Riemannian Conjugate Gradient Descent Method for Third-Order Tensor Completion

TL;DR

This paper introduces a Riemannian conjugate gradient descent method tailored for third-order tensor completion, leveraging low-rank assumptions and manifold optimization to effectively recover missing tensor entries.

Contribution

It develops a novel Riemannian optimization framework for third-order tensor completion, with convergence guarantees and sample complexity analysis under incoherence conditions.

Findings

Method successfully recovers low-rank tensors from partial data.

Convergence is guaranteed under certain incoherence conditions.

Numerical experiments demonstrate effectiveness on synthetic and image data.

Abstract

The goal of tensor completion is to fill in missing entries of a partially known tensor under a low-rank constraint. In this paper, we mainly study low rank third-order tensor completion problems by using Riemannian optimization methods on the smooth manifold. Here the tensor rank is defined to be a set of matrix ranks where the matrices are the slices of the transformed tensor obtained by applying the Fourier-related transformation onto the tubes of the original tensor. We show that with suitable incoherence conditions on the underlying low rank tensor, the proposed Riemannian optimization method is guaranteed to converge and find such low rank tensor with a high probability. In addition, numbers of sample entries required for solving low rank tensor completion problem under different initialized methods are studied and derived. Numerical examples for both synthetic and image data sets are reported to demonstrate the proposed method is able to recover low rank tensors.