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Dual simplex Volume Maximization for simplex-structured matrix factorization

SIAM JOURNAL ON IMAGING SCIENCES 2024年第4期17卷 2362-2391页

作者： Abdolali, Maryam Barbarino, Giovanni Gillis, Nicolas KN Toosi Univ KNTU Tehran Iran Univ Mons Mons Belgium

simplex-structured matrix factorization (SSMF), closely related to nonnegative matrix factorization, is a fundamental interpretable data analysis model and has applications in hyperspectral unmixing and topic modeling. To obtain identifiable solutions, a standard approach is to find minimum- volume solutions. By taking advantage of the duality/polarity concept for polytopes, we convert minimum-volume SSMF in the primal space to a maximum-volume problem in the dual space. We first prove the identifiability of this maximum-volume dual problem. Then, we use this dual formulation to provide a novel optimization approach which bridges the gap between two existing families of algorithms for SSMF, namely volume minimization and facet identification. Numerical experiments show that the proposed approach performs favorably compared to the state-of-the-art SSMF algorithms.

关键词： simplex-structured matrix factorization nonnegative matrix factorization minimum volume sparsity polarity/duality hyperspectral imaging

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Bounded simplex-structured matrix factorization: Algorithms, Identifiability and Applications

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2023年 71卷 2434-2447页

作者： Thanh, Olivier Vu Gillis, Nicolas Lecron, Fabian Univ Mons Math & Operat Res B-7000 Mons Belgium

In this article, we propose a new low-rank matrix factorization model dubbed bounded simplex-structured matrix factorization (BSSMF). Given an input matrix X and a factorization rank r, BSSMF looks for a matrix W with r columns and a matrix H with r rows such that X approximate to WH where the entries in each column of W are bounded, that is, they belong to given intervals, and the columns of H belong to the probability simplex, that is, H is column stochastic. BSSMF generalizes nonnegative matrix factorization (NMF), and simplex-structured matrix factorization (SSMF). BSSMF is particularly well suited when the entries of the input matrix X belong to a given interval;for example when the rows of X represent images, or X is a rating matrix such as in the Netflix and MovieLens datasets where the entries of X belong to the interval [1,5]. The simplex-structured matrix H not only leads to an easily understandable decomposition providing a soft clustering of the columns ofX, but implies that the entries of each column of WH belong to the same intervals as the columns of W. In this article, we first propose a fast algorithm for BSSMF, even in the presence of missing data in X. Then we provide identifiability conditions for BSSMF, that is, we provide conditions under which BSSMF admits a unique decomposition, up to trivial ambiguities. Finally, we illustrate the effectiveness of BSSMF on two applications: extraction of features in a set of images, and the matrix completion problem for recommender systems.

关键词： simplex-structured matrix factorization nonnegative matrix factorization identifiability algorithms

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BOUNDED simplex-structured matrix factorization 47

BOUNDED SIMPLEX-STRUCTURED MATRIX FACTORIZATION

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47th IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

作者： Thanh, Olivier Vu Gillis, Nicolas Lecron, Fabian Univ Mons Fac Polytech Dept Math & Operat Res Rue Houdain 9 B-7000 Mons Belgium

ISBN: (纸本)9781665405409

In this paper, we propose a new low-rank matrix factorization model, dubbed bounded simplex-structured matrix factorization (BSSMF). Given an input matrix X and a factorization rank r, BSSMF looks for a matrix W with r columns and a matrix H with r rows such that X approximate to WH where the entries in each column of W are bounded, that is, they belong to given intervals, and the columns of H belong to the unit simplex, that is, H is column stochastic. BSSMF generalizes non-negative matrix factorization (NMF), and simplex-structured matrix factorization (SSMF). BSSMF is particularly well suited when the entries of the input matrix X themselves belong to a given interval;for example when the columns of X represent images. In this paper, we first provide identifiability conditions for BSSMF, that is, we provide conditions under which BSSMF admits a unique decomposition, up to trivial ambiguities. Then we propose a fast inertial algorithm for BSSMF. Finally, we illustrate the effectiveness of BSSMF to obtain interpretable features in the MNIST dataset.

关键词： simplex-structured matrix factorization nonnegative matrix factorization identifiability inertial first-order methods

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simplex-structured matrix factorization: Sparsity-Based Identifiability and Provably Correct Algorithms

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SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE 2021年第2期3卷 593-623页

作者： Abdolali, Maryam Gillis, Nicolas Univ Mons Dept Math & Operat Res Fac Polytech B-7000 Mons Belgium

In this paper, we provide novel algorithms with identifiability guarantees for simplex-structured matrix factorization (SSMF), a generalization of nonnegative matrix factorization. Current state-of-the-art algorithms that provide identifiability results for SSMF rely on the sufficiently scattered condition (SSC), which requires the data points to be well spread within the convex hull of the basis vectors. The conditions under which our proposed algorithms recover the unique decomposition are in most cases much weaker than the SSC. We only require having d points on each facet of the convex hull of the basis vectors whose dimension is d - 1. The key idea is based on extracting facets containing the largest number of points. We illustrate the effectiveness of our approach on synthetic data sets and hyperspectral images, showing that it outperforms state-of-the-art SSMF algorithms, as it is able to handle higher noise levels, rank deficient matrices, outliers, and input data that highly violates the SSC.

