Seung, H. (1999). 2085:Approximate non-negative matrix factorization 944: 8: 6834:Computational Statistics & Data Analysis 6134:: CS1 maint: multiple names: authors list ( 6035: 6033: 5848: 5846: 5020:Computational Statistics & Data Analysis 4904: 4902: 3972:In astronomy, NMF is a promising method for 3382:. Specific approaches include the projected 3136:We note that the multiplicative factors for 2335:Different cost functions and regularizations 1672:that minimize the error function (using the 8174:Computational Intelligence and Neuroscience 7573:Abdalah; Boutchko; Mitra; Gullberg (2015). 5972:"Computing nonnegative rank factorizations" 3334:: in each step of such an algorithm, first 2291:. Furthermore, the resulting matrix factor 973:is represented by the two smaller matrices 6379:Wei Xu; Xin Liu & Yihong Gong (2003). 5906: 5904: 5902: 5205: 5203: 4733:(1971). "Self modeling curve resolution". 4393:Dhillon, Inderjit S.; Sra, Suvrit (2005). 3818:{\displaystyle \mathbf {{\tilde {W}}=WB} } 3595:from a probability distribution with mean 1190:using coefficients supplied by columns of 951: 937: 29: 8196: 8186: 8072: 7953: 7906: 7859: 7763: 7643: 7549: 7281: 7223: 7166: 7115: 7072: 7062: 7011: 6928: 6874: 6816: 6795: 6630: 6592: 6574: 6517: 6480: 6462: 6300: 6169:Eric Gaussier & Cyril Goutte (2005). 6051: 5987: 5946: 5928: 5866: 5834: 5794: 5724: 5678: 5588: 5530: 5477: 5275: 5265: 5238:Leo Taslaman & Björn Nilsson (2012). 5219: 5146: 4815:Pentti Paatero; Unto Tapper (June 1994). 4702: 4692: 4626: 4624: 4604: 4586: 4536: 4518: 4448: 4388: 4386: 4384: 4163: 4142: 4136: 4116: 3921: 3920: 3918: 3892: 3891: 3889: 3862: 3853: 3848: 3833: 3832: 3830: 3794: 3793: 3792: 3790: 3767: 3758: 3747: 3735: 3733: 3629: 3616: 3606: 3600: 3487: 3455: 3312: 3307: 3299: 3297: 3266: 3265: 3260: 3254: 3249: 3240: 3239: 3234: 3228: 3225: 3223: 3221: 3198: 3193: 3186: 3185: 3180: 3172: 3165: 3164: 3159: 3155: 3153: 3084: 3074: 3058: 3053: 3037: 3032: 3025: 3020: 2996: 2986: 2970: 2965: 2956: 2950: 2944: 2927: 2922: 2906: 2889: 2884: 2881: 2841: 2831: 2826: 2819: 2814: 2807: 2797: 2792: 2765: 2756: 2750: 2740: 2735: 2725: 2719: 2702: 2697: 2681: 2664: 2659: 2656: 2634: 2474: 2469: 2456: 2448: 2431: 2423: 2415: 2392: 2390: 2266: 2247: 2238: 2043: 2038: 2032: 2030: 2007: 2002:-th column gives the cluster centroid of 1987: 1967: 1947: 1926: 1920: 1888: 1883: 1870: 1865: 1862: 1842: 1808: 1803: 1797: 1795: 1775: 1773: 1732: 1708: 1683: 1657: 1637: 1617: 1612: 1604: 1602: 1582: 1580: 1553: 1534: 1519: 1517: 1244: 1238: 1233: 1227: 1218: 1213: 1210: 1161: 1156: 1151: 1143: 1141: 1050:NMF finds applications in such fields as 6347:Generalized Coupled Tensor Factorization 6003:Kalofolias, V.; Gallopoulos, E. (2012). 5824: 5822: 5450: 5448: 3419: 2331:, if it exists, is known to be NP-hard. 2200:Convex non-negative matrix factorization 1720:{\displaystyle \left\|V-WH\right\|_{F},} 1575:More specifically, the approximation of 1278:-th column vector of the product matrix 1095:researchers in the 1990s under the name 962: 5050: 5048: 4564: 4562: 4560: 4558: 4556: 4496: 4494: 4492: 4490: 4488: 4486: 4484: 4380: 3985:, especially for the direct imaging of 1982:gives the cluster centroids, i.e., the 1857:gives the cluster membership, i.e., if 37: 7744:IEEE Transactions on Signal Processing 7096:Hyunsoo Kim & Haesun Park (2007). 6557:Amara, Adam; Quanz, Sascha P. (2012). 6127: 5614:IEEE Transactions on Signal Processing 3655:probabilistic latent semantic analysis 3446:) was firstly used to relate NMF with 3267: 3241: 3187: 3166: 2074:probabilistic latent semantic analysis 2068:When the error function to be used is 5299:Hsieh, C. J.; Dhillon, I. S. (2011). 4237:NMF has been successfully applied in 4107:Scalable Internet distance prediction 3935:{\displaystyle \mathbf {\tilde {H}} } 3906:{\displaystyle \mathbf {\tilde {W}} } 2343:for measuring the divergence between 1414:with 10000 rows and 10 columns and a 7: 6324:Jingu Kim & Haesun Park (2012). 5658:SIAM Journal on Scientific Computing 5519:IEEE Transactions on Neural Networks 3638:{\displaystyle \sum _{a}W_{ia}h_{a}} 2582:There are several ways in which the 2297:becomes more sparse and orthogonal. 2278:{\displaystyle (v_{1},\dots ,v_{n})} 1915:, this suggests that the input data 1365:can be significantly less than both 6404:. Vol. 4. pp. 2529–2533. 6387:Association for Computing Machinery 5970:Campbell, S.L.; G.D. Poole (1981). 4997:C. Ding, X. He, H.D. Simon (2005). 2101:in NMF are selected so the product 906:Glossary of artificial intelligence 27:Algorithms for matrix decomposition 3705:NMF is an instance of nonnegative 2025:When the orthogonality constraint 25: 8335:Non-negative matrix factorization 6617:circumstellar disks with MLOCI". 6506:The Astrophysical Journal Letters 6220:Max Welling; et al. (2004). 5855:The Astrophysical Journal Letters 5752:. Vol. 4. pp. 606–610. 5091:A. Berman; R.J. Plemmons (1994). 4667:Ben Murrell; et al. (2011). 