Tensor Ring Decomposition

Abstract

Tensor networks have in recent years emerged as the powerful tools for solving the large-scale optimization problems. One of the most popular tensor network is tensor train (TT) decomposition that acts as the building blocks for the complicated tensor networks. However, the TT decomposition highly depends on permutations of tensor dimensions, due to its strictly sequential multilinear products over latent cores, which leads to difficulties in finding the optimal TT representation. In this paper, we introduce a fundamental tensor decomposition model to represent a large dimensional tensor by a circular multilinear products over a sequence of low dimensional cores, which can be graphically interpreted as a cyclic interconnection of 3rd-order tensors, and thus termed as tensor ring (TR) decomposition. The key advantage of TR model is the circular dimensional permutation invariance which is gained by employing the trace operation and treating the latent cores equivalently. TR model can be viewed as a linear combination of TT decompositions, thus obtaining the powerful and generalized representation abilities. For optimization of latent cores, we present four different algorithms based on the sequential SVDs, ALS scheme, and block-wise ALS techniques. Furthermore, the mathematical properties of TR model are investigated, which shows that the basic multilinear algebra can be performed efficiently by using TR representaions and the classical tensor decompositions can be conveniently transformed into the TR representation. Finally, the experiments on both synthetic signals and real-world datasets were conducted to evaluate the performance of different algorithms.

References in corpus (2)

• Computations in Quantum Tensor Networks
• Tensor networks-a new tool for old problems

Cited by in corpus (49)

• Tensor Methods in Computer Vision and Deep Learning
• Lecture Notes of Tensor Network Contractions
• Guaranteed Tensor Recovery Fused Low-rankness and Smoothness
• Tensorized Embedding Layers for Efficient Model Compression
• Tensor Train Neighborhood Preserving Embedding
• Gauging tensor networks with belief propagation
• Tensor Decomposition for Model Reduction in Neural Networks: A Review
• Fourth-order Tensors with Multidimensional Discrete Transforms
• Wide Compression: Tensor Ring Nets
• Randomized Tensor Ring Decomposition and Its Application to Large-scale Data Reconstruction
• Tensor networks for interpretable and efficient quantum-inspired machine learning
• Efficient Visual Recognition with Deep Neural Networks: A Survey on Recent Advances and New Directions
• Reduced-Rank Tensor-on-Tensor Regression and Tensor-variate Analysis of Variance
• Higher-dimension Tensor Completion via Low-rank Tensor Ring Decomposition
• Quantum State Tomography for Matrix Product Density Operators
• Tensor Ring Decomposition with Rank Minimization on Latent Space: An Efficient Approach for Tensor Completion
• High-dimension Tensor Completion via Gradient-based Optimization Under Tensor-train Format
• The trouble with tensor ring decompositions
• Hyperspectral Super-Resolution via Coupled Tensor Ring Factorization
• Interaction Decompositions for Tensor Network Regression
• Efficient construction of tensor ring representations from sampling
• Shortcut Matrix Product States and its applications
• Adaptive Learning of Tensor Network Structures
• Minimizing low-rank models of high-order tensors: Hardness, span, tight relaxation, and applications
• Efficient Low Rank Tensor Ring Completion
• SurReal: Complex-Valued Learning as Principled Transformations on a Scaling and Rotation Manifold
• Tensor-Ring Decomposition with Index-Splitting
• Lower and Upper Bounds on the VC-Dimension of Tensor Network Models
• Concatenated image completion via tensor augmentation and completion
• Searching to Sparsify Tensor Decomposition for N-ary Relational Data
• Hierarchical Tensor Ring Completion
• Robust Low-tubal-rank Tensor Completion based on Tensor Factorization and Maximum Correntopy Criterion
• Generalizing Tensor Decomposition for N-ary Relational Knowledge Bases
• H-OWAN: Multi-distorted Image Restoration with Tensor 1x1 Convolution
• Dictionary Learning with Low-rank Coding Coefficients for Tensor Completion
• Semi-tensor Product-based TensorDecomposition for Neural Network Compression
• Tensor Completion via Tensor Networks with a Tucker Wrapper
• POIRot: A rotation invariant omni-directional pointnet
• Beyond Unfolding: Exact Recovery of Latent Convex Tensor Decomposition under Reshuffling
• Tensor Ring Decomposition: Optimization Landscape and One-loop Convergence of Alternating Least Squares
• Tensor Completion via Convolutional Sparse Coding Regularization
• MARS: Masked Automatic Ranks Selection in Tensor Decompositions
• Fast Hypergraph Regularized Nonnegative Tensor Ring Factorization Based on Low-Rank Approximation
• A Sampling-Based Method for Tensor Ring Decomposition
• An Efficient Tensor Completion Method via New Latent Nuclear Norm
• Low-Rank Embedding of Kernels in Convolutional Neural Networks under Random Shuffling
• Block-term Tensor Neural Networks
• Protecting Big Data Privacy Using Randomized Tensor Network Decomposition and Dispersed Tensor Computation
• Multi-Tensor Network Representation for High-Order Tensor Completion