# Provable Tensor Ring Completion

## TL;DR

This paper proves that tensors can be exactly recovered from limited measurements using tensor ring decomposition under certain conditions, with theoretical guarantees and improved empirical performance.

## Contribution

It introduces a provable tensor completion method based on tensor ring decomposition, providing theoretical recovery guarantees and demonstrating superior empirical results.

## Key findings

- Exact recovery with high probability given specified sample complexity
- Proposed TR incoherence condition similar to matrix incoherence
- Improved recovery performance over state-of-the-art methods

## Abstract

Tensor completion recovers a multi-dimensional array from a limited number of measurements. Using the recently proposed tensor ring (TR) decomposition, in this paper we show that a d-order tensor of dimensional size n and TR rank r can be exactly recovered with high probability by solving a convex optimization program, given n^{d/2} r^2 ln^7(n^{d/2})samples. The proposed TR incoherence condition under which the result holds is similar to the matrix incoherence condition. The experiments on synthetic data verify the recovery guarantee for TR completion. Moreover, the experiments on real-world data show that our method improves the recovery performance compared with the state-of-the-art methods.