4049:applications. In this process, a 3996:of the NMF modeling process; the 3573:) are connected to hidden units ( 2629:by computing the following, with 2089:Usually the number of columns of 1101:non-negative matrix factorization 1099:. It became more widely known as 1008:non-negative matrix approximation 996:Non-negative matrix factorization 8317: 6693:10.1016/j.neuroimage.2005.04.034 6594:10.1111/j.1365-2966.2012.21918.x 4285: 3923: 3894: 3863: 3849: 3835: 3811: 3808: 3805: 3796: 3768: 3754: 3751: 3748: 3739: 3736: 3313: 3308: 3300: 3261: 3255: 3250: 3235: 3229: 3199: 3194: 3181: 3173: 3160: 3054: 3033: 3021: 2966: 2957: 2923: 2885: 2827: 2815: 2793: 2757: 2736: 2698: 2660: 2568:. By spatio-temporal pooling of 2460: 2457: 2449: 2432: 2424: 2393: 2107:will become an approximation to 2039: 2033: 1884: 1866: 1804: 1798: 1776: 1758:{\displaystyle W\geq 0,H\geq 0.} 1618: 1613: 1605: 1583: 1520: 1301:-th column vector of the matrix 1234: 1228: 1214: 1157: 1152: 1144: 18:Nonnegative matrix factorization 8092:IEEE Signal Processing Magazine 6939:10.1109/tnsm.2011.110911.100079 4192:Non-stationary speech denoising 3983:methods of detecting exoplanets 3688:NMF can be seen as a two-layer 3678:contains cluster centroids and 2204:In standard NMF, matrix factor 1121:be the product of the matrices 7926:J. Shen; G. W. Israël (1989). 5712:Journal of Global Optimization 5066:Linear and Multilinear Algebra 4650:Yang Bao; et al. (2014). 4631:Rainer Gemulla; Erik Nijkamp; 4174: 4168: 3469: 3457: 3097: 3085: 3081: 3071: 3049: 3016: 3009: 2997: 2993: 2983: 2961: 2953: 2940: 2928: 2918: 2902: 2890: 2854: 2842: 2838: 2804: 2788: 2785: 2778: 2766: 2762: 2747: 2731: 2728: 2715: 2703: 2693: 2677: 2665: 2594:may be found: Lee and Seung's 2465: 2444: 2436: 2420: 2301:Nonnegative rank factorization 2272: 2240: 1704: 1687: 1559: 1527: 326:Relevance vector machine (RVM) 1: 7982:10.1016/S0169-7439(96)00044-5 7117:10.1093/bioinformatics/btm134 6722:Cohen, William (2005-04-04). 6311:10.1016/S0167-8655(01)00070-8 5011:Ding C, Li Y, Peng W (2008). 4786:10.1016/S0021-8502(05)80089-8 4350:Multilinear subspace learning 3482:-th component with the first 2649:as an index of the iteration. 2529:fashion. One such use is for 2512:due to the similarity to the 2315:is equal to its actual rank, 2113:. The full decomposition of 1423:with 10 rows and 500 columns. 1097:positive matrix factorization 815:Computational learning theory 379:Expectation–maximization (EM) 7955:10.1016/0004-6981(89)90190-X 7654:10.1016/j.patcog.2007.09.010 7208:10.1016/j.celrep.2012.12.008 7064:10.1371/journal.pcbi.1000029 6619:Astronomy & Astrophysics 6536:10.1088/0004-637X/694/2/L148 6451:Astronomy & Astrophysics 5989:10.1016/0024-3795(81)90272-x 5488:10.1162/neco.2007.19.10.2756 5267:10.1371/journal.pone.0046331 5194:10.1016/j.neucom.2008.01.022 4893:10.1016/1352-2310(94)00367-T 4694:10.1371/journal.pone.0028898 3558:principal component analysis 3544:Relation to other techniques 3448:Principal Component Analysis 2400:{\displaystyle \mathbf {V} } 1783:{\displaystyle \mathbf {H} } 1590:{\displaystyle \mathbf {V} } 1202:can be computed as follows: 1035:into (usually) two matrices 772:Coefficient of determination 619:Convolutional neural network 331:Support vector machine (SVM) 8364:Machine learning algorithms 8144:10.1162/neco.2008.04-08-771 7004:10.1534/genetics.113.160572 6780:Nielsen, Finn Årup (2008). 6649:10.1051/0004-6361/201525837 6280:Pattern Recognition Letters 5948:10.3847/0004-637X/824/2/117 5885:10.1088/2041-8205/755/2/L28 5750:Proc. SIAM Data Mining Conf 3651:Kullback–Leibler divergence 2380:Kullback–Leibler divergence 2070:Kullback–Leibler divergence 1196:. That is, each column of 923:Outline of machine learning 820:Empirical risk minimization 8385: 8167:Ali Taylan Cemgil (2009). 8005:10.1162/neco.2007.19.3.780 7870:10.1137/1.9781611972832.28 7042:PLOS Computational Biology 6846:10.1016/j.csda.2006.11.006 6482:10.1051/0004-6361:20066282 6410:10.1109/IJCNN.2004.1381036 5805:10.1007/s41060-022-00370-9 5758:10.1137/1.9781611972757.70 5421:10.1109/IJCNN.2003.1224004 5378:10.1109/TNNLS.2012.2197827 5212:Non-negative sparse coding 5032:10.1016/j.csda.2008.01.011 4773:Journal of Aerosol Science 4635:; Yannis Sismanis (2011). 4371:Sources and external links 3332:non-negative least squares 2617:Then update the values in 2596:multiplicative update rule 2510:non-negative sparse coding 2233:of the input data vectors 2095:and the number of rows of 2072:, NMF is identical to the 1962:-th cluster. The computed 1837:Furthermore, the computed 560:Feedforward neural network 311:Artificial neural networks 8052:10.1007/s11434-005-1109-6 7403:10.1109/TCBB.2021.3091814 7360:10.1109/JBHI.2020.2991763 7266:10.1016/j.tig.2018.07.003 7159:10.1007/s00401-012-1077-2 6759:10.1007/s10588-005-5380-5 6024:10.1016/j.laa.2011.03.016 5917:The Astrophysical Journal 5726:10.1007/s10898-013-0035-4 5210:Hoyer, Patrik O. (2002). 