Multi-Tensor Network Representation for High-Order Tensor Completion

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.

References in corpus (15)

• An overview of gradient descent optimization algorithms
• Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
• Tensor Decomposition for Signal Processing and Machine Learning
• Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis
• Tensor renormalization group approach to 2D classical lattice models
• Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination
• Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges PART 1
• A Gradient Descent Algorithm on the Grassman Manifold for Matrix Completion
• Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks
• Tensor Ring Decomposition
• Incoherence-Optimal Matrix Completion
• Multiarray Signal Processing: Tensor decomposition meets compressed sensing
• Higher-dimension Tensor Completion via Low-rank Tensor Ring Decomposition
• Tensor-Ring Nuclear Norm Minimization and Application for Visual Data Completion
• Adaptive Learning of Tensor Network Structures