A Convergent Gradient Descent Algorithm for Rank Minimization and Semidefinite Programming from Random Linear Measurements

Abstract: We propose a simple, scalable, and fast gradient descent algorithm to optimize a nonconvex objective for the rank minimization problem and a closely related family of semidefinite programs. With $O(r^3 \kappa^2 n \log n)$ random measurements of a positive semidefinite $n \times n$ matrix of rank $r$ and condition number $\kappa$, our method is guaranteed to converge linearly to the global optimum.

• The paper introduces a convergent gradient descent algorithm designed for efficient rank minimization and a subset of semidefinite programming problems under random linear measurements.
• The algorithm proves linear convergence to a global optimum and demonstrates superior runtime efficiency over traditional methods such as nuclear norm and SVP.
• The algorithm provides an efficient solution for large-scale, underdetermined problems with practical applications in image compression, matrix completion, and metric embedding.

A Convergent Gradient Descent Algorithm for Rank Minimization and Semidefinite Programming from Random Linear Measurements

This paper introduces a gradient descent algorithm specifically designed to address rank minimization problems and a related subset of semidefinite programming (SDP) challenges. Emphasizing simplicity and scalability, the authors propose a method that guarantees linear convergence to a global optimum, provided the problem settings meet specific conditions.

The paper begins by framing the significance of semidefinite programming within applied mathematics and machine learning, highlighting the computational limitations of traditional algorithms employing interior point methods. The proposed approach utilizes a first-order gradient descent algorithm tailored to optimize a nonconvex objective function under random measurement conditions. The objective is to solve affine rank minimization problems and related semidefinite programs efficiently.

The core problem tackled involves finding a minimum rank matrix $X^\star$ within affine constraints. Formulated as a nonconvex optimization task, existing solutions have frequently relied on principles like nuclear norm relaxation and SVP, contingent upon certain assumptions (e.g., restricted isometry property). However, these conventional methods require substantial computational resources, typically involving singular value decompositions (SVD) that are burdensome for large-scale datasets.

Here, the authors capitalize on properties derived from phase retrieval and extend these concepts to affine rank minimization, proposing an algorithm that engages gradient descent to minimize a squared residual function. Significant contributions of this study include a proven linear convergence rate and reduced computational overhead compared to existing methods.

The theoretical backbone of the paper emphasizes establishing conditions under which the gradient descent method operates efficiently. The authors introduce concepts such as an initial spectral estimate based on random measurements, and comprehensively address local curvature and smoothness conditions, akin to strong convexity and Lipschitz conditions in classical optimization analysis.

Numerical experiments validate the effectiveness of the method, demonstrating superior runtime performance compared to prominent algorithms like nuclear norm methods and SVP, especially under cases involving sparse measurements. The empirical sample complexity testifies that the proposed algorithm can potentially match the rank-based measurement scaling $O(rn \log n)$, aligning with optimal bounds suggested for SDP problems.

This advancement holds implications for a variety of applications, including image compression, matrix completion, and metric embedding in machine learning. Practically, implementing the algorithm could substantially improve computational efficiency, especially where SDP formulations intersect with large-scale, underdetermined problems.

From a theoretical perspective, the work suggests promising directions for leveraging nonconvex formulations to approximate convex optimizations within machine learning. As demonstrated, refactoring matrix variables and deploying first-order optimization techniques can yield potent algorithms with rigorous performance guarantees. Future investigations may explore weakening the stringent assumptions about measurement matrices and examining broader classes of semidefinite programs.

In summary, this paper presents a constructive step toward bridging theoretical efficacy and practical applicability in SDP and rank minimization problems, establishing a pathway for efficient algorithmic solutions in high-dimensional and sparse contexts.