关键词： simplex-structured matrix factorization nonnegative matrix factorization sparsity identifiability uniqueness minimum volume

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STOCHASTIC ML simplex-structured matrix factorization UNDER THE DIRICHLET MIXTURE MODEL 44

STOCHASTIC ML SIMPLEX-STRUCTURED MATRIX FACTORIZATION UNDER ...

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44th IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

作者： Wu, Ruiyuan Li, Qiang Ma, Wing-Kin Chinese Univ Hong Kong Dept Elect Engn Shatin Hong Kong Peoples R China Univ Elect Sci & Technol China Sch Info & Comm Engn Hefei Anhui Peoples R China

ISBN: (纸本)9781479981311

simplex-structured matrix factorization (SSMF) is a problem of recovering a basis matrix and the corresponding coefficient vectors from data, where the coefficient vectors are constrained to lie in the unit simplex. SSMF has attracted growing attention in recent years, with numerous applications such as hyperspectral unmixing and document clustering. In this work, we develop a maximum likelihood (ML) approach for SSMF. Specifically, by modeling the coefficient vectors as random variables following a Dirichlet mixture distribution which allows us to model more complex data distributions in real-life data, a probabilistic model for SSMF is employed. We consider a marginalized likelihood with respect to the coefficient vectors, and use ML estimation to learn the basis matrix and unknown Dirichlet mixture parameters. The marginalized likelihood does not admit a closed form and is non-concave, and this makes the problem challenging to solve. To handle this challenge, an effective algorithm using sample average approximation and block successive upper-bound minimization is proposed. We consider the aforementioned two real-world applications by simulations. Numerical results show that the proposed algorithm delivers appealing performance in both applications.

关键词： simplex-structured matrix factorization Dirichlet mixture model maximum-likelihood estimation sample average approximation block successive upper-bound minimization

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Probabilistic simplex Component Analysis by Importance Sampling

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IEEE SIGNAL PROCESSING LETTERS 2023年 30卷 683-687页

作者： Granot, Nerya Diskin, Tzvi Dobigeon, Nicolas Wiesel, Ami Hebrew Univ Jerusalem Rachel & Selim Benin Sch Comp Sci & Engn Dept Comp Sci & Computat Biol IL-91904 Jerusalem Israel Univ Toulouse F-31500 Toulouse France

In this letter we consider the problem of linear unmixing hidden random variables defined over the simplex with additive Gaussian noise, also known as probabilistic simplex component analysis (PRISM). Previous solutions to tackle this challenging problem were based on geometrical approaches or computationally intensive variational methods. In contrast, we propose a conventional expectation maximization (EM) algorithm which embeds importance sampling. For this purpose, the proposal distribution is chosen as a simple surrogate distribution of the target posterior that is guaranteed to lie in the simplex. It is based on fitting the Dirichlet parameters to the linear minimum mean squared error (LMMSE) approximation, which is accurate at high signal-to-noise ratio. Numerical experiments in different settings demonstrate the advantages of this adaptive surrogate over state-of-the-art methods.

关键词： Expectation maximization importance sampling simplex-structured matrix factorization

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SISAL Revisited

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SIAM JOURNAL ON IMAGING SCIENCES 2022年第2期15卷 591-624页

作者： Huang, Chujun Shao, Mingjie Ma, Wing-Kin So, Anthony Man-Cho Chinese Univ Hong Kong Dept Elect Engn Shatin Hong Kong Peoples R China Chinese Univ Hong Kong Dept Syst Engn & Engn Management Shatin Hong Kong Peoples R China

simplex identification via split augmented Lagrangian (SISAL) is a popularly used algorithm in blind unmixing of hyperspectral images. Developed by Jose'\ M. Bioucas-Dias in 2009, the algorithm is fundamentally relevant to tackling simplex-structured matrix factorization and, by extension, non negative matrix factorization, which have many applications under their umbrellas. In this article, we revisit SISAL and provide new meanings to this quintessential algorithm. The formulation of SISAL was motivated from a geometric perspective, with no noise. We show that SISAL can be explained as an approximation scheme from a probabilistic simplex component analysis framework, which is statistical and is principally more powerful in accommodating the presence of noise. The algorithm for SISAL was designed based on a successive convex approximation method, with a focus on practical utility. It was not known, by analyses, whether the SISAL algorithm has any kind of guarantee of convergence to a stationary point. By establishing associations between the SISAL algorithm and a line search--based proximal gradient method, we confirm that SISAL can indeed guarantee convergence to a stationary point. Our re-explanation of SISAL also reveals new formulations and algorithms. The performance of these new possibilities is demonstrated by numerical experiments.

