Paper Info
Title
Randomized Alternating Least Squares for Canonical Tensor Decompositions: Application to A PDE With Random Data.
Abstract
This paper introduces a randomized variation of the alternating least squares (ALS) algorithm for rank reduction of canonical tensor formats. The aim is to address the potential numerical ill-conditioning of least squares matrices at each ALS iteration. The proposed algorithm, dubbed randomized ALS, mitigates large condition numbers via projections onto random tensors, a technique inspired by well-established randomized projection methods for solving overdetermined least squares problems in a matrix setting. A probabilistic bound on the condition numbers of the randomized ALS matrices is provided, demonstrating reductions relative to their standard counterparts. Additionally, results are provided that guarantee comparable accuracy of the randomized ALS solution at each iteration. The performance of the randomized algorithm is studied with three examples, including manufactured tensors and an elliptic PDE with random inputs. In particular, for the latter, tests illustrate not only improvements in condition numbers but also improved accuracy of the iterative solver for the PDE solution represented in a canonical tensor format.
Year
DOI
Venue
2016
10.1137/15M1042802
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Keywords
Field
DocType
separated representations,tensor decomposition,randomized projection,alternating least squares,canonical tensors,stochastic PDE
Least squares,Randomized algorithm,Overdetermined system,Mathematical optimization,Tensor,Mathematical analysis,Matrix (mathematics),Solver,Non-linear least squares,Probabilistic logic,Mathematics
Journal
Volume
Issue
ISSN
38
5
1064-8275
Citations 
PageRank 
References 
4
0.42
9
Authors
3
Name
Order
Citations
PageRank
Matthew J. Reynolds140.76
Alireza Doostan218815.57
Gregory Beylkin323430.77