Implicit Multiple Tensor Decomposition

TL;DR

This paper introduces Implicit Multiple Tensor Decomposition (IMTD), a flexible framework that generalizes tensor decomposition to arbitrary orders, employs neural representations for continuous factor tensors, and provides convergence analysis without KL property.

Contribution

It proposes a novel generalization of tensor decomposition to arbitrary orders with neural representation integration and KL-free convergence analysis.

Findings

Effective tensor reconstruction demonstrated in experiments

Flexible handling of tensors with varying dimensions

Convergence of the proposed algorithm validated

Abstract

Recently, triple decomposition has attracted increasing attention for decomposing third-order tensors into three factor tensors. However, this approach is limited to third-order tensors and enforces uniformity in the lower dimensions across all factor tensors, which restricts its flexibility and applicability. To address these issues, we propose the Multiple decomposition, a novel framework that generalizes triple decomposition to arbitrary order tensors and allows the short dimensions of the factor tensors to differ. We establish its connections with other classical tensor decompositions. Furthermore, implicit neural representation (INR) is employed to continuously represent the factor tensors in Multiple decomposition, enabling the method to generalize to non-grid data. We refer to this INR-based Multiple decomposition as Implicit Multiple Tensor Decomposition (IMTD). Then, the Proximal Alternating Least Squares (PALS) algorithm is utilized to solve the IMTD-based tensor reconstruction models. Since the objective function in IMTD-based models often lacks the Kurdyka-Lojasiewicz (KL) property, we establish a KL-free convergence analysis for the algorithm. Finally, extensive numerical experiments further validate the effectiveness of the proposed method.