关键词： simplex-structured matrix factorization probabilistic simplex component analysis hyperspectral unmixing first-order optimization

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Probabilistic simplex Component Analysis

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2022年 70卷 582-599页

作者： Wu, Ruiyuan Ma, Wing-Kin Li, Yuening So, Anthony Man-Cho Sidiropoulos, Nicholas D. Chinese Univ Hong Kong Dept Elect Engn Hong Kong Peoples R China Chinese Univ Hong Kong Dept Syst Engn & Engn Management Hong Kong Peoples R China Univ Virginia Dept Elect Comp Engn Charlottesville VA 22904 USA

This study presents PRISM, a probabilistic simplex component analysis approach to identifying the vertices of a data-circumscribing simplex from data. The problem has a rich variety of applications, the most notable being hyperspectral unmixing in remote sensing and non-negative matrix factorization in machine learning. PRISM uses a simple probabilistic model, namely, uniform simplex data distribution and additive Gaussian noise, and it carries out inference by maximum likelihood. The inference model is sound in the sense that the vertices are provably identifiable under some assumptions, and it suggests that PRISM can be effective in combating noise when the number of data points is large. PRISM has strong, but hidden, relationships with simplex volume minimization, a powerful geometric approach for the same problem. We study these fundamental aspects, and we also consider algorithmic schemes based on importance sampling and variational inference. In particular, the variational inference scheme is shown to resemble a matrix factorization problem with a special regularizer, which draws an interesting connection to the matrix factorization approach. Numerical results are provided to demonstrate the potential of PRISM.

关键词： Probabilistic logic Noise measurement Signal processing algorithms Machine learning Inference algorithms Random variables Principal component analysis simplex-structured matrix factorization maximum likelihood identifiability simplex volume minimization variational inference hyperspectral unmixing

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Identifiability-Guaranteed simplex-structured Post-Nonlinear Mixture Learning via Autoencoder

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2021年 69卷 4921-4936页

作者： Lyu, Qi Fu, Xiao Oregon State Univ Sch Elect Engn & Comp Sci Corvallis OR 97331 USA

This work focuses on the problem of unraveling nonlinearly mixed latent components in an unsupervised manner. The latent components are assumed to reside in the probability simplex, and are transformed by an unknown post-nonlinear mixing system. This problem finds various applications in signal and data analytics, e.g., nonlinear hyperspectral unmixing, image embedding, and nonlinear clustering. Linear mixture learning problems are already ill-posed, as identifiability of the target latent components is hard to establish in general. With unknown nonlinearity involved, the problem is even more challenging. Prior work offered a function equation-based formulation for provable latent component identification. However, the identifiability conditions are somewhat stringent and unrealistic. In addition, the identifiability analysis is based on the infinite sample (i.e., population) case, while the understanding for practical finite sample cases has been elusive. Moreover, the algorithm in the prior work trades model expressiveness with computational convenience, which often hinders the learning performance. Our contribution is threefold. First, new identifiability conditions are derived under largely relaxed assumptions. Second, comprehensive sample complexity results are presented-which are the first of the kind. Third, a constrained autoencoder-based algorithmic framework is proposed for implementation, which effectively circumvents the challenges in the existing algorithm. Synthetic and real experiments corroborate our theoretical analyses.

关键词： Unified modeling language Nonlinear distortion Hyperspectral imaging Computational modeling Brain modeling Mixture models Analytical models Identifiability post-nonlinear mixture model sample complexity simplex-structured matrix factorization autoencoder neural networks

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STOCHASTIC ML ESTIMATION FOR HYPERSPECTRAL UNMIXING UNDER ENDMEMBER VARIABILITY AND NONLINEAR MODELS

STOCHASTIC ML ESTIMATION FOR HYPERSPECTRAL UNMIXING UNDER EN...

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IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)

作者： Li, Yuening Wu, Ruiyuan Ma, Wing-Kin Chinese Univ Hong Kong Dept Elect Engn Hong Kong Peoples R China

ISBN: (纸本)9781509066315

Hyperspectral unmixing (HU) is a problem of blindly identifying the underlying materials, in form of spectral signatures, in the captured hyperspectral image. HU has received tremendous interest in remote sensing, and fundamentally the problem can be regarded as solving a simplex-structured matrix factorization problem. The majority of HU research has been focused on the linear mixture model. On the other hand, remote sensing research has long indicated that real-life hyperspectral images can exhibit spatial variant and nonlinear effects. In this study we introduce a probabilistic approach for HU under two different models, namely, the normal composition model for modeling spatial endmember variability and the generalized bilinear model for modeling multi-path nonlinear effects. Our approach formulates HU as a maximum-likelihood (ML) model parameter estimation problem, and we demonstrate that the ML formulation can flexibly handle the two models. The ML problem is technical challenging, and we apply a combination of techniques, namely, sample average approximation, block coordinate descent and majorization-minimization, to tackle the problem. Our empirical study suggests that the ML method gives competitive recovery performance.

关键词： hyperspectral unmixing simplex-structured matrix factorization stochastic maximum-likelihood normal compositional model bilinear model

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