Guaranteed Sampling Flexibility for Low-tubal-rank Tensor Completion

TL;DR

This paper introduces t-CCS, a flexible sampling model for low-tubal-rank tensor completion, along with an efficient algorithm, validated through theoretical analysis and experiments on synthetic and real data.

Contribution

We propose t-CCS, a novel sampling method that enhances flexibility in tensor completion, and develop the ITCURTC algorithm with theoretical guarantees.

Findings

t-CCS improves sampling flexibility over existing methods.

Theoretical conditions for successful tensor recovery are established.

ITCURTC algorithm demonstrates effectiveness on various datasets.

Abstract

While Bernoulli sampling is extensively studied in tensor completion, t-CUR sampling approximates low-tubal-rank tensors via lateral and horizontal subtensors. However, both methods lack sufficient flexibility for diverse practical applications. To address this, we introduce Tensor Cross-Concentrated Sampling (t-CCS), a novel and straightforward sampling model that advances the matrix cross-concentrated sampling concept within a tensor framework. t-CCS effectively bridges the gap between Bernoulli and t-CUR sampling, offering additional flexibility that can lead to computational savings in various contexts. A key aspect of our work is the comprehensive theoretical analysis provided. We establish a sufficient condition for the successful recovery of a low-rank tensor from its t-CCS samples. In support of this, we also develop a theoretical framework validating the feasibility of t-CUR via uniform random sampling and conduct a detailed theoretical sampling complexity analysis for tensor completion problems utilizing the general Bernoulli sampling model. Moreover, we introduce an efficient non-convex algorithm, the Iterative t-CUR Tensor Completion (ITCURTC) algorithm, specifically designed to tackle the t-CCS-based tensor completion. We have intensively tested and validated the effectiveness of the t-CCS model and the ITCURTC algorithm across both synthetic and real-world datasets.