Multi-Tensor Network Representation for High-Order Tensor Completion

TL;DR

This paper introduces Multi-Tensor Network Representation (MTNR), a novel tensor decomposition framework that combines multiple tensor network models with adaptive topology learning for high-order tensor completion.

Contribution

The paper proposes a new MTNR framework that automatically generates tensor network topologies and combines multiple models for improved tensor completion performance.

Findings

MTNR outperforms state-of-the-art methods on synthetic and real datasets.

The adaptive topology learning algorithm effectively captures tensor structures.

Theoretical analysis reveals structural transformation of MTNR to a single tensor network.

Abstract

This work studies the problem of high-dimensional data (referred to as tensors) completion from partially observed samplings. We consider that a tensor is a superposition of multiple low-rank components. In particular, each component can be represented as multilinear connections over several latent factors and naturally mapped to a specific tensor network (TN) topology. In this paper, we propose a fundamental tensor decomposition (TD) framework: Multi-Tensor Network Representation (MTNR), which can be regarded as a linear combination of a range of TD models, e.g., CANDECOMP/PARAFAC (CP) decomposition, Tensor Train (TT), and Tensor Ring (TR). Specifically, MTNR represents a high-order tensor as the addition of multiple TN models, and the topology of each TN is automatically generated instead of manually pre-designed. For the optimization phase, an adaptive topology learning (ATL) algorithm is presented to obtain latent factors of each TN based on a rank incremental strategy and a projection error measurement strategy. In addition, we theoretically establish the fundamental multilinear operations for the tensors with TN representation, and reveal the structural transformation of MTNR to a single TN. Finally, MTNR is applied to a typical task, tensor completion, and two effective algorithms are proposed for the exact recovery of incomplete data based on the Alternating Least Squares (ALS) scheme and Alternating Direction Method of Multiplier (ADMM) framework. Extensive numerical experiments on synthetic data and real-world datasets demonstrate the effectiveness of MTNR compared with the start-of-the-art methods.