Incoherence-Optimal Matrix Completion (1310.0154v4)

Abstract: This paper considers the matrix completion problem. We show that it is not necessary to assume joint incoherence, which is a standard but unintuitive and restrictive condition that is imposed by previous studies. This leads to a sample complexity bound that is order-wise optimal with respect to the incoherence parameter (as well as to the rank $r$ and the matrix dimension $n$ up to a log factor). As a consequence, we improve the sample complexity of recovering a semidefinite matrix from $O(nr^{2}\log^{2}n)$ to $O(nr\log^{2}n)$, and the highest allowable rank from $\Theta(\sqrt{n}/\log n)$ to $\Theta(n/\log^{2}n)$. The key step in proof is to obtain new bounds on the $\ell_{\infty,2}$-norm, defined as the maximum of the row and column norms of a matrix. To illustrate the applicability of our techniques, we discuss extensions to SVD projection, structured matrix completion and semi-supervised clustering, for which we provide order-wise improvements over existing results. Finally, we turn to the closely-related problem of low-rank-plus-sparse matrix decomposition. We show that the joint incoherence condition is unavoidable here for polynomial-time algorithms conditioned on the Planted Clique conjecture. This means it is intractable in general to separate a rank-$\omega(\sqrt{n})$ positive semidefinite matrix and a sparse matrix. Interestingly, our results show that the standard and joint incoherence conditions are associated respectively with the information (statistical) and computational aspects of the matrix decomposition problem.

• The paper demonstrates that removing the joint incoherence condition enables optimal low-rank matrix completion using only the standard incoherence criterion.
• It achieves a reduction in sample complexity from O(nr² log² n) to O(nr log² n) while increasing the permissible matrix rank.
• The study introduces the ℓ∞,2 norm to better capture singular vector distribution, paving the way for advancements in SVD-based and structured matrix applications.

Incoherence-Optimal Matrix Completion: A Review

This paper presents a pivotal examination of the constraints typically imposed in the matrix completion problem, offering a novel perspective on incoherence. Matrix completion involves recovering a low-rank matrix from a subset of its observed entries. Traditional methods have relied on two forms of incoherence: the standard and joint incoherence conditions, with the latter often being seen as unintuitive and unnecessarily restrictive.

At the core of the discussion is the removal of the joint incoherence condition. Previous studies mandated the alignment of the matrix’s singular vectors to meet specific parameters, which limited applicability in various real-world matrix types, including positive semidefinite matrices. The crux of this paper is demonstrating that only the standard incoherence condition is necessary, characterized by the distribution of singular vectors across rows and columns of the matrix, thus broadening the scope of matrices that can be optimally completed.

Quantitative Improvements and Theoretical Implications

Quantitatively, this paper reduces the sample complexity from $O(nr^2\log^2n)$ to $O(nr\log^2n)$, given the standard incoherence condition without enforcing joint incoherence. The highest permissible rank for matrix completion is also improved from $\Theta(\sqrt{n}/\log n)$ to $\Theta(n/\log^2n)$. These results aren't just statistically robust but also theoretically optimal, up to a logarithmic factor. This demonstrates a refinement in the underlying theoretical framework governing matrix completion, allowing for more efficient utilization of observations.

The paper leverages a revised analytical framework, utilizing the $\ell_{\infty,2}$ norm—a measure focusing on the maximum of row and column norms—as opposed to the previously used $\ell_{\infty}$ norm. This subtle shift provides a more appropriate metric for capturing the distribution of information in the rows and columns of a matrix, which aligns better with the goal of matrix completion.

Extensions and Computational Complexity

Additionally, the paper explores extensions to related areas, such as the error bounds in Singular Value Decomposition (SVD) projection algorithms, structured matrix completion, and semi-supervised clustering. These areas benefit significantly from the reduced complexity requirements established, enabling better handling of matrices with inherent subspace structures or where partial information is already available about the row or column spaces.

A further critical point of discussion in the paper is the distinction between matrix completion and matrix decomposition, where joint incoherence remains unavoidable due to computational intractabilities. The paper argues, under the Planted Clique complexity conjecture, that decomposing a matrix into low-rank and sparse components necessitates joint incoherence, showing that this condition bridges a gap between computational feasibility and intrinsic matrix properties.

Future Directions

This research potentially reshapes the landscape of matrix completion, especially in settings prevalent in machine learning, signal processing, and data science, where data is prone to sparsity and incomplete observations. The abolition of the joint incoherence prerequisite broadens the spectrum of applicable matrices and encourages the development of more inclusive algorithms that leverage the efficiency gains discussed herein.

For future research directions, the application of the $\ell_{\infty,2}$ norm could inspire new algorithms across a variety of matrix-based applications outside basic completion, including error correction in corrupted data contexts or more efficient recommendations with sparse matrices. Additionally, the interplay between statistical and computational constraints could foster further understanding of limits imposed by real-world data structures. Overall, the paper contributes significantly to both the theoretical knowledge and practical execution of matrix completion tasks.