Robust to Rank Selection: Low-Rank Sparse Tensor-Ring Completion

Neural Network Compression Based on Tensor Ring Decomposition

Deep neural networks (DNNs) have made great breakthroughs and seen applications in many domains. However, the incomparable accuracy of DNNs is achieved with the cost of considerable memory consumption and high computational complexity, which restricts their deployment on conventional desktops and portable devices. To address this issue, low-rank factorization, which decomposes the neural network parameters into smaller sized matrices or tensors, has emerged as a promising technique for network compression. In this article, we propose leveraging the emerging tensor ring (TR) factorization to compress the neural network. We investigate the impact of both parameter tensor reshaping and TR decomposition (TRD) on the total number of compressed parameters. To achieve the maximal parameter compression, we propose an algorithm based on prime factorization that simultaneously identifies the optimal tensor reshaping and TRD. In addition, we discover that different execution orders of the core tensors result in varying computational complexities. To identify the optimal execution order, we construct a novel tree structure. Based on this structure, we propose a top-to-bottom splitting algorithm to schedule the execution of core tensors, thereby minimizing computational complexity. We have performed extensive experiments using three kinds of neural networks with three different datasets. The experimental results demonstrate that, compared with the three state-of-the-art algorithms for low-rank factorization, our algorithm can achieve better performance with much lower memory consumption and lower computational complexity.

Balanced Unfolding Induced Tensor Nuclear Norms for High-Order Tensor Completion

The recently proposed tensor tubal rank has been witnessed to obtain extraordinary success in real-world tensor data completion. However, existing works usually fix the transform orientation along the third mode and may fail to turn multidimensional low-tubal-rank structure into account. To alleviate these bottlenecks, we introduce two unfolding induced tensor nuclear norms (TNNs) for the tensor completion (TC) problem, which naturally extends tensor tubal rank to high-order data. Specifically, we show how multidimensional low-tubal-rank structure can be captured by utilizing a novel balanced unfolding strategy, upon which two TNNs, namely, overlapped TNN (OTNN) and latent TNN (LTNN), are developed. We also show the immediate relationship between the tubal rank of unfolding tensor and the existing tensor network (TN) rank, e.g., CANDECOMP/PARAFAC (CP) rank, Tucker rank, and tensor ring (TR) rank, to demonstrate its efficiency and practicality. Two efficient TC models are then proposed with theoretical guarantees by analyzing a unified nonasymptotic upper bound. To solve optimization problems, we develop two alternating direction methods of multipliers (ADMM) based algorithms. The proposed models have been demonstrated to exhibit superior performance based on experimental findings involving synthetic and real-world tensors, including facial images, light field images, and video sequences.

Tensor Completion Using Bilayer Multimode Low-Rank Prior and Total Variation

In this article, we propose a novel bilayer low-rankness measure and two models based on it to recover a low-rank (LR) tensor. The global low rankness of underlying tensor is first encoded by LR matrix factorizations (MFs) to the all-mode matricizations, which can exploit multiorientational spectral low rankness. Presumably, the factor matrices of all-mode decomposition are LR, since local low-rankness property exists in within-mode correlation. In the decomposed subspace, to describe the refined local LR structures of factor/subspace, a new low-rankness insight of subspace: a double nuclear norm scheme is designed to explore the so-called second-layer low rankness. By simultaneously representing the bilayer low rankness of the all modes of the underlying tensor, the proposed methods aim to model multiorientational correlations for arbitrary N -way ( N ≥ 3 ) tensors. A block successive upper-bound minimization (BSUM) algorithm is designed to solve the optimization problem. Subsequence convergence of our algorithms can be established, and the iterates generated by our algorithms converge to the coordinatewise minimizers in some mild conditions. Experiments on several types of public datasets show that our algorithm can recover a variety of LR tensors from significantly fewer samples than its counterparts.

Online subspace learning and imputation by Tensor-Ring decomposition

This paper considers the completion problem of a partially observed high-order streaming data, which is cast as an online low-rank tensor completion problem. Though the online low-rank tensor completion problem has drawn lots of attention in recent years, most of them are designed based on the traditional decomposition method, such as CP and Tucker. Inspired by the advantages of Tensor Ring decomposition over the traditional decompositions in expressing high-order data and its superiority in missing values estimation, this paper proposes two online subspace learning and imputation methods based on Tensor Ring decomposition. Specifically, we first propose an online Tensor Ring subspace learning and imputation model by formulating an exponentially weighted least squares with Frobenium norm regularization of TR-cores. Then, two commonly used optimization algorithms, i.e. alternating recursive least squares and stochastic-gradient algorithms, are developed to solve the proposed model. Numerical experiments show that the proposed methods are more effective to exploit the time-varying subspace in comparison with the conventional Tensor Ring completion methods. Besides, the proposed methods are demonstrated to be superior to obtain better results than state-of-the-art online methods in streaming data completion under varying missing ratios and noise.