5078:10.1080/03081087408817055 4575:The Astrophysical Journal 4507:The Astrophysical Journal 4294:This section needs to be 4204:is suitable for additive 543:Artificial neural network 8113:10.1109/MSP.2008.4408452 8031:Chinese Science Bulletin 7774:10.1109/tsp.2012.2222381 7593:10.1109/TMI.2014.2352033 7456:10.1109/tns.1982.4332188 6885:10.1109/JSAC.2006.884026 5634:10.1109/TSP.2012.2190406 4606:10.3847/1538-4357/ab7024 4538:10.3847/1538-4357/aaa1f2 4437:The Astronomical Journal 3785:If the two new matrices 3502:components constructed. 852:Journals and conferences 799:Mathematical foundations 709:Temporal difference (TD) 565:Recurrent neural network 485:Conditional random field 408:Dimensionality reduction 156:Dimensionality reduction 118:Quantum machine learning 113:Neuromorphic engineering 73:Self-supervised learning 68:Semi-supervised learning 7933:Atmospheric Environment 7315:10.1145/1150402.1150420 6641:2015A&A...581A..24W 6473:2007A&A...469..575B 5911:Pueyo, Laurent (2016). 5541:10.1109/TNN.2007.895831 5353:– via dl.acm.org. 5314:10.1145/2020408.2020577 4872:Atmospheric Environment 4198:audio signal processing 4081:English Knowledge (XXG) 2531:collaborative filtering 1632:is achieved by finding 1072:audio signal processing 1064:missing data imputation 261:Apprenticeship learning 7580:IEEE Trans Med Imaging 7480:IEEE Trans Med Imaging 7035:Devarajan, K. (2008). 5455:Lin, Chih-Jen (2007). 4833:10.1002/ENV.3170050203 4220:Sparse NMF is used in 4181: 4152: 4125: 4099:Spectral data analysis 4068:of related documents. 3936: 3907: 3884:The non-negativity of 3871: 3819: 3776: 3711:support vector machine 3646: 3639: 3507:Karhunen–Loève theorem 3496: 3476: 3426: 3321: 3282: 3210: 3109: 2866: 2643: 2535:recommendation systems 2484: 2401: 2279: 2178:into smaller matrices 2131:as well as a residual 2059: 2016: 1996: 1976: 1956: 1936: 1901: 1851: 1824: 1784: 1759: 1721: 1666: 1646: 1626: 1591: 1566: 1402:represents a document. 1252: 1169: 992: 810:Bias–variance tradeoff 692:Reinforcement learning 668:Spiking neural network 78:Reinforcement learning 8085:; Rafal Zdunek & 7542:10.1038/jcbfm.2015.69 7146:Acta Neuropathologica 7139:Schwalbe, E. (2013). 6724:"Enron Email Dataset" 6149:Wray Buntine (2002). 5095:. Philadelphia: SIAM. 4182: 4153: 4151:{\displaystyle N^{2}} 4126: 3937: 3908: 3872: 3820: 3777: 3707:quadratic programming 3640: 3566: 3497: 3477: 3475:{\displaystyle (n+1)} 3423: 3322: 3283: 3211: 3110: 2867: 2644: 2485: 2402: 2280: 2060: 2017: 1997: 1977: 1957: 1937: 1935:{\displaystyle v_{j}} 1902: 1852: 1825: 1785: 1760: 1722: 1667: 1647: 1627: 1592: 1567: 1253: 1170: 1016:multivariate analysis 966: 646:Neural radiance field 468:Structured prediction 191:Structured prediction 63:Unsupervised learning 7854:. pp. 252–260. 7309:. pp. 126–135. 5188:(10–12): 1824–1831. 4909:Daniel D. Lee & 4360:Tensor decomposition 4180:{\displaystyle O(N)} 4162: 4135: 4115: 4045:NMF can be used for 3942:applies at least if 3917: 3888: 3829: 3789: 3732: 3672:: the matrix factor 3599: 3486: 3454: 3296: 3220: 3152: 2880: 2655: 2633: 2495:total variation norm 2414: 2389: 2237: 2029: 2006: 1986: 1966: 1946: 1919: 1861: 1841: 1794: 1772: 1731: 1682: 1656: 1636: 1601: 1579: 1516: 1209: 1140: 835:Statistical learning 733:Learning with humans 525:Local outlier factor 8188:10.1155/2009/785152 8105:2008ISPM...25R.142C 8044:2006ChSBu..51....7L 7946:1989AtmEn..23.2289S 7756:2013ITSP...61...44W 7636:2008PatRe..41.1350B 7624:Pattern Recognition 7448:1982ITNS...29.1310D 7435:IEEE Trans Nucl Sci 7055:2008PLSCB...4E0029D 6585:2012MNRAS.427..948A 6528:2009ApJ...694L.148L 6389:. pp. 267–273. 6293:2001PaReL..22.1255W 6098:1999Natur.401..788L 6062:2012arXiv1212.4777A 6012:Linear Algebra Appl 5976:Linear Algebra Appl 5939:2016ApJ...824..117P 5877:2012ApJ...755L..28S 5671:2011SJSC...33.3261K 5626:2012ITSP...60.2882G 5407:Behnke, S. (2003). 5258:2012PLoSO...746331T 4982:. pp. 556–562. 4929:1999Natur.401..788L 4885:1995AtmEn..29.1705A 4731:Edward A. Sylvestre 4685:2011PLoSO...628898M 4597:2020ApJ...892...74R 4529:2018ApJ...852..104R 4459:2007AJ....133..734B 4401:. pp. 283–290. 4345:Multilinear algebra 4222:Population genetics 4216:Population genetics 4085:scientific journals 4006:circumstellar disks 3987:circumstellar disks 3974:dimension reduction 3554:vector quantization 2949: 2917: 2724: 2692: 2566:convolution kernels 2479: 2231:convex combinations 1508:Clustering property 1416:coefficients matrix 1076:recommender systems 1060:document clustering 678:Electrochemical RAM 585:reservoir computing 316:Logistic regression 235:Supervised learning 221:Multimodal learning 196:Feature engineering 141:Generative modeling 103:Rule-based learning 98:Curriculum learning 58:Supervised learning 33:Part of a series on 8131:Neural Computation 7992:Neural Computation 7254:Trends in Genetics 5562:Hyunsoo Kim & 5465:Neural Computation 4911:H. Sebastian Seung 4177: 4148: 4121: 3979:parallel computing 3948:is a non-negative 3932: 3903: 3867: 3815: 3772: 3690:directed graphical 3670:K-means clustering 3663:latent class model 3659:maximum likelihood 3647: 3635: 3611: 3579:) through weights 3492: 3472: 3427: 3317: 3278: 3276: 3206: 3105: 2921: 2883: 2862: 2696: 2658: 2639: 2545:If the columns of 2480: 2442: 2397: 2275: 2055: 2012: 1992: 1972: 1952: 1932: 1897: 1847: 1832:K-means clustering 1820: 1780: 1755: 1717: 1662: 1642: 1622: 1587: 1562: 1248: 1165: 993: 246: • 161:Density estimation 8308:978-1-611976-40-3 7940:(10): 2289–2298. 