We study the Sparse Plus Low-Rank decomposition problem (SLR), which is the problem of decomposing a corrupted data matrix into a sparse matrix of perturbations plus a low-rank matrix containing the ground truth. SLR is a fundamental problem in Operations Research and Machine Learning which arises in various applications, including data compression, latent semantic indexing, collaborative filtering, and medical imaging. We introduce a novel formulation for SLR that directly models its underlying discreteness. For this formulation, we develop an alternating minimization heuristic that computes high-quality solutions and a novel semidefinite relaxation that provides meaningful bounds for the solutions returned by our heuristic. We also develop a custom branch-and-bound algorithm that leverages our heuristic and convex relaxations to solve small instances of SLR to certifiable (near) optimality. Given an input $n$-by-$n$ matrix, our heuristic scales to solve instances where $n=10000$ in minutes, our relaxation scales to instances where $n=200$ in hours, and our branch-and-bound algorithm scales to instances where $n=25$ in minutes. Our numerical results demonstrate that our approach outperforms existing state-of-the-art approaches in terms of rank, sparsity, and mean-square error while maintaining a comparable runtime.

翻译：我们研究了稀疏加低秩分解问题（SLR），该问题旨在将受污染的数据矩阵分解为一个稀疏的扰动矩阵与一个包含真实信息的低秩矩阵之和。SLR是运筹学与机器学习中的基础问题，广泛应用于数据压缩、潜在语义索引、协同过滤和医学成像等领域。我们提出了一种新型SLR建模方法，直接刻画了其潜在的离散性。针对该模型，我们开发了交替最小化启发式算法以计算高质量解，并引入了一种新型半定松弛方法，为启发式算法返回的解提供有意义的界限。此外，我们设计了定制的分支定界算法，结合启发式与凸松弛方法，可在可验证的（近似）最优性下求解小规模SLR实例。给定一个$n \times n$矩阵，我们的启发式算法可在数分钟内求解$n=10000$的实例，松弛方法可在数小时内求解$n=200$的实例，分支定界算法可在数分钟内求解$n=25$的实例。数值结果表明，与现有最先进方法相比，我们的方法在秩、稀疏性和均方误差指标上表现更优，同时保持了可比的运行时间。