When Does Non-Negative Matrix Factorization Give a Correct Decomposition into Parts?

By David L. Donoho, Victoria Stodden. 2003

964 citations recorded by the source corpus.

Subfields: Computer Vision and Pattern Recognition

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Citations in the corpus

Group invariance principles for causal generative models (2017, methodology): “ sparse matrices V by selecting non-zero coefﬁcients by sampling from i.i.d. Bernoulli variables with probability :1 (sparsity of one factor matrix tends to increase the identiﬁability of NMF models [Donoho and Stodden, 2004]), the value of the selected non-vanishing coefﬁcients were drawn from i.i.”
Projected Gradient Methods for Nonnegative Matrix Factorization (2007, methodology): “NMF has been applied to many areas such as finding basis vectors of images (Lee and Seung, 1999), document clustering (Xu et al., 2003), molecular pattern discovery (Brunet et al., 2004), etc. Donoho and Stodden (2004) have addressed the theoretical issues associated with the NMF approach.”
A Block Coordinate Descent Method for Regularized Multiconvex Optimization with Applications to Nonnegative Tensor Factorization and Completion (2013, methodology): “Another one has the size of 32 × 32 × 256 and is formed with the 256 images in the Swimmer dataset [20].”
Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework (2013, background): “For discussion on nonnegative rank as well as the geometric interpretation of NMF, see Lin and Chu [72], Gillis [34], and Donoho and Stodden [27].”
Hyperspectral Unmixing via $L_{1/2}$ Sparsity-Constrained Nonnegative Matrix Factorization (2011, background): “In [22], Donoho and Stodden analyzed the assumptions required for NMF to generate unique solutions and lead to a well-defined answer.”
Linear discriminant analysis: A detailed tutorial (2017, neutral): “There are many unsupervised dimensionality reduction techniques such as Independent Component Analysis (ICA) [28,31] and Non-negative Matrix Factorization (NMF) [14], but the most famous technique of the unsupervised approach is the Principal Component Analysis (PCA) [4,62,67,71].”
Nonsmooth nonnegative matrix factorization (nsNMF) (2006, methodology): “4.2 Swimmers Data Set The “swimmer” data set is described in [ 20 ].”
A Practical Algorithm for Topic Modeling with Provable Guarantees (2012, methodology): “Arora et al. [2012b] present an algorithm that provably recovers the parameters of topic models provided that the topics meet a certain separability assumption [Donoho & Stodden, 2003].”
On the Geometric Interpretation of the Nonnegative Rank (2010, methodology): “We rely on the geometric interpretation of the nonnegative rank, which was first introduced in [29]; see also [22,12,42] where related results are presented.”
On reduced rank nonnegative matrix factorization for symmetric nonnegative matrices (2004, background): “(2.9) In the next section, we shall show that the Lagrange multipliers ν and µ are always zero when r = 1 which enables us to answer the three questions we posed in Section 1 in the rank one case.”
Generalized Nonnegative Matrix Approximations with Bregman Divergences (2005, background): “Donoho and Stodden [9] mull over criteria tha t enable one to determine when does NNMA give a correct decomposition into parts for the original dat a.”
Manifold Regularized Discriminative Nonnegative Matrix Factorization With Fast Gradient Descent (2011, background): “1) Donoho and Stodden [16] theoretically proved that NMF does not necessarily decompose an object into parts, i.e., NMF could fail to obtain the parts-based representation for some datasets.”