On Tensor Train Rank Minimization: Statistical Efficiency and Scalable Algorithm

Abstract

Tensor train (TT) decomposition provides a space-efficient representation for higher-order tensors. Despite its advantage, we face two crucial limitations when we apply the TT decomposition to machine learning problems: the lack of statistical theory and of scalable algorithms. In this paper, we address the limitations. First, we introduce a convex relaxation of the TT decomposition problem and derive its error bound for the tensor completion task. Next, we develop an alternating optimization method with a randomization technique, in which the time complexity is as efficient as the space complexity is. In experiments, we numerically confirm the derived bounds and empirically demonstrate the performance of our method with a real higher-order tensor.

24 pages

References in corpus (6)

• Tensorizing Neural Networks
• Equivalence of restricted Boltzmann machines and tensor network states
• Estimation of low-rank tensors via convex optimization
• Tensor Completion by Alternating Minimization under the Tensor Train (TT) Model
• Efficient tensor completion: Low-rank tensor train
• Alternating Least Squares Tensor Completion in The TT-Format

Cited by in corpus (1)

• Provable Tensor-Train Format Tensor Completion by Riemannian Optimization