A multilevel approach for nonnegative matrix factorization

Gillis, Nicolas; Glineur, François

doi:10.1016/j.cam.2011.10.002

Cited by 42 publications

(39 citation statements)

References 29 publications

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“…Under the assumption of asymptotic normality, the decision whether to increase the size is made based on statistical hypothesis testing. Gillis and Glineur [38] proposed a multi-level approach, which also gradually increases the problem size based on a multi-grid representation. The method in [38] is applicable not only to the ANLS methods, but also to the HALS/RRI method and the multiplicative updating method.…”

Section: Accelerated Methodsmentioning

confidence: 99%

Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework

2013

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We review algorithms developed for nonnegative matrix factorization (NMF) and nonnegative tensor factorization (NTF) from a unified view based on the block coordinate descent (BCD) framework. NMF and NTF are low-rank approximation methods for matrices and tensors in which the low-rank factors are constrained to have only nonnegative elements. The nonnegativity constraints have been shown to enable natural interpretations and allow better solutions in numerous applications including text analysis, computer vision, and bioinformatics. However, the computation of NMF and NTF remains challenging and expensive due the constraints. Numerous algorithmic approaches have been proposed to efficiently compute NMF and NTF. The BCD framework in constrained non-linear optimization readily explains the theoretical convergence properties of several efficient NMF and NTF algorithms, which are consistent with experimental observations reported in literature. In addition, we discuss algorithms that do not fit in the BCD framework contrasting them from those based on the BCD framework. With insights acquired from the unified perspective, we also propose efficient algorithms for updating NMF when there is a small change in the reduced dimension or in the data. The effectiveness of the proposed updating algorithms are validated experimentally with synthetic and real-world data sets.

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Section: Accelerated Methodsmentioning

confidence: 99%

Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework

2013

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“…Where A ∈ R B×P + and S ∈ R P×N + are all constrained to be nonnegative, and usually P min(B, N). As a result of the nonnegative constraint, the NMF has a part-based representation [42], which makes the results more physically meaningful than other statistical methods, such as the principal component analysis and ICA. To solve the NMF problem, the object function is usually constructed as follows:…”

Section: Nmfmentioning

confidence: 99%

Bilateral Filter Regularized L2 Sparse Nonnegative Matrix Factorization for Hyperspectral Unmixing

et al. 2018

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Hyperspectral unmixing (HU) is one of the most active hyperspectral image (HSI) processing research fields, which aims to identify the materials and their corresponding proportions in each HSI pixel. The extensions of the nonnegative matrix factorization (NMF) have been proved effective for HU, which usually uses the sparsity of abundances and the correlation between the pixels to alleviate the non-convex problem. However, the commonly used L 1/2 sparse constraint will introduce an additional local minima because of the non-convexity, and the correlation between the pixels is not fully utilized because of the separation of the spatial and structural information. To overcome these limitations, a novel bilateral filter regularized L 2 sparse NMF is proposed for HU. Firstly, the L 2 -norm is utilized in order to improve the sparsity of the abundance matrix. Secondly, a bilateral filter regularizer is adopted so as to explore both the spatial information and the manifold structure of the abundance maps. In addition, NeNMF is used to solve the object function in order to improve the convergence rate. The results of the simulated and real data experiments have demonstrated the advantage of the proposed method.

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“…Most of them run in O(nmr). We refer the reader to [10] for a comprehensive survey of these algorithms. In the experiments, the alternating non-negative least squares algorithm has shown to be a good trade-off between convergence speed and quality of the approximation.…”

Section: Algorithm 1 Spectral Algorithm For Ir-mamentioning

confidence: 99%

Non-negative Spectral Learning for Linear Sequential Systems

Glaude

Enderli

Pietquin

2015

Neural Information Processing

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Abstract. Method of moments (MoM) has recently become an appealing alternative to standard iterative approaches like Expectation Maximization (EM) to learn latent variable models. In addition, MoM-based algorithms come with global convergence guarantees in the form of finite sample bounds. However, given enough computation time, by using restarts and heuristics to avoid local optima, iterative approaches often achieve better performance. We believe that this performance gap is in part due to the fact that MoM-based algorithms can output negative probabilities. By constraining the search space, we propose a non-negative spectral algorithm (NNSpectral) avoiding computing negative probabilities by design. NNSpectral is compared to other MoM-based algorithms and EM on synthetic problems of the PAutomaC challenge. Not only, NNSpectral outperforms other MoM-based algorithms, but also, achieves very competitive results in comparison to EM.

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A multilevel approach for nonnegative matrix factorization

Cited by 42 publications

References 29 publications

Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework

Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework

Bilateral Filter Regularized L2 Sparse Nonnegative Matrix Factorization for Hyperspectral Unmixing

Non-negative Spectral Learning for Linear Sequential Systems

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