Robustness analysis of preconditioned successive projection algorithm for general form of separable NMF problem

Mizutani, Tomohiko

doi:10.1016/j.laa.2016.02.016

Cited by 5 publications

(10 citation statements)

References 11 publications

(53 reference statements)

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“…Gillis and Vavasis showed the robustness of PSPA in [16]. A later study [27] was conducted under weaker conditions than those assumed by them.…”

Section: : Compute the Top-k Truncated Svdmentioning

confidence: 91%

“…Since L * is positive definite, such a C exists and it can be constructed by using the eigenvalue decomposition of L * . One may be concerned as to how this C serves as a restriction on the condition number of F in A. Intuitive explanations are given in Section 2 of [16] and Section 2.2.1 of [27].…”

Section: : Compute the Top-k Truncated Svdmentioning

confidence: 99%

“…Then, even if we replace Algorithm 3 with Algorithm 4 in the statement of Theorem 2, the theorem is still valid as long as we increase the constant values 1225, 432, and 4 involved in it. The theorem can be proven by following the arguments in the proof of Theorem 2 given in Section 3 of [27].…”

Section: : Compute the Top-k Truncated Svdmentioning

confidence: 99%

“…Accordingly, Gillis and Vavasis proposed the preconditioned SPA (PSPA) in [16]. The results in [16,27] imply that PSPA is more robust than SPA.…”

Section: Introductionmentioning

confidence: 99%

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Efficient preconditioning for noisy separable nonnegative matrix factorization problems by successive projection based low-rank approximations

Mizutani

Tanaka

2017

Mach Learn

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The successive projection algorithm (SPA) can quickly solve a nonnegative matrix factorization problem under a separability assumption. Even if noise is added to the problem, SPA is robust as long as the perturbations caused by the noise are small. In particular, robustness against noise should be high when handling the problems arising from real applications. The preconditioner proposed by Gillis and Vavasis (2015) makes it possible to enhance the noise robustness of SPA. Meanwhile, an additional computational cost is required. The construction of the preconditioner contains a step to compute the top-k truncated singular value decomposition of an input matrix. It is known that the decomposition provides the best rank-k approximation to the input matrix; in other words, a matrix with the smallest approximation error among all matrices of rank less than k. This step is an obstacle to an efficient implementation of the preconditioned SPA.To address the cost issue, we propose a modification of the algorithm for constructing the preconditioner. Although the original algorithm uses the best rank-k approximation, instead of it, our modification uses an alternative. Ideally, this alternative should have high approximation accuracy and low computational cost. To ensure this, our modification employs a rank-k approximation produced by an SPA based algorithm. We analyze the accuracy of the approximation and evaluate the computational cost of the algorithm. We then present an empirical study revealing the actual performance of the SPA based rank-k approximation algorithm and the modified preconditioned SPA.

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“…Gillis and Vavasis showed the robustness of PSPA in [16]. A later study [27] was conducted under weaker conditions than those assumed by them.…”

Section: : Compute the Top-k Truncated Svdmentioning

confidence: 91%

Section: : Compute the Top-k Truncated Svdmentioning

confidence: 99%

Section: : Compute the Top-k Truncated Svdmentioning

confidence: 99%

“…Accordingly, Gillis and Vavasis proposed the preconditioned SPA (PSPA) in [16]. The results in [16,27] imply that PSPA is more robust than SPA.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Efficient preconditioning for noisy separable nonnegative matrix factorization problems by successive projection based low-rank approximations

Mizutani

Tanaka

2017

Mach Learn

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“…The only unspecified part of the algorithm is the application of successive projection algorithm (SPA) to the matrixÛ T . We will briefly describe this algorithm in Section 3.2 below, see also the detailed discussions in [6,7,17].…”

Section: Algorithm 1 Spocmentioning

confidence: 99%

Consistent Estimation of Mixed Memberships with Successive Projections

Panov

Slavnov

Ushakov

2017

Studies in Computational Intelligence

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This paper considers the parameter estimation problem in Mixed Membership Stochastic Block Model (MMSB), which is a quite general instance of random graph model allowing for overlapping community structure. We present the new algorithm successive projection overlapping clustering (SPOC) which combines the ideas of spectral clustering and geometric approach for separable nonnegative matrix factorization. The proposed algorithm is provably consistent under MMSB with general conditions on the parameters of the model. SPOC is also shown to perform well experimentally in comparison to other algorithms.

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