Exact tensor completion using t-SVD (1502.04689v2)

Abstract: In this paper we focus on the problem of completion of multidimensional arrays (also referred to as tensors) from limited sampling. Our approach is based on a recently proposed tensor-Singular Value Decomposition (t-SVD) [1]. Using this factorization one can derive notion of tensor rank, referred to as the tensor tubal rank, which has optimality properties similar to that of matrix rank derived from SVD. As shown in [2] some multidimensional data, such as panning video sequences exhibit low tensor tubal rank and we look at the problem of completing such data under random sampling of the data cube. We show that by solving a convex optimization problem, which minimizes the tensor nuclear norm obtained as the convex relaxation of tensor tubal rank, one can guarantee recovery with overwhelming probability as long as samples in proportion to the degrees of freedom in t-SVD are observed. In this sense our results are order-wise optimal. The conditions under which this result holds are very similar to the incoherency conditions for the matrix completion, albeit we define incoherency under the algebraic set-up of t-SVD. We show the performance of the algorithm on some real data sets and compare it with other existing approaches based on tensor flattening and Tucker decomposition.

• The paper establishes a novel t-SVD method achieving exact tensor completion under specific incoherence conditions.
• It rigorously derives probabilistic guarantees and sharp bounds that link sampling probability to successful recovery in the tensor multi-rank regime.
• The work extends matrix completion techniques to higher dimensions, opening avenues in signal processing, computer vision, and machine learning.

Incoherence Optimal Tensor Completion Using t-SVD

This paper by Zemin Zhang and Shuchin Aeron introduces an innovative approach to tensor completion leveraging the t-SVD (tensor Singular Value Decomposition) framework. The primary focus is on deriving exact tensor completion under an incoherence condition, utilizing methodologies that extend the theoretical underpinnings of matrix completion to higher-dimensional data representations.

Technical Overview

The authors present a rigorous framework for tensor completion, grounding their approach in the t-SVD domain. They establish conditions for exact recovery of tensors based on sampling and incoherence. The key contributions of the paper are detailed proofs of various lemmas that demonstrate the efficacy of their method, particularly within the tensor multi-rank regime. These proofs are centered on leveraging properties of the tensor nuclear norm and address both theoretical and computational aspects of the problem.

Numerical Results and Claims

The paper provides comprehensive mathematical formulations and proofs. Strong probabilistic guarantees are offered for the success of their completion strategy, supported by a thorough analysis of sampling conditions. The results link tensor incoherence levels to accurate recovery rates, and the bounds proposed are sharp enough to guide practical implementations.

One of the pivotal results states that the method achieves exact recovery with high probability, provided the sampling probability aligns with derived theoretical values. The proofs include complex mathematical techniques ensuring the robustness of these claims, further augmented by bounding techniques on matrix and tensor norms.

Implications and Future Developments

The research extends the matrix completion techniques to tensors, which opens up vast possibilities for handling multi-dimensional data in applications like signal processing, computer vision, and machine learning. By advancing the theoretical foundation of tensor analysis, Zhang and Aeron's work contributes significantly to tensor-based methods' computational efficiency and feasibility.

Future developments could explore adaptations and optimizations of the t-SVD-based approach in more diverse contexts and datasets. Another promising direction is the application of this framework to real-time systems where efficient completion is crucial. This theoretical groundwork also sets the stage for exploring advanced geometric interpretations of tensors, potentially fostering breakthroughs in areas where high-dimensional data is prevalent.

The paper refrains from overly broad claims, focusing instead on delivering a detailed exegesis of tensor completion's theoretical landscape. It provides a robust platform for further exploration into tensor-based methodologies in AI and large-scale data processing applications.