Matrix Completion from Fewer Entries: Spectral Detectability and Rank Estimation
This addresses matrix completion for applications needing efficient rank estimation and reconstruction from limited data, representing an incremental improvement with specific gains.
The paper tackles the problem of completing low-rank matrices from few entries by proposing a spectral algorithm (MaCBetH) that estimates rank and minimizes reconstruction error, showing it can detect rank from about r√(nm) entries with constant close to 1 and achieves competitive root-mean-square error compared to other methods.
The completion of low rank matrices from few entries is a task with many practical applications. We consider here two aspects of this problem: detectability, i.e. the ability to estimate the rank $r$ reliably from the fewest possible random entries, and performance in achieving small reconstruction error. We propose a spectral algorithm for these two tasks called MaCBetH (for Matrix Completion with the Bethe Hessian). The rank is estimated as the number of negative eigenvalues of the Bethe Hessian matrix, and the corresponding eigenvectors are used as initial condition for the minimization of the discrepancy between the estimated matrix and the revealed entries. We analyze the performance in a random matrix setting using results from the statistical mechanics of the Hopfield neural network, and show in particular that MaCBetH efficiently detects the rank $r$ of a large $n\times m$ matrix from $C(r)r\sqrt{nm}$ entries, where $C(r)$ is a constant close to $1$. We also evaluate the corresponding root-mean-square error empirically and show that MaCBetH compares favorably to other existing approaches.