MATLAB tensor classes for fast algorithm prototyping.

Bader, Brett W.; Kolda, Tamara G.

doi:10.2172/974890

Cited by 227 publications

(394 citation statements)

References 9 publications

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“…The supporting software is Matlab 7.8.0 (R2009a) as a platform. We use Matlab Tensor Toolbox Version 2.4 [3] whenever tensor operations are called, and we use GloptiPoly 3 [25] for general polynomial optimization for the purpose of comparison and set the relaxation order of GloptiPoly 3 by default. To simplify our implementation, we use cvx v1.2 (Grant and Boyd [17]) as a modeling tool for our MBI subroutine.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

160

177

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Abstract. In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we only accept a block update that achieves the maximum improvement, hence the name of our new search method: Maximum Block Improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proven. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem, thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.Key words. block coordinate descent, polynomial optimization problem, tensor form AMS subject classifications. 90C26, 90C30, 15A69, 49M271. Introduction. The optimization models whose objective and constraints are polynomial functions have recently attracted much research attention. This is in part due to an increased demand on the application side (cf. the sample applications in numerical linear algebra [50,26,28], material sciences [57], quantum physics [9,18], and signal processing [16,4,53]), and in part due to its own strong theoretical appeal. Indeed, polynomial optimization is a challenging task; at the same time it is rich enough to be fruitful. For instance, even the simplest instances of polynomial optimization, such as maximizing a cubic polynomial over a sphere, is NP-hard (Nesterov [42]). However, the problem is so elementary that it can be even attempted in an undergraduate calculus class. For readers interested in polynomial optimization with simple constraints, we refer to De Klerk [12] for a survey on the computational complexity of optimizing various classes of polynomial functions over a simplex, hypercube or sphere. In particular, De Klerk et al.[13] designed a polynomial-time approximation scheme (PTAS) for minimizing polynomials of fixed degree over the simplex.So far, a few results have been obtained for approximation algorithms with guaranteed worst-case performance ratios for higher degree polynomial optimization problems. Luo and Zhang [39] derived a polynomial-time approximation algorithm to optimize a multivariate quartic polynomial over a region defined by quadratic inequalities.

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Section: Numerical Experimentsmentioning

confidence: 99%

Maximum Block Improvement and Polynomial Optimization

Chen¹,

He²,

Li³

et al. 2012

SIAM J. Optim.

160

177

View full text Add to dashboard Cite

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“…Then, we obtain reconstructed tensor. The HALS NTD algorithm was compared with the HONMF [16] algorithm, and also with the HOOI algorithm for Tucker decomposition [50]. All algorithms were evaluated under the same condition for fit (with difference of the fitness less than 1e-6) using Peak Signal to Noise Ratio (PSNR) for all frontal slices.…”

Section: Denoising Tensor Datamentioning

confidence: 99%

Extended HALS algorithm for nonnegative Tucker decomposition and its applications for multiway analysis and classification

Phan

Cichocki

2011

Neurocomputing

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“…Definition 2 (Tensor unfolding [9]). Unfolding a tensor Y ∈ R I 1 ×I 2 ×···×I N along modes r = [r 1 , r 2 , .…”

Section: Notation and Basic Multilinear Algebramentioning

confidence: 99%

On Revealing Replicating Structures in Multiway Data: A Novel Tensor Decomposition Approach

Phan

Cichocki

Tichavský

et al. 2012

Latent Variable Analysis and Signal Separation

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Abstract. A novel tensor decomposition called pattern or P-decomposition is proposed to make it possible to identify replicating structures in complex data, such as textures and patterns in music spectrograms. In order to establish a computational framework for this paradigm, we adopt a multiway (tensor) approach. To this end, a novel tensor product is introduced, and the analysis of its properties shows a perfect match to the identification of recurrent data structures. Out of a whole class of possible algorithms, we illuminate those derived so as to cater for orthogonal and nonnegative patterns. Simulations on texture images and music sequence confirm the benefits of the proposed model and of the associated learning algorithms.Key words: tensor decomposition, tensor product, pattern analysis, nonnegative matrix decomposition, structural complexity Problem FormulationEstimation problems for data with self-replicating structures, such as images, various textures and music spectrograms require specifically designed approaches to identify, approximate, and retrieve the dynamical structures present in the data. By modeling data via summations of Kronecker products of two matrices (scaling and pattern matrices), Loan and Pitsianis [1] established an approximation to address this problem. Subsequently, Nagy and Kilmer [2] addressed 3-D image reconstruction from realworld imaging systems in which the point spread function was decomposed into a Kronecker product form, Bouhamidi and Jbilou [3] used Kronecker approximation for image restoration, Ford and Tyrtyshnikov focused on sparse matrices in the wavelet domain [4], while the extension to tensor data was addressed in [5].It is important to note that at present, the Kronecker approximation [1] is limited to 2-D structures which are required to have the same dimension. In this paper, we generalize this problem by considering replicas (or similar structures) for multiway data ⋆ Also affiliated with the EE

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MATLAB tensor classes for fast algorithm prototyping.

Cited by 227 publications

References 9 publications

Maximum Block Improvement and Polynomial Optimization

Maximum Block Improvement and Polynomial Optimization

Extended HALS algorithm for nonnegative Tucker decomposition and its applications for multiway analysis and classification

On Revealing Replicating Structures in Multiway Data: A Novel Tensor Decomposition Approach

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