On Tensor Train Rank Minimization: Statistical Efficiency and Scalable Algorithm

Imaizumi, Masaaki; Maehara, Takanori; Hayashi, Kohei

doi:10.48550/arxiv.1708.00132

Cited by 2 publications

(3 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let T 0 be initialized by the sequential second-order method as Algorithm 3 and {T l } lmax l=1 be the iterates produced by Algorithm 2 where l max is the maximum number of iterations, and the stepsize α = 0.12n −1 d * . There exist absolute constants C m,1 , C m,2 > 0 depending only on m such that if the sample size n satisfies This improves the existing result (Imaizumi et al, 2017) based on matricization and matrix nuclear norm penalization. Moreover, for the case m = 3, Barak and Moitra (2016) conjectures that, based on the reduction to Boolean satisfiability problem, O(d 3/2 ) is a lower bound for the sample size such that polynomial-time algorithm exists for exact tensor completion.…”

Section: Exact Recovery and Convergence Analysismentioning

confidence: 51%

See 1 more Smart Citation

Provable Tensor-Train Format Tensor Completion by Riemannian Optimization

Cai¹,

Li²,

Dong³

2021

Preprint

View full text Add to dashboard Cite

The tensor train (TT) format enjoys appealing advantages in handling structural high-order tensors. The recent decade has witnessed the wide applications of TT-format tensors from diverse disciplines, among which tensor completion has drawn considerable attention. Numerous fast algorithms, including the Riemannian gradient descent (RGrad) algorithm, have been proposed for the TT-format tensor completion. However, the theoretical guarantees of these algorithms are largely missing or sub-optimal, partly due to the complicated and recursive algebraic operations in TT-format decomposition. Moreover, existing results established for the tensors of other formats, for example, Tucker and CP, are inapplicable because the algorithms treating TT-format tensors are substantially different and more involved. In this paper, we provide, to our best knowledge, the first theoretical guarantees of the convergence of RGrad algorithm for TT-format tensor completion, under a nearly optimal sample size condition. The RGrad algorithm converges linearly with a constant contraction rate that is free of tensor condition number without the necessity of re-conditioning. We also propose a novel approach, referred to as the sequential second-order moment method, to attain a warm initialization under a similar sample size requirement. As a byproduct, our result even significantly refines the prior investigation of RGrad algorithm for matrix completion. Numerical experiments confirm our theoretical discovery and showcase the computational speedup gained by the TT-format decomposition.

show abstract

Section: Exact Recovery and Convergence Analysismentioning

confidence: 51%

“…These prior works mostly focus on the methodology and algorithm designs without, or with rather limited, theoretical justification. Towards that end, Imaizumi et al (2017) 2020) established the statistically optimal convergence rates of tensor SVD by the fast higher-order orthogonal iteration algorithm in the TT-format.…”

Section: Introductionmentioning

confidence: 99%

Provable Tensor-Train Format Tensor Completion by Riemannian Optimization

Cai¹,

Li²,

Dong³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…This involves the derivation of an analytic expression for geodesics, as well as an expression for the Riemannian Hessian in the respective product manifolds. Another important motivation for phrasing GST as a randomly subsampled tensor completion problem is to bring it closer to potential analytical recovery guarantees common for related tensor completion problems [50][51][52][53][54][55][56][57], opening up a new research direction.…”

Section: Introductionmentioning

confidence: 99%