7879:978-1-61197-262-7 7716:mahout.apache.org 7493:10.1109/42.996340 7354:(11): 3162–3172. 7110:(12): 1495–1502. 6869:(12): 2273–2284. 6419:978-0-7803-8359-3 6287:(12): 1255–1261. 6092:(6755): 788–791. 5767:978-0-89871-593-4 5689:10.1137/110821172 5599:10.1137/07069239x 5472:(10): 2756–2779. 5430:978-0-7803-7898-8 5157:10.1137/070709967 4923:(6755): 788–791. 4879:(14): 1705–1718. 4727:William H. Lawton 4315: 4314: 4226:genetic admixture 4124:{\displaystyle N} 3929: 3900: 3841: 3802: 3602: 3495:{\displaystyle n} 3274: 3204: 3103: 2860: 2642:{\displaystyle n} 2541:Convolutional NMF 2502:L1 regularization 2166:are smaller than 2015:{\displaystyle k} 1995:{\displaystyle k} 1975:{\displaystyle W} 1955:{\displaystyle k} 1850:{\displaystyle H} 1665:{\displaystyle H} 1645:{\displaystyle W} 961: 960: 766:Model diagnostics 749:Human-in-the-loop 592:Boltzmann machine 505:Anomaly detection 301:Linear regression 216:Ontology learning 211:Grammar induction 186:Semantic analysis 181:Association rules 166:Anomaly detection 108:Neuro-symbolic AI 16:(Redirected from 8376: 8321: 8320: 8210: 8200: 8190: 8163: 8124: 8083:Andrzej Cichocki 8078: 8076: 8063: 8024: 7985: 7959: 7957: 7913: 7912: 7910: 7898: 7892: 7891: 7863: 7853: 7842: 7836: 7835: 7833: 7832: 7826: 7820:. Archived from 7815: 7806: 7800: 7799: 7797: 7796: 7790: 7784:. Archived from 7767: 7741: 7732: 7726: 7725: 7723: 7722: 7708: 7702: 7701: 7695: 7686: 7680: 7679: 7673: 7664: 7658: 7657: 7647: 7630:(4): 1350–1362. 7619: 7613: 7612: 7570: 7564: 7563: 7553: 7519: 7513: 7512: 7474: 7468: 7467: 7429: 7423: 7422: 7397:(4): 1956–1967. 7386: 7380: 7379: 7343: 7337: 7336: 7302: 7296: 7295: 7285: 7244: 7238: 7237: 7227: 7187: 7181: 7180: 7170: 7136: 7130: 7129: 7119: 7093: 7087: 7086: 7076: 7066: 7032: 7026: 7025: 7015: 6981: 6975: 6964: 6958: 6957: 6955: 6949:. Archived from 6932: 6912: 6903: 6897: 6896: 6878: 6856: 6850: 6849: 6829: 6823: 6822: 6820: 6808: 6802: 6801: 6799: 6777: 6771: 6770: 6740: 6734: 6733: 6731: 6730: 6719: 6713: 6712: 6676: 6667: 6661: 6660: 6634: 6613: 6607: 6606: 6596: 6578: 6554: 6548: 6547: 6521: 6501: 6495: 6494: 6484: 6466: 6464:astro-ph/0703072 6438: 6432: 6431: 6397: 6391: 6390: 6376: 6370: 6369: 6361: 6355: 6354: 6352: 6341: 6335: 6334: 6332: 6321: 6315: 6314: 6304: 6274: 6268: 6267: 6234: 6228: 6227: 6217: 6211: 6200: 6194: 6193: 6191: 6190: 6184: 6177: 6166: 6160: 6159: 6157: 6146: 6140: 6139: 6133: 6125: 6081: 6072: 6066: 6065: 6055: 6037: 6028: 6027: 6009: 6000: 5994: 5993: 5991: 5967: 5961: 5960: 5950: 5932: 5908: 5897: 5896: 5870: 5850: 5841: 5840: 5838: 5826: 5817: 5816: 5798: 5778: 5772: 5771: 5745: 5739: 5738: 5728: 5708: 5699: 5693: 5692: 5682: 5665:(6): 3261–3281. 5652: 5646: 5645: 5620:(6): 2882–2898. 5609: 5603: 5602: 5592: 5572: 5559: 5553: 5552: 5534: 5525:(6): 1589–1596. 5514: 5508: 5507: 5481: 5461: 5452: 5443: 5442: 5404: 5398: 5397: 5372:(7): 1087–1099. 5361: 5355: 5354: 5334: 5328: 5327: 5307: 5296: 5290: 5289: 5279: 5269: 5235: 5226: 5225: 5223: 5207: 5198: 5197: 5175: 5169: 5168: 5150: 5141:(3): 1364–1377. 5130: 5124: 5123: 5103: 5097: 5096: 5088: 5082: 5081: 5061: 5055: 5052: 5043: 5042: 5040: 5034:. Archived from 5026:(8): 3913–3927. 5017: 5008: 5002: 4995: 4984: 4983: 4977: 4966: 4957: 4956: 4906: 4897: 4896: 4859: 4853: 4852: 4812: 4806: 4805: 4767: 4761: 4760: 4723: 4717: 4716: 4706: 4696: 4664: 4658: 4657: 4647: 4641: 4640: 4628: 4619: 4618: 4608: 4590: 4566: 4551: 4550: 4540: 4522: 4498: 4479: 4478: 4452: 4450:astro-ph/0606170 4432: 4421: 4420: 4418: 4409: 4403: 4402: 4390: 4310: 4307: 4301: 4289: 4288: 4281: 4277:Current research 4255:drug repurposing 4186: 4184: 4183: 4178: 4157: 4155: 4154: 4149: 4147: 4146: 4130: 4128: 4127: 4122: 4062:feature-document 3947: 3941: 3939: 3938: 3933: 3931: 3930: 3922: 3912: 3910: 3909: 3904: 3902: 3901: 3893: 3876: 3874: 3873: 3868: 3866: 3861: 3860: 3852: 3843: 3842: 3834: 3824: 3822: 3821: 3816: 3814: 3804: 3803: 3795: 3781: 3779: 3778: 3773: 3771: 3766: 3765: 3757: 3742: 3709:, just like the 3683: 3677: 3644: 3642: 3641: 3636: 3634: 3633: 3624: 3623: 3610: 3590: 3584: 3578: 3572: 3538: 3534: 3528: 3522: 3501: 3499: 3498: 3493: 3481: 3479: 3478: 3473: 3445: 3439: 3412: 3406: 3384:gradient descent 3381: 3375: 3369: 3363: 3357: 3351: 3345: 3339: 3326: 3324: 3323: 3318: 3316: 3311: 3303: 3290:matrices of ones 3287: 3285: 3284: 3279: 3277: 3275: 3273: 3272: 3271: 3270: 3264: 3258: 3253: 3247: 3246: 3245: 3244: 3238: 3232: 3226: 3215: 3213: 3212: 3207: 3205: 3203: 3202: 3197: 3192: 3191: 3190: 3184: 3177: 3176: 3171: 3170: 3169: 3163: 3156: 3147: 3141: 3128: 3122: 3114: 3112: 3111: 3106: 3104: 3102: 3101: 3100: 3079: 3078: 3069: 3068: 3057: 3048: 3047: 3036: 3030: 3029: 3024: 3014: 3013: 3012: 2991: 2990: 2981: 2980: 2969: 2960: 2951: 2948: 2943: 2926: 2916: 2905: 2888: 2871: 2869: 2868: 2863: 