Tensor Train Factorization and Completion under Noisy Data with Prior Analysis and Rank Estimation

TL;DR

This paper introduces a Bayesian tensor train decomposition method with automatic rank determination, effectively handling noisy and incomplete data for improved tensor analysis and completion.

Contribution

It proposes a probabilistic TT model with a Gaussian-product-Gamma prior for automatic rank estimation and develops an efficient variational inference algorithm.

Findings

Successfully recovers TT structure from noisy incomplete data

Outperforms existing TT methods in image completion

Achieves better image classification accuracy

Abstract

Tensor train (TT) decomposition, a powerful tool for analyzing multidimensional data, exhibits superior performance in many machine learning tasks. However, existing methods for TT decomposition either suffer from noise overfitting, or require extensive fine-tuning of the balance between model complexity and representation accuracy. In this paper, a fully Bayesian treatment of TT decomposition is employed to avoid noise overfitting, by endowing it with the ability of automatic rank determination. In particular, theoretical evidence is established for adopting a Gaussian-product-Gamma prior to induce sparsity on the slices of the TT cores, so that the model complexity is automatically determined even under incomplete and noisy observed data. Furthermore, based on the proposed probabilistic model, an efficient learning algorithm is derived under the variational inference framework. Simulation results on synthetic data show the success of the proposed model and algorithm in recovering the ground-truth TT structure from incomplete noisy data. Further experiments on real-world data demonstrate the proposed algorithm performs better in image completion and image classification, compared to other existing TT decomposition algorithms.