Matrix computations (3rd ed.)
Algorithm 805: computation and uses of the semidiscrete matrix decomposition
ACM Transactions on Mathematical Software (TOMS)
The Centroid Decomposition: Relationships between Discrete Variational Decompositions and SVDs
SIAM Journal on Matrix Analysis and Applications
Document clustering based on non-negative matrix factorization
Proceedings of the 26th annual international ACM SIGIR conference on Research and development in informaion retrieval
Non-negative Matrix Factorization with Sparseness Constraints
The Journal of Machine Learning Research
Nonsmooth Nonnegative Matrix Factorization (nsNMF)
IEEE Transactions on Pattern Analysis and Machine Intelligence
Nonorthogonal decomposition of binary matrices for bounded-error data compression and analysis
ACM Transactions on Mathematical Software (TOMS)
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
Low-Dimensional Polytope Approximation and Its Applications to Nonnegative Matrix Factorization
SIAM Journal on Scientific Computing
IEEE Transactions on Knowledge and Data Engineering
SIAM Journal on Matrix Analysis and Applications
The reverse greedy algorithm for the metric k-median problem
Information Processing Letters
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The well-known Eckart-Young theorem asserts that the truncated singular value decomposition, obtained by discarding all but the first $k$ largest singular values and their corresponding left and right singular vectors, is the best rank-$k$ approximation in the sense of least squares to the original matrix. In other words, singular values alone serve well as unambiguous indicators of proximity to the data matrix. Unlike continuous data, the decomposition of a matrix with discrete data which is subject to the requirement that its approximations have the same type of data is a harder task and it is even harder when it comes to ranking these approximations. This work generalizes the notion of singular value decomposition via a sequence of variational formulations to discrete-type data. The process itself can guarantee neither the orthogonality, as is expected of discrete data, nor the ordering of best approximations. However, at the end of the undertaking, it is shown that a quantity analogous to the singular values and a truncated low rank factorization for discrete data analogous to the truncated singular value decomposition for continuous data are attainable. Our empirical study shows the applicability of our method to cluster analysis and pattern discovery using real-life data.