2861: 2859: 2858: 2857: 2836: 2835: 2830: 2824: 2823: 2818: 2812: 2811: 2802: 2801: 2796: 2783: 2782: 2781: 2760: 2755: 2754: 2745: 2744: 2739: 2726: 2723: 2718: 2701: 2691: 2680: 2663: 2648: 2646: 2645: 2640: 2628: 2622: 2613: 2607: 2593: 2587: 2573: 2563: 2557: 2550: 2489: 2487: 2486: 2481: 2478: 2473: 2468: 2464: 2463: 2452: 2435: 2427: 2406: 2404: 2403: 2398: 2396: 2370: 2364: 2355:and possibly by 2354: 2348: 2330: 2324: 2314: 2307:nonnegative rank 2296: 2290: 2284: 2282: 2281: 2276: 2271: 2270: 2252: 2251: 2228: 2222: 2216: 2195: 2189: 2183: 2177: 2171: 2165: 2159: 2150: 2136: 2130: 2124: 2118: 2112: 2106: 2100: 2094: 2064: 2062: 2061: 2056: 2048: 2047: 2042: 2036: 2021: 2019: 2018: 2013: 2001: 1999: 1998: 1993: 1981: 1979: 1978: 1973: 1961: 1959: 1958: 1953: 1941: 1939: 1938: 1933: 1931: 1930: 1906: 1904: 1903: 1898: 1896: 1895: 1887: 1878: 1877: 1869: 1856: 1854: 1853: 1848: 1829: 1827: 1826: 1821: 1813: 1812: 1807: 1801: 1789: 1787: 1786: 1781: 1779: 1764: 1762: 1761: 1756: 1726: 1724: 1723: 1718: 1713: 1712: 1707: 1703: 1671: 1669: 1668: 1663: 1651: 1649: 1648: 1643: 1631: 1629: 1628: 1623: 1621: 1616: 1608: 1596: 1594: 1593: 1588: 1586: 1571: 1569: 1568: 1563: 1558: 1557: 1539: 1538: 1523: 1503: 1497: 1490: 1483: 1470: 1464: 1458: 1449: 1443: 1437: 1431: 1422: 1413: 1401: 1395: 1389: 1376: 1370: 1364: 1358: 1348: 1342: 1332: 1326: 1316: 1306: 1300: 1294: 1283: 1277: 1271: 1257: 1255: 1254: 1249: 1243: 1242: 1237: 1231: 1223: 1222: 1217: 1201: 1195: 1189: 1183: 1174: 1172: 1171: 1166: 1160: 1155: 1147: 1132: 1126: 1120: 1046: 1040: 1030: 990: 984: 978: 972: 953: 946: 939: 900:Related articles 777:Confusion matrix 530:Isolation forest 475:Graphical models 254: 253: 206:Learning to rank 201:Feature learning 39:Machine learning 30: 21: 8384: 8383: 8379: 8378: 8377: 8375: 8374: 8373: 8344: 8343: 8342: 8341: 8340: 8322: 8318: 8313: 8166: 8127: 8087:Shun-ichi Amari 8081: 8066: 8038:(17–18): 7–18. 8027: 7988: 7962: 7925: 7921: 7916: 7900: 7899: 7895: 7880: 7861:10.1.1.301.1771 7851: 7844: 7843: 7839: 7830: 7828: 7824: 7813: 7808: 7807: 7803: 7794: 7792: 7788: 7765:10.1.1.707.7348 7739: 7734: 7733: 7729: 7720: 7718: 7712:"Apache Mahout" 7710: 7709: 7705: 7693: 7688: 7687: 7683: 7671: 7666: 7665: 7661: 7645:10.1.1.137.8281 7621: 7620: 7616: 7572: 7571: 7567: 7521: 7520: 7516: 7476: 7475: 7471: 7431: 7430: 7426: 7388: 7387: 7383: 7345: 7344: 7340: 7325: 7304: 7303: 7299: 7260:(10): 790–805. 7246: 7245: 7241: 7189: 7188: 7184: 7138: 7137: 7133: 7095: 7094: 7090: 7049:(7): e1000029. 7034: 7033: 7029: 6983: 6982: 6978: 6965: 6961: 6953: 6930:10.1.1.300.2851 6910: 6905: 6904: 6900: 6876:10.1.1.136.3837 6858: 6857: 6853: 6831: 6830: 6826: 6810: 6809: 6805: 6779: 6778: 6774: 6742: 6741: 6737: 6728: 6726: 6721: 6720: 6716: 6674: 6669: 6668: 6664: 6615: 6614: 6610: 6556: 6555: 6551: 6503: 6502: 6498: 6440: 6439: 6435: 6420: 6399: 6398: 6394: 6378: 6377: 6373: 6363: 6362: 6358: 6350: 6343: 6342: 6338: 6330: 6323: 6322: 6318: 6276: 6275: 6271: 6256:10.2307/1390831 6236: 6235: 6231: 6219: 6218: 6214: 6201: 6197: 6188: 6186: 6182: 6175: 6168: 6167: 6163: 6155: 6148: 6147: 6143: 6126: 6079: 6074: 6073: 6069: 6039: 6038: 6031: 6007: 6002: 6001: 5997: 5969: 5968: 5964: 5910: 5909: 5900: 5852: 5851: 5844: 5828: 5827: 5820: 5780: 5779: 5775: 5768: 5747: 5746: 5742: 5706: 5701: 5700: 5696: 5654: 5653: 5649: 5611: 5610: 5606: 5570: 5561: 5560: 5556: 5516: 5515: 5511: 5479:10.1.1.308.9135 5459: 5454: 5453: 5446: 5431: 5406: 5405: 5401: 5363: 5362: 5358: 5351: 5336: 5335: 5331: 5324: 5305: 5298: 5297: 5293: 5237: 5236: 5229: 5209: 5208: 5201: 5177: 5176: 5172: 5132: 5131: 5127: 5120:10.1137/1016064 5105: 5104: 5100: 5090: 5089: 5085: 5063: 5062: 5058: 5053: 5046: 5038: 5015: 5010: 5009: 5005: 4996: 4987: 4975: 4968: 4967: 4960: 4908: 4907: 4900: 4861: 4860: 4856: 4814: 4813: 4809: 4769: 4768: 4764: 4749:10.2307/1267173 4725: 4724: 4720: 4666: 4665: 4661: 4649: 4648: 4644: 4630: 4629: 4622: 4568: 4567: 4554: 4500: 4499: 4482: 4434: 4433: 4424: 4416: 4411: 4410: 4406: 4392: 4391: 4382: 4378: 4373: 4365:Tensor software 4341: 4311: 4305: 4302: 4299: 4290: 4286: 4279: 4263: 4261:Nuclear imaging 4247:DNA methylation 4243:gene expression 4241:for clustering 4235: 4218: 4194: 4160: 4159: 4138: 4133: 4132: 4113: 4112: 4109: 4101: 4043: 4026:data imputation 4022: 4020:Data imputation 3970: 3965: 3950:monomial matrix 3943: 3915: 3914: 3886: 3885: 3847: 3827: 3826: 3787: 3786: 3746: 3730: 3729: 3719: 3679: 3673: 3625: 3612: 3597: 3596: 3586: 3580: 3574: 3568: 3546: 3536: 3530: 3524: 3518: 3515: 3484: 3483: 3452: 3451: 3441: 3435: 3432: 3408: 3402: 3377: 3371: 3365: 3359: 3353: 3347: 3341: 3335: 3294: 3293: 3259: 3248: 3233: 3227: 3218: 3217: 3179: 3178: 3158: 3157: 3150: 3149: 3143: 3137: 3124: 3118: 3080: 3070: 3052: 3031: 3019: 3015: 2992: 2982: 2964: 2952: 2878: 2877: 2837: 2825: 2813: 2803: 2791: 2784: 2761: 2746: 2734: 2727: 2653: 2652: 2631: 2630: 2624: 2618: 2609: 2603: 2589: 2583: 2580: 2569: 2564:, representing 2559: 2553: 2546: 2543: 2522: 2447: 2443: 2412: 2411: 