Adaptive Rank-One Matrix Completion Using Sum of Outer Products

Matrix completion refers to recovering a matrix from a small subset of its entries. It is an important topic because numerous real-world data can be modeled as low-rank matrices. One popular approach for matrix completion is based on low-rank matrix factorization, but it requires knowing the matrix rank, which is difficult to accurately determine in many practical scenarios. We propose a novel algorithm based on rank-one approximation that a matrix can be decomposed as a sum of outer products. The key idea is to find the basis vectors of the underlying matrix according to the observed entries, and gradually increase the vector number until an appropriate rank estimate is reached. In contrast to the conventional rank-one schemes that employ unchanging rank-one basis matrices, our algorithm performs completion from the vector viewpoint and is able to generate continuously updated rank-one basis matrices. Besides, we theoretically show that the developed method has a linear convergence rate and a smaller recovery error than existing rank-one based algorithms. Experimental results using both synthetic data and real-world images demonstrate that our solution has the best recovery performance among the competing algorithms when the observations are contaminated by Gaussian noise.