Quadratic regularization projected Barzilai–Borwein method for nonnegative matrix factorization

Huang, Yakui; Liu, Hongwei; Zhou, Shuisheng

doi:10.1007/s10618-014-0390-x

Cited by 37 publications

(42 citation statements)

References 32 publications

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“…By Lemma 1 in [20], we know that f (W ) is convex and its gradient ∇f (W ) is Lipschitz continuous with constant L = ∥H k (H k ) T ∥ 2 . Since H k (H k ) T is an r × r matrix and r ≪ min{m, n}, the Lipschitz constant L is not expensive to obtain.…”

Section: Monotone Projected Barzilai-borwein Methods and Its Convergencementioning

confidence: 99%

“…Since ⟨∇f (Z t ), D t ⟩ ≤ 0 (see [20]) and σ ∈ (0, 1), we have φ(1) ≤ 0. If δ t > 0, thenλ t defined by (7) is the minimizer of φ(λ).…”

Section: Algorithm 1 Monotone Projected Bb Methodsmentioning

confidence: 99%

“…They applied Nesterov's optimal gradient method (OGM) [19] to solve the subproblems without line search and developed the NeNMF solver. However, it was observed by Huang et al [20] that OGM may take thousands of iterations to reach a given tolerance which will degrade the efficiency of NeNMF. In [20], the authors proposed a quadratic regularization projected Barzilai-Borwein (QRPBB) method.…”

Section: Introductionmentioning

confidence: 99%

“…However, it was observed by Huang et al [20] that OGM may take thousands of iterations to reach a given tolerance which will degrade the efficiency of NeNMF. In [20], the authors proposed a quadratic regularization projected Barzilai-Borwein (QRPBB) method. By making use of the Lipschitz constant of the gradient, the QRPBB method improves the performance of the projected BB method significantly and outperforms other three solvers including PG, APBB2 [15], and NeNMF.…”

Section: Introductionmentioning

confidence: 99%

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An efficient monotone projected Barzilai–Borwein method for nonnegative matrix factorization

Huang

Liu

Zhou

2015

Applied Mathematics Letters

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a b s t r a c tIn this paper, we present an efficient method for nonnegative matrix factorization based on the alternating nonnegative least squares framework. Our approach adopts a monotone projected Barzilai-Borwein (MPBB) method as an essential subroutine where the step length is determined without line search. The Lipschitz constant of the gradient is exploited to accelerate convergence. Global convergence of the proposed MPBB method is established. Numerical results are reported to demonstrate the efficiency of our algorithm.

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Section: Monotone Projected Barzilai-borwein Methods and Its Convergencementioning

confidence: 99%

“…Since ⟨∇f (Z t ), D t ⟩ ≤ 0 (see [20]) and σ ∈ (0, 1), we have φ(1) ≤ 0. If δ t > 0, thenλ t defined by (7) is the minimizer of φ(λ).…”

Section: Algorithm 1 Monotone Projected Bb Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An efficient monotone projected Barzilai–Borwein method for nonnegative matrix factorization

Huang

Liu

Zhou

2015

Applied Mathematics Letters

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“…For more details on BB type methods and their applications, see [34,35,40,44,[46][47][48][49][50][51][52]66] and references therein.…”

Section: Algorithmmentioning

confidence: 99%

Smoothing projected Barzilai–Borwein method for constrained non-Lipschitz optimization

Huang

Liu

2016

Comput Optim Appl

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We present a smoothing projected Barzilai-Borwein (SPBB) algorithm for solving a class of minimization problems on a closed convex set, where the objective function is nonsmooth nonconvex, perhaps even non-Lipschitz. At each iteration, the SPBB algorithm applies the projected gradient strategy that alternately uses the two Barzilai-Borwein stepsizes to the smooth approximation of the original problem. Nonmonotone scheme is adopted to ensure global convergence. Under mild conditions, we prove convergence of the SPBB algorithm to a scaled stationary point of the original problem. When the objective function is locally Lipschitz continuous, we consider a general constrained optimization problem and show that any accumulation point generated by the SPBB algorithm is a stationary point associated with the smoothing function used in the algorithm. Numerical experiments on 2 -p problems, image restoration problems, and stochastic linear complementarity problems show that the SPBB algorithm is promising.

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Distributed geometric nonnegative matrix factorization and hierarchical alternating least squares–based nonnegative tensor factorization with the MapReduce paradigm

Zdunek

Fonał

2018

Concurrency and Computation

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Summary Nonnegative matrix factorization and its multilinear extension known as nonnegative tensor factorization are commonly used methods in machine learning and data analysis for feature extraction and dimensionality reduction for nonnegative high‐dimensional data. Dimensionality reduction for massive amounts of data usually involves distributed computation across multi‐node computer architectures. In this study, we propose various computational strategies for parallel and distributed computation of the latent factors in both factorization models, all of which are based on partitioning the computational tasks according to the MapReduce paradigm. We extend the previously reported distributed hierarchical alternating least squares algorithm to the multi‐way array factorization model, where we assume that the observed multi‐way data can be partitioned into chunks along one mode. Moreover, we propose a new geometry‐based distributed computational strategy for solving nonnegative matrix factorization problems. Numerical experiments performed using various large‐scale data sets demonstrated that these algorithms are efficient and robust to noisy data.

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Quadratic regularization projected Barzilai–Borwein method for nonnegative matrix factorization

Cited by 37 publications

References 32 publications

An efficient monotone projected Barzilai–Borwein method for nonnegative matrix factorization

An efficient monotone projected Barzilai–Borwein method for nonnegative matrix factorization

Smoothing projected Barzilai–Borwein method for constrained non-Lipschitz optimization

Distributed geometric nonnegative matrix factorization and hierarchical alternating least squares–based nonnegative tensor factorization with the MapReduce paradigm

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