HAL CCSD
Matrix Completion from Fewer Entries: Spectral Detectability and Rank Estimation
Saade, Alaa
Krzakala, Florent
Zdeborová, Lenka
Center for Soft Matter Research [New-York] (CSMR) ; New York University [New York] (NYU) ; NYU System (NYU)-NYU System (NYU)
Sorbonne Université (SU)
Institut de Physique Théorique - UMR CNRS 3681 (IPHT) ; Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)
LOC307RDYSPA
cea-01222302
https://cea.hal.science/cea-01222302
https://cea.hal.science/cea-01222302
2015
ARXIV: 1506.03498
info:eu-repo/semantics/altIdentifier/arxiv/1506.03498
en
[PHYS]Physics [physics]
info:eu-repo/semantics/preprint
Preprints, Working Papers, ...
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.
2015-10-29