Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent (1605.07051v2)

Abstract: We address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent scheme. With $O( \mu r^2 \kappa^2 n \max(\mu, \log n))$ random observations of a $n_1 \times n_2$ $\mu$-incoherent matrix of rank $r$ and condition number $\kappa$, where $n = \max(n_1, n_2)$, the algorithm linearly converges to the global optimum with high probability.

• The paper demonstrates that Burer-Monteiro factorization within a nonconvex framework achieves linear convergence for recovering low-rank rectangular matrices.
• The methodology employs spectral initialization followed by projected gradient descent to enforce incoherence and meet sample complexity bounds.
• The convergence analysis offers strong theoretical guarantees for exact recovery and suggests potential to reduce sample requirements to O(nr) observations.

Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent

The paper entitled "Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent," authored by Qinqing Zheng and John Lafferty from the University of Chicago, presents a novel approach to the problem of rectangular matrix completion. This paper extends the application of Burer-Monteiro factorization techniques, coupled with gradient descent, to address the completion of incoherent matrices. The authors provide a convergence analysis showing strong guarantees for their proposed algorithm.

Summary and Approach

The main focus of the paper is to address the challenge of completing a low-rank rectangular matrix, where only a small fraction of entries are observed. This is formulated as a constrained optimization problem over matrices, traditionally considered hard due to nonconvexity. To tackle this, the authors employ a "lifting" strategy whereby the unknown matrix is represented in a higher dimensional space as a positive semidefinite matrix. Through Burer-Monteiro factorization, this matrix is subsequently decomposed into a product of lower-dimensional factors, thereby transforming the problem into a nonconvex optimization framework solvable by gradient descent.

The algorithm proposed by the authors involves initializing with a spectral method and iteratively applying gradient descent with projections to ensure incoherence conditions. A key feature of this approach is its linear convergence rate to the global optimum under a probabilistic model of observations. The authors establish that given appropriate conditions on the number of observations, the algorithm can successfully recover the original matrix.

Key Results and Convergence

The core theoretical contribution of this work is its convergence analysis, which specifies the number of observations required to ensure the recovery of the matrix with high probability. The authors prove that the algorithm converges linearly to the global optimum when offered a sufficient number of observations, expressed as $O( \mu r^2 \kappa^2 n \max(\mu, \log n))$. Here, $\mu$ represents the incoherence parameter, $r$ the rank of the matrix, $\kappa$ the condition number, and $n$ is the maximum of the matrix dimensions.

Implications and Future Work

This research carries substantial implications for large-scale data analysis contexts that rely on matrix completion, such as collaborative filtering, computer vision, and signal processing. The ability to efficiently recover matrices from incomplete data holds promise for improved performance in these fields.

On a theoretical level, this paper builds upon recent advances in nonconvex optimization and expands the toolkit available to researchers working on related low-rank matrix recovery problems. The authors conjecture that the sample complexity may be further reduced to $O(nr)$ observations, pointing to potential areas for future research. The effectiveness of lifting techniques in broader contexts is suggested as a promising avenue for exploration.

Conclusion

In conclusion, this paper contributes significantly to the understanding of nonconvex optimization for matrix completion, offering a detailed convergence analysis and a practical algorithmic solution. Its rigorous treatment of sample complexity and guarantees of exact recovery highlight the potential for further application and development within the field of low-rank matrix problems. As the field advances, these insights will likely stimulate the development of even more efficient algorithms and broaden the applicability of such techniques across various domains.