High-dimension Tensor Completion via Gradient-based Optimization Under Tensor-train Format

In this paper, we propose a novel approach to recover the missing entries of incomplete data represented by a high-dimension tensor. Tensor-train decomposition, which has powerful tensor representation ability and is free from `the curse of dimensionality', is employed in our approach. By observed entries of incomplete data, we consider to find the factors which can capture the latent features of the data and then reconstruct the missing entries. With low-rank assumption to the original data, tensor completion problem is cast into solving optimization models. Gradient descent methods are applied to optimize the core tensors of tensor-train decomposition. We propose two algorithms: Tensor-train Weighted Optimization (TT-WOPT) and Tensor-train Stochastic Gradient Descent (TT-SGD) to solve tensor completion problems. A high-order tensorization method named visual data tensorization (VDT) is proposed to transform visual data to higher-order forms by which the performance of our algorithms can be improved. The synthetic data experiments and visual data experiments show that our algorithms outperform the state-of-the-art completion algorithms. Especially in high-dimension, high missing rate and large-scale data cases, significant performance can be obtained from our algorithms.