Tailed Low-Rank Matrix Factorization for Similarity Matrix Completion

Similarity matrix serves as a fundamental tool at the core of numerous downstream machine-learning tasks. However, missing data is inevitable and often results in an inaccurate similarity matrix. To address this issue, Similarity Matrix Completion (SMC) methods have been proposed, but they suffer from high computation complexity due to the Singular Value Decomposition (SVD) operation. To reduce the computation complexity, Matrix Factorization (MF) techniques are more explicit and frequently applied to provide a low-rank solution, but the exact low-rank optimal solution can not be guaranteed since it suffers from a non-convex structure. In this paper, we introduce a novel SMC framework that offers a more reliable and efficient solution. Specifically, beyond simply utilizing the unique Positive Semi-definiteness (PSD) property to guide the completion process, our approach further complements a carefully designed rank-minimization regularizer, aiming to achieve an optimal and low-rank solution. Based on the key insights that the underlying PSD property and Low-Rank property improve the SMC performance, we present two novel, scalable, and effective algorithms, SMCNN and SMCNmF, which investigate the PSD property to guide the estimation process and incorporate nonconvex low-rank regularizer to ensure the low-rank solution. Theoretical analysis ensures better estimation performance and convergence speed. Empirical results on real-world datasets demonstrate the superiority and efficiency of our proposed methods compared to various baseline methods.