2387: 2386: 2366: 2360: 2350: 2344: 2337: 2326: 2316: 2310: 2303: 2292: 2286: 2262: 2243: 2235: 2234: 2224: 2218: 2215: 2205: 2202: 2191: 2185: 2179: 2173: 2167: 2161: 2155: 2138: 2132: 2126: 2120: 2114: 2108: 2102: 2096: 2090: 2087: 2082: 2037: 2027: 2026: 2004: 2003: 1984: 1983: 1964: 1963: 1944: 1943: 1922: 1917: 1916: 1882: 1864: 1859: 1858: 1839: 1838: 1802: 1792: 1791: 1770: 1769: 1729: 1728: 1690: 1686: 1685: 1680: 1679: 1654: 1653: 1634: 1633: 1599: 1598: 1577: 1576: 1549: 1530: 1514: 1513: 1510: 1499: 1493: 1486: 1479: 1466: 1460: 1454: 1445: 1439: 1433: 1427: 1426:The product of 1418: 1409: 1407:features matrix 1397: 1391: 1385: 1372: 1366: 1360: 1350: 1344: 1334: 1328: 1318: 1312: 1302: 1296: 1293: 1285: 1279: 1273: 1270: 1262: 1232: 1212: 1207: 1206: 1197: 1191: 1185: 1179: 1138: 1137: 1128: 1122: 1116: 1113: 1088: 1056:computer vision 1042: 1036: 1026: 986: 980: 974: 968: 957: 928: 927: 901: 893: 892: 853: 845: 844: 805:Kernel machines 800: 792: 791: 767: 759: 758: 739:Active learning 734: 726: 725: 694: 684: 683: 609:Diffusion model 545: 535: 534: 507: 497: 496: 470: 460: 459: 415:Factor analysis 410: 400: 399: 383: 346: 336: 335: 256: 255: 239: 238: 237: 226: 225: 131: 123: 122: 88:Online learning 53: 41: 28: 23: 22: 15: 12: 11: 5: 8382: 8380: 8372: 8371: 8366: 8361: 8356: 8354:Linear algebra 8346: 8345: 8323: 8316: 8315: 8314: 8312: 8311: 8300: 8297:978-3030103033 8289: 8286:978-0128177969 8278: 8275:978-3844048148 8267: 8264:978-3662517000 8256: 8253:978-9812872265 8245: 8242:978-3844324891 8234: 8231:978-0470746660 8222: 8219:978-9774540455 8211: 8164: 8138:(3): 793–830. 8125: 8099:(1): 142–145. 8079: 8064: 8025: 7999:(3): 780–791. 7986: 7964:Pentti Paatero 7960: 7922: 7920: 7917: 7915: 7914: 7893: 7878: 7837: 7801: 7727: 7703: 7681: 7659: 7614: 7565: 7536:(7): 1104–11. 7514: 7469: 7442:(4): 1310–21. 7424: 7381: 7338: 7323: 7297: 7239: 7202:(1): 246–259. 7182: 7153:(3): 359–371. 7131: 7103:Bioinformatics 7088: 7027: 6998:(4): 973–983. 6976: 6959: 6956:on 2011-11-14. 6923:(4): 334–347. 6898: 6851: 6840:(1): 155–173. 6824: 6803: 6772: 6753:(3): 249–264. 6735: 6714: 6687:(3): 520–522. 6662: 6608: 6549: 6496: 6457:(2): 575–586. 6433: 6418: 6392: 6371: 6356: 6336: 6316: 6269: 6250:(4): 854–888. 6238:Pentti Paatero 6229: 6212: 6195: 6161: 6141: 6067: 6029: 6018:(2): 421–435. 5995: 5962: 5898: 5842: 5818: 5789:(1): 119–134. 5773: 5766: 5740: 5719:(2): 285–319. 5694: 5680:10.1.1.419.798 5647: 5604: 5590:10.1.1.70.3485 5583:(2): 713–730. 5554: 5532:10.1.1.407.318 5509: 5444: 5429: 5399: 5356: 5349: 5329: 5322: 5291: 5252:(11): e46331. 5227: 5199: 5181:Neurocomputing 5170: 5125: 5114:(3): 393–394. 5098: 5083: 5072:(2): 161–172. 5056: 5044: 5041:on 2016-03-04. 5003: 4985: 4958: 4898: 4867:Pentti Paatero 4854: 4827:(2): 111–126. 4821:Environmetrics 4807: 4762: 4743:(3): 617–633. 4718: 4679:(12): e28898. 4659: 4642: 4620: 4552: 4480: 4467:10.1086/510127 4443:(2): 734–754. 4422: 4404: 4379: 4377: 4374: 4372: 4369: 4368: 4367: 4362: 4357: 4352: 4347: 4340: 4337: 4336: 4335: 4332: 4329: 4326: 4325:Decomposition. 4322: 4313: 4312: 4293: 4291: 4284: 4278: 4275: 4262: 4259: 4239:bioinformatics 4234: 4233:Bioinformatics 4231: 4217: 4214: 4206:Gaussian noise 4193: 4190: 4176: 4173: 4170: 4167: 4145: 4141: 4120: 4108: 4105: 4100: 4097: 4042: 4039: 4021: 4018: 3969: 3966: 3964: 3961: 3928: 3925: 3899: 3896: 3865: 3859: 3856: 3851: 3846: 3840: 3837: 3813: 3810: 3807: 3801: 3798: 3783: 3782: 3770: 3764: 3761: 3756: 3753: 3750: 3745: 3741: 3738: 3718: 3715: 3632: 3628: 3622: 3619: 3615: 3609: 3605: 3545: 3542: 3514: 3511: 3491: 3471: 3468: 3465: 3462: 3459: 3431: 3430:Sequential NMF 3428: 3315: 3310: 3306: 3302: 3269: 3263: 3257: 3252: 3243: 3237: 3231: 3201: 3196: 3189: 3183: 3175: 3168: 3162: 3131: 3130: 3115: 3099: 3096: 3093: 3090: 3087: 3083: 3077: 3073: 3067: 3064: 3061: 3056: 3051: 3046: 3043: 3040: 3035: 3028: 3023: 3018: 3011: 3008: 3005: 3002: 2999: 2995: 2989: 2985: 2979: 2976: 2973: 2968: 2963: 2959: 2955: 2947: 2942: 2939: 2936: 2933: 2930: 2925: 2920: 2915: 2912: 2909: 2904: 2901: 2898: 2895: 2892: 2887: 2875: 2872: 2856: 2853: 2850: 2847: 2844: 2840: 2834: 2829: 2822: 2817: 2810: 2806: 2800: 2795: 2790: 2787: 2780: 2777: 2774: 2771: 2768: 2764: 2759: 2753: 2749: 2743: 2738: 2733: 2730: 2722: 2717: 2714: 2711: 2708: 2705: 2700: 2695: 2690: 2687: 2684: 2679: 2676: 2673: 2670: 2667: 2662: 2650: 2638: 2615: 2579: 2576: 2542: 2539: 2521: 2518: 2491: 2490: 2477: 2472: 2467: 2462: 2459: 2455: 2451: 2446: 2441: 2438: 2434: 2430: 2426: 2422: 2419: 2395: 2376:Frobenius norm 2357:regularization 2341:cost functions 2336: 2333: 2302: 2299: 2274: 2269: 2265: 2261: 2258: 2255: 2250: 2246: 2242: 2213: 2201: 2198: 2086: 2083: 2081: 2078: 2054: 2051: 2046: 2041: 2035: 2011: 1991: 1971: 1951: 1929: 1925: 1894: 1891: 1886: 1881: 1876: 1873: 1868: 1846: 1819: 1816: 1811: 1806: 1800: 1778: 1754: 1751: 1748: 1745: 1742: 1739: 1736: 1716: 1711: 1706: 1702: 1699: 1696: 1693: 1689: 1674:Frobenius norm 1661: 1641: 1620: 1615: 1611: 1607: 1585: 1561: 1556: 