Uniqueness of Low-Rank Matrix Completion by Rigidity Theory

The problem of completing a low-rank matrix from a subset of its entries is often encountered in the analysis of incomplete data sets exhibiting an underlying factor model with applications in collaborative filtering, computer vision and control. Most recent work had been focused on constructing efficient algorithms for exact or approximate recovery of the missing matrix entries and proving lower bounds for the number of known entries that guarantee a successful recovery with high probability. A related problem from both the mathematical and algorithmic point of view is the distance geometry problem of realizing points in a Euclidean space from a given subset of their pairwise distances. Rigidity theory answers basic questions regarding the uniqueness of the realization satisfying a given partial set of distances. We observe that basic ideas and tools of rigidity theory can be adapted to determine uniqueness of low-rank matrix completion, where inner products play the role that distanc...

• Type: Journal
• Year: 2010
• Where: SIAMMAX
• Authors: Amit Singer, Mihai Cucuringu
• Keywords: Low-rank Matrix, Matrix, Rigidity Theory, SIAMMAX 2010