Sharp results for NIEP and NMF

Jiashun Jin, Tzu-Kuo Huang

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

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Keywords: Non-negative matrix factorization, non-negative inverse eigenvalue problem, social network modeling, Discrete Fourier Transform, Haar basis, construction approach

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TL;DR: Sharp results for non-negative Inverse Eigenvalue Problem

Abstract: The orthodox Non-negative Inverse Eigenvalue Problem (oNIEP) has challenged mathematicians for over $70$ years. Motivated by applications in non-negative matrix factorization (NMF) and network modeling, we consider an NIEP as follows. Consider a $K \times K$ diagonal matrix $J_{K, m} = \diag(1 + a_{K, m}, 1, \ldots, 1, -1, \ldots, -1)$, where exactly $m$ entries are $-1$ and $a_{K, m} = \max\{0, (2m-K)\}$. We wish to determine for which $(K, m)$, there is a $K \times K$ orthogonal matrix $Q$ such that $Q J_{K, m} Q'$ is doubly stochastic. Using several approaches (especially a combined Haar and Discrete Fourier Transform (DFT) approach) we developed, we show that in most of the cases, the NIEP is solvable. We show that these results are sharp. Also, since these are construction approaches, they automatically provide an explicit way for computing matrix $Q$. As a result, these approaches give rise to both a computable NMF algorithm and sharp results for NMF. We also discuss the implication of our results for social network modeling.

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Submission Number: 7981