1552: 1548: 1545: 1542: 1537: 1533: 1529: 1526: 1522: 1509: 1506: 1473: 1472: 1451: 1424: 1403: 1289: 1266: 1259: 1258: 1247: 1241: 1236: 1230: 1226: 1221: 1216: 1176: 1175: 1164: 1159: 1154: 1150: 1146: 1112: 1109: 1103:after Lee and 1087: 1084: 1080:bioinformatics 1020:linear algebra 1010:is a group of 959: 958: 956: 955: 948: 941: 933: 930: 929: 926: 925: 920: 919: 918: 908: 902: 899: 898: 895: 894: 891: 890: 885: 880: 875: 870: 865: 860: 854: 851: 850: 847: 846: 843: 842: 837: 832: 827: 825:Occam learning 822: 817: 812: 807: 801: 798: 797: 794: 793: 790: 789: 784: 782:Learning curve 779: 774: 768: 765: 764: 761: 760: 757: 756: 751: 746: 741: 735: 732: 731: 728: 727: 724: 723: 722: 721: 711: 706: 701: 695: 690: 689: 686: 685: 682: 681: 675: 670: 665: 660: 659: 658: 648: 643: 642: 641: 636: 631: 626: 616: 611: 606: 601: 600: 599: 589: 588: 587: 582: 577: 572: 562: 557: 552: 546: 541: 540: 537: 536: 533: 532: 527: 522: 514: 508: 503: 502: 499: 498: 495: 494: 493: 492: 487: 482: 471: 466: 465: 462: 461: 458: 457: 452: 447: 442: 437: 432: 427: 422: 417: 411: 406: 405: 402: 401: 398: 397: 392: 387: 381: 376: 371: 363: 358: 353: 347: 342: 341: 338: 337: 334: 333: 328: 323: 318: 313: 308: 303: 298: 290: 289: 288: 283: 278: 268: 266:Decision trees 263: 257: 243:classification 233: 232: 231: 228: 227: 224: 223: 218: 213: 208: 203: 198: 193: 188: 183: 178: 173: 168: 163: 158: 153: 148: 143: 138: 136:Classification 132: 129: 128: 125: 124: 121: 120: 115: 110: 105: 100: 95: 93:Batch learning 90: 85: 80: 75: 70: 65: 60: 54: 51: 50: 47: 46: 35: 34: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 8381: 8370: 8369:Factorization 8367: 8365: 8362: 8360: 8359:Matrix theory 8357: 8355: 8352: 8351: 8349: 8338: 8337: 8336: 8330: 8326: 8309: 8305: 8301: 8298: 8294: 8290: 8287: 8283: 8279: 8276: 8272: 8268: 8265: 8261: 8257: 8254: 8250: 8246: 8243: 8239: 8235: 8232: 8228: 8223: 8220: 8216: 8212: 8208: 8204: 8199: 8194: 8189: 8184: 8180: 8176: 8175: 8170: 8165: 8161: 8157: 8153: 8149: 8145: 8141: 8137: 8133: 8132: 8126: 8122: 8118: 8114: 8110: 8106: 8102: 8098: 8094: 8093: 8088: 8084: 8080: 8075: 8070: 8065: 8061: 8057: 8053: 8049: 8045: 8041: 8037: 8033: 8032: 8026: 8022: 8018: 8014: 8010: 8006: 8002: 7998: 7994: 7993: 7987: 7983: 7979: 7975: 7971: 7970: 7965: 7961: 7956: 7951: 7947: 7943: 7939: 7935: 7934: 7929: 7924: 7923: 7918: 7909: 7904: 7897: 7894: 7889: 7885: 7881: 7875: 7871: 7867: 7862: 7857: 7850: 7849: 7841: 7838: 7827:on 2015-04-02 7823: 7819: 7812: 7805: 7802: 7791:on 2015-04-19 7787: 7783: 7779: 7775: 7771: 7766: 7761: 7757: 7753: 7749: 7745: 7738: 7731: 7728: 7717: 7713: 7707: 7704: 7699: 7692: 7685: 7682: 7677: 7670: 7663: 7660: 7655: 7651: 7646: 7641: 7637: 7633: 7629: 7625: 7618: 7615: 7610: 7606: 7602: 7598: 7594: 7590: 7587:(1): 216–18. 7586: 7582: 7581: 7576: 7569: 7566: 7561: 7557: 7552: 7547: 7543: 7539: 7535: 7531: 7530: 7525: 7518: 7515: 7510: 7506: 7502: 7498: 7494: 7490: 7487:(3): 216–25. 7486: 7482: 7481: 7473: 7470: 7465: 7461: 7457: 7453: 7449: 7445: 7441: 7437: 7436: 7428: 7425: 7420: 7416: 7412: 7408: 7404: 7400: 7396: 7392: 7385: 7382: 7377: 7373: 7369: 7365: 7361: 7357: 7353: 7349: 7342: 7339: 7334: 7330: 7326: 7320: 7316: 7312: 7308: 7301: 7298: 7293: 7289: 7284: 7279: 7275: 7271: 7267: 7263: 7259: 7255: 7251: 7243: 7240: 7235: 7231: 7226: 7221: 7217: 7213: 7209: 7205: 7201: 7197: 7193: 7186: 7183: 7178: 7174: 7169: 7164: 7160: 7156: 7152: 7148: 7147: 7142: 7135: 7132: 7127: 7123: 7118: 7113: 7109: 7105: 7104: 7099: 7092: 7089: 7084: 7080: 7075: 7070: 7065: 7060: 7056: 7052: 7048: 7044: 7043: 7038: 7031: 7028: 7023: 7019: 7014: 7009: 7005: 7001: 6997: 6993: 6992: 6987: 6980: 6977: 6973: 6969: 6963: 6960: 6952: 6948: 6944: 6940: 6936: 6931: 6926: 6922: 6918: 6917: 6909: 6902: 6899: 6894: 6890: 6886: 6882: 6877: 6872: 6868: 6864: 6863: 6855: 6852: 6847: 6843: 6839: 6835: 6828: 6825: 6819: 6814: 6807: 6804: 6798: 6793: 6789: 6785: 6784: 6776: 6773: 6768: 6764: 6760: 6756: 6752: 6748: 6747: 6739: 6736: 6725: 6718: 6715: 6710: 6706: 6702: 6698: 6694: 6690: 6686: 6682: 6681: 6673: 6666: 6663: 6658: 6654: 6650: 6646: 6642: 6638: 6633: 6628: 6624: 6620: 6612: 6609: 6604: 6600: 6595: 6590: 6586: 6582: 6577: 6572: 6568: 6564: 6560: 6553: 6550: 6545: 6541: 6537: 6533: 6529: 6525: 6520: 6515: 6511: 6507: 6500: 6497: 6492: 6488: 6483: 6478: 6474: 6470: 6465: 6460: 6456: 6452: 6448: 6444: 6437: 6434: 6429: 6425: 6421: 6415: 6411: 6407: 6403: 6396: 6393: 6388: 6384: 6383: 6375: 6372: 6367: 6360: 6357: 6349: 6348: 6340: 6337: 6329: 6328: 6320: 6317: 6312: 6308: 6303: 6298: 6294: 6290: 6286: 6282: 6281: 6273: 6270: 6265: 6261: 6257: 6253: 6249: 6245: 6244: 6239: 6233: 6230: 6225: 6224: 6216: 6213: 6209: 6205: 6204:Amnon Shashua 6202:Ron Zass and 6199: 6196: 6185:on 2007-09-28 6181: 6174: 6173: 6165: 6162: 6154: 6153: 6145: 6142: 6137: 6131: 6123: 6119: 6115: 6111: 6107: 6106:10.1038/44565 6103: 6099: 6095: 6091: 6087: 6086: 6078: 6071: 6068: 6063: 6059: 6054: 6049: 6045: 6044: 6036: 6034: 6030: 6025: 6021: 6017: 6013: 6006: 5999: 5996: 5990: 