An L-DEIM Induced High Order Tensor Interpolatory Decomposition

TL;DR

This paper introduces a new tensor decomposition method based on L-DEIM, combining randomized sampling and empirical interpolation to achieve efficient, structure-preserving, and interpretable tensor factorizations with reduced computational cost.

Contribution

It develops a novel L-DEIM based tensor CUR-like factorization and randomized algorithms for efficient, interpretable tensor decompositions with theoretical error analysis.

Findings

Efficient algorithms for tensor decomposition with reduced computational cost.

Hybrid decompositions outperform traditional tensor CUR in accuracy.

Numerical results confirm the effectiveness of the proposed methods.

Abstract

This paper derives the CUR-type factorization for tensors in the Tucker format based on a new variant of the discrete empirical interpolation method known as L-DEIM. This novel sampling technique allows us to construct an efficient algorithm for computing the structure-preserving decomposition, which significantly reduces the computational cost. For large-scale datasets, we incorporate the random sampling technique with the L-DEIM procedure to further improve efficiency. Moreover, we propose randomized algorithms for computing a hybrid decomposition, which yield interpretable factorization and provide a smaller approximation error than the tensor CUR factorization. We provide comprehensive analysis of probabilistic errors associated with our proposed algorithms, and present numerical results that demonstrate the effectiveness of our methods.