2013/06/20 by David J. Biagioni, Biagioni, David J., Daniel Beylkin +3
Computer Science · Engineering · Mathematics · #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Sparse and Compressive Sensing Techniques #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.1306.5013
openalex publication_date 2013/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce tensor Interpolative Decomposition (tensor ID) for the reduction of the separation rank of Canonical Tensor Decompositions (CTDs). Tensor ID selects, for a user-defined accuracy ε, a near optimal subset of terms of a CTD to represent the remaining terms via a linear combination of the selected terms. Tensor ID can be used as an alternative to or a step of the Alternating Least Squares (ALS) algorithm. In addition, we briefly discuss Q-factorization to reduce the size of components within an ALS iteration. Combined, tensor ID and Q-factorization lead to a new paradigm for the reduction of the separation rank of CTDs. In this context, we also discuss the spectral norm as a computational alternative to the Frobenius norm. We reduce the problem of finding tensor IDs to that of constructing Interpolative Decompositions of certain matrices. These matrices are generated via either randomized projection or randomized sampling of the given tensor. We provide cost estimates and several examples of the new approach to the reduction of separation rank.