On Tensor Completion via Nuclear Norm Minimization (1405.1773v1)

Abstract: Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. To overcome these difficulties, existing approaches often proceed by unfolding tensors into matrices and then apply techniques for matrix completion. We show here that such matricization fails to exploit the tensor structure and may lead to suboptimal procedure. More specifically, we investigate a convex optimization approach to tensor completion by directly minimizing a tensor nuclear norm and prove that this leads to an improved sample size requirement. To establish our results, we develop a series of algebraic and probabilistic techniques such as characterization of subdifferetial for tensor nuclear norm and concentration inequalities for tensor martingales, which may be of independent interests and could be useful in other tensor related problems.

• The paper proposes a direct convex optimization approach for tensor completion based on minimizing the tensor nuclear norm, avoiding information loss from matricization.
• The authors establish that this method requires considerably fewer samples for perfect tensor recovery compared to previous matricization-based methods.
• The research develops new tensor-specific frameworks like subgradient characterization and concentration inequalities, beneficial for tensor completion and other problems.

On Tensor Completion via Nuclear Norm Minimization

The paper authored by Ming Yuan and Cun-Hui Zhang addresses the complex problem of tensor completion, a crucial task with applications across various domains such as hyper-spectral image analysis, multi-energy computed tomography, and text mining. The challenge in tensor completion arises primarily due to the intricate nature of higher-order tensor decompositions, which become increasingly nontrivial as the tensor order exceeds two.

The authors critique prevailing methods which rely on transforming tensors into matrices, arguing convincingly that this technique forfeits the intrinsic structure of tensors, often leading to suboptimal performance. Instead, they propose a direct convex optimization method by leveraging the minimization of the tensor nuclear norm, thus preserving tensor organization and achieving more efficient sample size requirements.

Central to their solution is the development of new algebraic and probabilistic frameworks catered specifically to tensors, such as novel characterizations of the subdifferential of the tensor nuclear norm and concentration inequalities applicable to tensor martingales. These advancements not only facilitate the proposed tensor completion approach but also possess potential utility in broader tensor-related problems.

The paper highlights a significant finding: the authors establish that the previously accepted matricization-based sample size requirements for tensor reconstruction are not ideal. Their innovative nuclear norm minimization formulation requires considerably fewer samples, resulting in a more streamlined approach to achieving perfect tensor recovery. Specifically, the authors demonstrate that the sample size requirement is $$|\Omega| \ge C\left(r^2(d_1 + d_2 + d_3) + \sqrt{rd_1 d_2 d_3}\right)\left(d_1 + d_2 + d_3\right)$$, with $r$ representing a composite rank-like measure and $C$ being a constant.

The theoretical implications of such a shift from matrix to tensor-focused methods suggest potential improvements in computational efficiency and accuracy across various applications that utilize high-dimensional data. The authors' methods present a decisive step forward in tensor completion techniques and could stimulate further research into more generalized solutions for higher-order tensors beyond third-order formulations.

Considering future AI developments, one could speculate that these enhanced tensor completion techniques might integrate deeply with machine learning models, particularly in environments characterized by vast multidimensional datasets. The potential to accurately complete tensors with fewer samples could lead to more reliable models and systems, particularly in data-scarce applications like medical imaging and remote sensing.

In conclusion, this paper proposes an approach that represents an evolution of tensor completion methodologies. By minimizing the tensor nuclear norm directly, Yuan and Zhang offer a valuable new perspective that is likely to influence future research in both theoretical and applied computing areas dealing with high-order tensors.