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Identification of NMF by choosing maximum-volume basis vectors

Qianqian Qi, Zhongming Chen, Peter G. M. van der Heijden

Abstract

In nonnegative matrix factorization (NMF), minimum-volume-constrained NMF is a widely used framework for identifying the solution of NMF by making basis vectors as similar as possible. This typically induces sparsity in the coefficient matrix, with each row containing zero entries. Consequently, minimum-volume-constrained NMF may fail for highly mixed data, where such sparsity does not hold. Moreover, the estimated basis vectors in minimum-volume-constrained NMF may be difficult to interpret as they may be mixtures of the ground truth basis vectors. To address these limitations, in this paper we propose a new NMF framework, called maximum-volume-constrained NMF, which makes the basis vectors as distinct as possible. We further establish an identifiability theorem for maximum-volume-constrained NMF and provide an algorithm to estimate it. Experimental results demonstrate the effectiveness of the proposed method.

Identification of NMF by choosing maximum-volume basis vectors

Abstract

In nonnegative matrix factorization (NMF), minimum-volume-constrained NMF is a widely used framework for identifying the solution of NMF by making basis vectors as similar as possible. This typically induces sparsity in the coefficient matrix, with each row containing zero entries. Consequently, minimum-volume-constrained NMF may fail for highly mixed data, where such sparsity does not hold. Moreover, the estimated basis vectors in minimum-volume-constrained NMF may be difficult to interpret as they may be mixtures of the ground truth basis vectors. To address these limitations, in this paper we propose a new NMF framework, called maximum-volume-constrained NMF, which makes the basis vectors as distinct as possible. We further establish an identifiability theorem for maximum-volume-constrained NMF and provide an algorithm to estimate it. Experimental results demonstrate the effectiveness of the proposed method.
Paper Structure (8 sections, 1 theorem, 22 equations, 5 figures, 6 tables, 1 algorithm)

This paper contains 8 sections, 1 theorem, 22 equations, 5 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Uniqueness theorem for maximum-volume-constrained NMF
  3. A simple approach for solving maximum-volume-constrained NMF
  4. Numerical experiments
  5. Sedimentary geology: Grain-size distributions dataset
  6. Image processing: Human face images dataset
  7. Social science: Time allocation dataset
  8. Conclusion

Key Result

Theorem 1

Assume that (1) $\text{rank}(\bm{M}) = \text{rank}(\bm{H}) = K$, (2) $\bm{M}^T$ satisfies SSC. Then MAV-NMF uniquely identifies $\tilde{\bm{M}}$ and $\tilde{\bm{H}}$ up to permutation. That is, any optimal solution $(\bm{M}_{\#}, \bm{H}_{\#})$ of MAV-NMF (E: maxvolcriteria) can be expressed as: where $\bm{\Gamma}$ is a permutation matrix.

Figures (5)

  • Figure 1: A graphical view of columns of the matrix $\bm{X}$ (blue dots) and $\text{cone}(\bm{M})$ (dashed triangle), sliced by the unit-sum hyperplane $\bm{y}^T\bm{1} = 1$: (a) nonuniqueness; (b) minimum-volume-constrained NMF; (c) maximum-volume-constrained NMF.
  • Figure 2: (a, b): A graphical representation of rows of the estimated basis matrix $\bm{MS}$ (blue dots), $\text{cone}(\bm{e}_1, \bm{e}_2, \bm{e}_3)$ (solid triangle) with $\bm{e}_1 = (1, 0, 0)^T$, $\bm{e}_2 = (0, 1, 0)^T$, and $\bm{e}_3 = (0, 0, 1)^T$, and $\text{cone}((\bm{MS})^T)$ (dashed polygon), sliced by the unit-sum hyperplane $\bm{y}^T\bm{1} = 1$; (c, d): A graphical representation of columns of the estimated coefficient matrix $\bm{S}^{-1}\bm{H}$ (blue dots), $\text{cone}(\bm{e}_1, \bm{e}_2, \bm{e}_3)$ (solid triangle), and $\text{cone}(\bm{S}^{-1}\bm{H})$ (dashed polygon), sliced by the unit-sum hyperplane $\bm{y}^T\bm{1} = 1$.
  • Figure 3: Artificial grain-size distributions (GSDs) datasets from ZHANG2020106656 (a) The four ground truth basis vectors used from ZHANG2020106656, which were from the calculated EMs from the grain-size data of the surface sediment in the South Yellow Sea by zhang2016end; (b) Test Dataset 1; (c) Test Dataset 2.
  • Figure 4: CBCL face data set: (a) MVC-NMF; (b) MAV-NMF.
  • Figure B.1: Data with no zero entries in all rows of the ground truth coefficient matrix $\bm{H}$: (a) one row without zero entries; (b) two rows without zero entries; (c) all three rows without zero entries.

Theorems & Definitions (2)

  • Definition 1
  • Theorem 1