5985: 5981: 5977: 5973: 5966: 5963: 5958: 5954: 5949: 5944: 5940: 5936: 5931: 5926: 5922: 5918: 5914: 5907: 5905: 5903: 5899: 5894: 5890: 5886: 5882: 5878: 5874: 5869: 5864: 5860: 5856: 5849: 5847: 5843: 5837: 5832: 5825: 5823: 5819: 5814: 5810: 5806: 5802: 5797: 5792: 5788: 5784: 5777: 5774: 5769: 5763: 5759: 5755: 5751: 5744: 5741: 5736: 5732: 5727: 5722: 5718: 5714: 5713: 5705: 5698: 5695: 5690: 5686: 5681: 5676: 5672: 5668: 5664: 5660: 5659: 5651: 5648: 5643: 5639: 5635: 5631: 5627: 5623: 5619: 5615: 5608: 5605: 5600: 5596: 5591: 5586: 5582: 5578: 5577: 5569: 5565: 5558: 5555: 5550: 5546: 5542: 5538: 5533: 5528: 5524: 5520: 5513: 5510: 5505: 5501: 5497: 5493: 5489: 5485: 5480: 5475: 5471: 5467: 5466: 5458: 5451: 5449: 5445: 5440: 5436: 5432: 5426: 5422: 5418: 5414: 5410: 5403: 5400: 5395: 5391: 5387: 5383: 5379: 5375: 5371: 5367: 5360: 5357: 5352: 5350:9780769530284 5346: 5342: 5341: 5333: 5330: 5325: 5323:9781450308137 5319: 5315: 5311: 5304: 5303: 5295: 5292: 5287: 5283: 5278: 5273: 5268: 5263: 5259: 5255: 5251: 5247: 5246: 5241: 5234: 5232: 5228: 5222: 5217: 5213: 5206: 5204: 5200: 5195: 5191: 5187: 5183: 5182: 5174: 5171: 5166: 5162: 5158: 5154: 5149: 5144: 5140: 5136: 5135:SIAM J. Optim 5129: 5126: 5121: 5117: 5113: 5109: 5102: 5099: 5094: 5087: 5084: 5079: 5075: 5071: 5067: 5060: 5057: 5051: 5049: 5045: 5037: 5033: 5029: 5025: 5021: 5014: 5007: 5004: 5000: 4994: 4992: 4990: 4986: 4981: 4974: 4973: 4965: 4963: 4959: 4954: 4950: 4946: 4942: 4938: 4937:10.1038/44565 4934: 4930: 4926: 4922: 4918: 4917: 4912: 4905: 4903: 4899: 4894: 4890: 4886: 4882: 4878: 4874: 4873: 4868: 4864: 4858: 4855: 4850: 4846: 4842: 4838: 4834: 4830: 4826: 4822: 4818: 4811: 4808: 4803: 4799: 4795: 4791: 4787: 4783: 4780:: S273–S276. 4779: 4775: 4774: 4766: 4763: 4758: 4754: 4750: 4746: 4742: 4738: 4737: 4736:Technometrics 4732: 4728: 4722: 4719: 4714: 4710: 4705: 4700: 4695: 4690: 4686: 4682: 4678: 4674: 4670: 4663: 4660: 4655: 4654: 4646: 4643: 4638: 4634: 4633:Peter J. Haas 4627: 4625: 4621: 4616: 4612: 4607: 4602: 4598: 4594: 4589: 4584: 4580: 4576: 4572: 4565: 4563: 4561: 4559: 4557: 4553: 4548: 4544: 4539: 4534: 4530: 4526: 4521: 4516: 4512: 4508: 4504: 4497: 4495: 4493: 4491: 4489: 4487: 4485: 4481: 4476: 4472: 4468: 4464: 4460: 4456: 4451: 4446: 4442: 4438: 4431: 4429: 4427: 4423: 4415: 4408: 4405: 4400: 4396: 4389: 4387: 4385: 4381: 4375: 4370: 4366: 4363: 4361: 4358: 4356: 4353: 4351: 4348: 4346: 4343: 4342: 4338: 4333: 4330: 4327: 4323: 4320: 4319: 4318: 4309: 4306:February 2024 4297: 4292: 4283: 4282: 4276: 4274: 4272: 4268: 4260: 4258: 4256: 4251: 4248: 4244: 4240: 4232: 4230: 4227: 4223: 4215: 4213: 4209: 4207: 4203: 4202:Wiener filter 4199: 4191: 4189: 4171: 4165: 4143: 4139: 4118: 4106: 4104: 4098: 4096: 4092: 4088: 4086: 4083:articles and 4082: 4078: 4074: 4069: 4067: 4066:data clusters 4063: 4059: 4055: 4053: 4052:document-term 4048: 4040: 4038: 4034: 4030: 4027: 4019: 4017: 4015: 4009: 4007: 4003: 3999: 3995: 3990: 3988: 3984: 3980: 3975: 3967: 3962: 3960: 3957: 3955: 3951: 3946: 3882: 3880: 3857: 3854: 3844: 3762: 3759: 3743: 3728: 3727: 3726: 3724: 3716: 3714: 3712: 3708: 3703: 3700: 3698: 3693: 3691: 3686: 3682: 3676: 3671: 3666: 3664: 3660: 3657:, trained by 3656: 3652: 3630: 3626: 3620: 3617: 3613: 3607: 3603: 3594: 3589: 3583: 3577: 3571: 3565: 3561: 3559: 3555: 3551: 3543: 3541: 3533: 3527: 3521: 3512: 3510: 3508: 3503: 3489: 3466: 3463: 3460: 3449: 3444: 3438: 3429: 3422: 3418: 3416: 3411: 3405: 3399: 3397: 3391: 3389: 3386:methods, the 3385: 3380: 3374: 3368: 3362: 3356: 3352:is fixed and 3350: 3344: 3340:is fixed and 3338: 3333: 3328: 3304: 3291: 3146: 3140: 3134: 3127: 3121: 3116: 3094: 3091: 3088: 3075: 3065: 3062: 3059: 3044: 3041: 3038: 3026: 3006: 3003: 3000: 2987: 2977: 2974: 2971: 2945: 2937: 2934: 2931: 2913: 2910: 2907: 2899: 2896: 2893: 2876: 2873: 2851: 2848: 2845: 2832: 2820: 2808: 2798: 2775: 2772: 2769: 2751: 2741: 2720: 2712: 2709: 2706: 2688: 2685: 2682: 2674: 2671: 2668: 2651: 2636: 2627: 2621: 2616: 2614:non negative. 2612: 2606: 2601: 2600: 2599: 2597: 2592: 2586: 2577: 2575: 2572: 2567: 2562: 2556: 2549: 2540: 2538: 2536: 2532: 2528: 2519: 2517: 2515: 2514:sparse coding 2511: 2507: 2503: 2498: 2496: 2475: 2470: 2453: 2439: 2428: 2417: 2410: 2409: 2408: 2383: 2381: 2377: 2372: 2369: 2363: 2358: 2353: 2347: 2342: 2334: 2332: 2329: 2323: 2319: 2313: 2308: 2300: 2298: 2295: 2289: 2267: 2263: 2259: 2256: 2253: 2248: 2244: 2232: 2227: 2221: 2212: 2208: 2199: 2197: 2194: 2188: 2182: 2176: 2170: 2164: 2158: 2152: 2149: 2145: 2141: 2137:, such that: 2135: 2129: 2123: 2117: 2111: 2105: 2099: 2093: 2084: 2079: 2077: 2075: 2071: 2066: 2052: 2049: 2044: 2023: 2009: 1989: 1969: 1949: 1927: 1923: 1914: 1910: 1892: 1889: 1879: 1874: 1871: 1844: 1835: 1833: 1817: 1814: 1809: 1766: 1752: 1749: 1746: 1743: 1740: 1737: 1734: 1714: 1709: 1700: 1697: 1694: 1691: 1677: 1675: 1659: 1639: 1609: 1573: 1554: 1550: 1546: 1543: 1540: 1535: 1531: 1524: 1507: 1505: 1502: 1496: 1489: 1482: 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