A Note on Alternating Minimization Algorithm for the Matrix Completion Problem
This provides theoretical guarantees for matrix completion algorithms in a specific constrained setting, but is incremental as it builds on existing Alternating Minimization methods.
The authors analyzed two variants of the Alternating Minimization algorithm for reconstructing low-rank matrices from partial entries, proving that both succeed for rank-1 matrices with specific conditions in polynomial time. Simulation results showed the message-passing variant performs significantly better.
We consider the problem of reconstructing a low rank matrix from a subset of its entries and analyze two variants of the so-called Alternating Minimization algorithm, which has been proposed in the past. We establish that when the underlying matrix has rank $r=1$, has positive bounded entries, and the graph $\mathcal{G}$ underlying the revealed entries has bounded degree and diameter which is at most logarithmic in the size of the matrix, both algorithms succeed in reconstructing the matrix approximately in polynomial time starting from an arbitrary initialization. We further provide simulation results which suggest that the second algorithm which is based on the message passing type updates, performs significantly better.