Hierarchical ALS Algorithms for Nonnegative Matrix and 3D Tensor Factorization

Cichocki, Andrzej; Zdunek, Rafał; Амари, Шун-ичи

doi:10.1007/978-3-540-74494-8_22

Cited by 242 publications

(255 citation statements)

References 16 publications

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“…Other BSS approaches that can deal with statistically dependent sources include: independent subspace analysis (ISA) [24][25], nonnegative matrix and tensor factorization (NMF/NTF) [27][28][29][30], and the blind Richardson-Lucy (BRL) algorithm [33][34][35][36], which are used for comparison purpose in this paper. They are briefly described as follows.…”

Section: Algorithms For Comparisonmentioning

confidence: 99%

“…NMF/NTF algorithms may yield physically useful solutions by imposing the nonnegativity, sparseness or smoothness constraints on the sources [27][28][29][30]. In [27], the NMF algorithm was first derived to minimize two cost functions: the squared Euclidean distance and the Kullback-Leibler divergence.…”

Section: Nonnegative Matrix and Tensor Factorizationmentioning

confidence: 99%

“…Generalization of the NMF algorithms [27] has been done in [28][29][30]. The gradient-based NMF algorithm with a sparseness constraint being incorporated into the cost function leads to the regularized alternating least square (RALS) algorithm [28]:…”

Section: Nonnegative Matrix and Tensor Factorizationmentioning

confidence: 99%

“…Extension of this approach, known as local ALS, to 3D tensor factorization is given in [29], which is referred to as the NTF algorithm in this paper. In this case G and S in model (2) become 3D tensors:…”

Section: Nonnegative Matrix and Tensor Factorizationmentioning

confidence: 99%

“…Unlike a majority of NTF/NMF algorithms that estimate the source matrix/tensor globally, the local ALS algorithm [29] performs it at the source level:…”

Section: Nonnegative Matrix and Tensor Factorizationmentioning

confidence: 99%

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Dependent component analysis for blind restoration of images degraded by turbulent atmosphere

Kopriva

2009

Neurocomputing

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In our previous research, we applied independent component analysis (ICA) for the restoration of image sequences degraded by atmospheric turbulence. The original high-resolution image and turbulent sources were considered independent sources from which the degraded image is composed of. Although the result was promising, the assumption of source independence may not be true in practice. In this paper, we propose to apply the concept of dependent component analysis (DCA), which can relax the independence assumption, to image restoration. In addition, the restored image can be further enhanced by employing a recently developed Gabor-filter-bank-based single channel blind image deconvolution algorithm. Both simulated and real data experiments demonstrate that DCA outperforms ICA, resulting in the flexibility in the use of adjacent image frames. The contribution of this research is to convert the original multi-frame blind deconvolution problem into blind source separation problem without 3 the assumption on source independence; as a result, there is no a priori information, such as sensor bandwidth, point-spread-function, or statistics of source images, that is required.

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Section: Algorithms For Comparisonmentioning

confidence: 99%

Section: Nonnegative Matrix and Tensor Factorizationmentioning

confidence: 99%

Section: Nonnegative Matrix and Tensor Factorizationmentioning

confidence: 99%

Section: Nonnegative Matrix and Tensor Factorizationmentioning

confidence: 99%

“…Unlike a majority of NTF/NMF algorithms that estimate the source matrix/tensor globally, the local ALS algorithm [29] performs it at the source level:…”

Section: Nonnegative Matrix and Tensor Factorizationmentioning

confidence: 99%

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Dependent component analysis for blind restoration of images degraded by turbulent atmosphere

Kopriva

2009

Neurocomputing

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Nonnegative Matrix and Tensor Factorizations

Cichocki¹,

Zdunek²,

Phan³

et al. 2009

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Introduction – Problem Statements and Models

Cichocki

Zdunek

Phan

et al. 2009

Nonnegative Matrix and Tensor Factorizations

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Introduction -Problem Statements and ModelsMatrix factorization is an important and unifying topic in signal processing and linear algebra, which has found numerous applications in many other areas. This chapter introduces basic linear and multi-linear 1 models for matrix and tensor factorizations and decompositions, and formulates the analysis framework for the solution of problems posed in this book. The workhorse in this book is Nonnegative Matrix Factorization (NMF) for sparse representation of data and its extensions including the multi-layer NMF, semi-NMF, sparse NMF, tri-NMF, symmetric NMF, orthogonal NMF, non-smooth NMF (nsNMF), overlapping NMF, convolutive NMF (CNMF), and large-scale NMF. Our particular emphasis is on NMF and semi-NMF models and their extensions to multi-way models (i.e., multi-linear models which perform multi-way array (tensor) decompositions) with nonnegativity and sparsity constraints, including, Nonnegative Tucker Decompositions (NTD), Constrained Tucker Decompositions, Nonnegative and semi-nonnegative Tensor Factorizations (NTF) that are mostly based on a family of the TUCKER, PARAFAC and PARATUCK models.As the theory and applications of NMF, NTF and NTD are still being developed, our aim is to produce a unified, state-of-the-art framework for the analysis and development of efficient and robust algorithms. In doing so, our main goals are to:1. Develop various working tools and algorithms for data decomposition and feature extraction based on nonnegative matrix factorization (NMF) and sparse component analysis (SCA) approaches. We thus integrate several emerging techniques in order to estimate physically, physiologically, and neuroanatomically meaningful sources or latent (hidden) components with morphological constraints. These constraints include nonnegativity, sparsity, orthogonality, smoothness, and semi-orthogonality. 2. Extend NMF models to multi-way array (tensor) decompositions, factorizations, and filtering, and to derive efficient learning algorithms for these models. 3. Develop a class of advanced blind source separation (BSS), unsupervised feature extraction and clustering algorithms, and to evaluate their performance using a priori knowledge and morphological constraints. 4. Develop computational methods to efficiently solve the bi-linear system Y = AX + E for noisy data, where Y is an input data matrix, A and X represent unknown matrix factors to be estimated, and the matrix E represents error or noise (which should be minimized using suitably designed cost function).1 A function in two or more variables is said to be multi-linear if it is linear in each variable separately.

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Hierarchical ALS Algorithms for Nonnegative Matrix and 3D Tensor Factorization

Cited by 242 publications

References 16 publications

Dependent component analysis for blind restoration of images degraded by turbulent atmosphere

Dependent component analysis for blind restoration of images degraded by turbulent atmosphere

Nonnegative Matrix and Tensor Factorizations

Introduction – Problem Statements and Models

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