Nonnegative Matrix Factorization with Toeplitz Penalty
Matthew Corsetti,Ernest Fokoué +1 more
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A new NMF algorithm is proposed that makes use of non-datadependent auxiliary constraints which incorporate a Toeplitz matrix into the multiplicative updating of the basis and weight matrices and is compared to the Zellner Nonnegative Matrix Factorization algorithm.Abstract:
Nonnegative Matrix Factorization (NMF) is an unsupervised learning algorithm that produces a linear, parts-based approximation of a data matrix. NMF constructs a nonnegative low rank basis matrix and a nonnegative low rank matrix of weights which, when multiplied together, approximate the data matrix of interest using some cost function. The NMF algorithm can be modified to include auxiliary constraints which impose task-specific penalties or restrictions on the cost function of the matrix factorization. In this paper we propose a new NMF algorithm that makes use of non-data dependent auxiliary constraints which incorporate a Toeplitz matrix into the multiplicative updating of the basis and weight matrices. We compare the facial recognition performance of our new Toeplitz Nonnegative Matrix Factorization (TNMF) algorithm to the performance of the Zellner Nonnegative Matrix Factorization (ZNMF) algorithm which makes use of data-dependent auxiliary constraints. We also compare the facial recognition performance of the two aforementioned algorithms with the performance of several preexisting constrained NMF algorithms that have non-data-dependent penalties. The facial recognition performances are evaluated using the Cambridge ORL Database of Faces and the Yale Database of Faces.read more
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References
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Journal ArticleDOI
Learning the parts of objects by non-negative matrix factorization
TL;DR: An algorithm for non-negative matrix factorization is demonstrated that is able to learn parts of faces and semantic features of text and is in contrast to other methods that learn holistic, not parts-based, representations.
Learning parts of objects by non-negative matrix factorization
TL;DR: In this article, non-negative matrix factorization is used to learn parts of faces and semantic features of text, which is in contrast to principal components analysis and vector quantization that learn holistic, not parts-based, representations.
Proceedings Article
Algorithms for Non-negative Matrix Factorization
Daniel D. Lee,H. Sebastian Seung +1 more
TL;DR: Two different multiplicative algorithms for non-negative matrix factorization are analyzed and one algorithm can be shown to minimize the conventional least squares error while the other minimizes the generalized Kullback-Leibler divergence.
Journal ArticleDOI
Sparse bayesian learning and the relevance vector machine
TL;DR: It is demonstrated that by exploiting a probabilistic Bayesian learning framework, the 'relevance vector machine' (RVM) can derive accurate prediction models which typically utilise dramatically fewer basis functions than a comparable SVM while offering a number of additional advantages.
Journal ArticleDOI
Positive matrix factorization: A non-negative factor model with optimal utilization of error estimates of data values†
Pentti Paatero,Unto Tapper +1 more
TL;DR: In this paper, a new variant of Factor Analysis (PMF) is described, where the problem is solved in the weighted least squares sense: G and F are determined so that the Frobenius norm of E divided (element-by-element) by σ is minimized.