Fixed Point and Bregman Iterative Methods for Matrix Rank Minimization (0905.1643v2)

Abstract: The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast as a semidefinite programming problem, such an approach is computationally expensive to solve when the matrices are large. In this paper, we propose fixed point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 x 1000 matrices of rank 50 with a relative error of 1e-5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms.

• The paper presents fixed point and Bregman iterative methods to efficiently address matrix rank minimization via nuclear norm minimization.
• It integrates an approximate SVD technique with a homotopy approach to reduce computational complexity and enhance convergence.
• Numerical results demonstrate that FPCA significantly outperforms SDP solvers, recovering large-scale low-rank matrices with high accuracy.

Fixed Point and Bregman Iterative Methods for Matrix Rank Minimization

In the paper titled "Fixed Point and Bregman Iterative Methods for Matrix Rank Minimization," authors Shiqian Ma, Donald Goldfarb, and Lifeng Chen address the computational complexities involved in matrix rank minimization problems. This problem has wide applicability, including control systems, signal processing, and system identification. The authors develop and analyze algorithms that offer more efficient solutions compared to traditional semidefinite programming (SDP) approaches.

Problem Formulation

The linearly constrained matrix rank minimization problem is formally expressed as: $$\min \quad \text{rank}(X) \quad \text{s.t.} \quad A(X) = b,$$ where $X$ is a matrix variable, $A$ is a linear map, and $b$ is a given vector. Given the NP-hard nature of rank minimization, the authors instead consider its convex relaxation: $$\min \quad \|X\|_* \quad \text{s.t.} \quad A(X) = b,$$ where $\|X\|_*$ denotes the nuclear norm of $X$. Though this problem can be formulated as an SDP, such approaches are computationally prohibitive for large matrices.

Proposed Algorithms

The authors propose fixed point and Bregman iterative algorithms to solve nuclear norm minimization problems efficiently. They introduce the Fixed Point Continuation with Approximate SVD (FPCA) algorithm, which integrates a homotopy approach to accelerate convergence and employs a Monte Carlo-based approximate singular value decomposition (SVD) technique to reduce computational costs.

Fixed Point Iterative Algorithm

The fixed point iterative algorithm is designed as follows:

1. Iteration Update:

   $Y^k = X^k - \tau g(X^k)$

   $X^{k+1} = S_{\tau \mu}(Y^k),$

where $S_\nu$ denotes the matrix shrinkage operator.

1. Convergence Proof: The authors prove that this iterative scheme converges to an optimal solution, leveraging the non-expansive property of the shrinkage operator.

Bregman Iterative Algorithm

For enhanced performance, particularly in matrix completion, the authors extend the fixed point algorithm using Bregman iterative regularization. This modification involves iteratively refining the problem:

1. Subproblem Solution: Solve

   $$X^{k+1} \leftarrow \arg \min_X \mu \|X\|_* + \frac{1}{2}\|A(X)-b^{k+1}\|_2^2$$

   using the fixed point algorithm.

2. Update: Modify $b$ at each step to improve recovery.

Numerical Results

The FPCA algorithm shows substantial computational benefits. Notably, the algorithm outperforms traditional SDP solvers, such as SDPT3, under various problem sizes and ranks. For example, FPCA can recover a $1000 \times 1000$ matrix of rank 50 from only 20% of its elements in approximately 3 minutes, yielding a relative error of $10^{-5}$.

Comparative Analysis

The performance of FPCA is also compared against the Singular Value Thresholding (SVT) algorithm. Results indicate that FPCA is faster and yields better recoverability, especially for challenging low-rank matrix completion problems.

Application on Real Data

To validate the practicality of their methods, the authors apply FPCA to real-world data sets, including online recommendation systems and DNA microarray data. The matrices in these applications are typically large and sparse, making traditional methods infeasible. FPCA, however, demonstrates effective recovery of the underlying low-rank structure.

Conclusion and Future Implications

The proposed fixed point and Bregman iterative algorithms for matrix rank minimization are effective alternatives to conventional SDP solvers. They offer notable reductions in computational cost and improve recoverability of low-rank matrices. The theoretical underpinnings and numerical results presented promise significant advancements in fields requiring high-dimensional data completion and compression.

Future developments could focus on further refinement of the approximate SVD procedures for increased efficiency and robustness. Extending these methodologies to incorporate adaptive rank estimation techniques might provide insights into dynamic rank determination during iterations.

Overall, the contributions of this paper offer substantial advancements in optimizing matrix rank minimization problems, presenting broad implications for artificial intelligence and data science, where efficient handling of large-scale matrix data is critical.