I like it, from the paper:

In particular, we shall analyze the algorithm using tools from spin glass theory in statistical mechanics, and show that there exists a phase transition between a phase where it is able to detect the rank, and a phase where it is unable to do so.

Matrix Completion from Fewer Entries: Spectral Detectability and Rank Estimation by Alaa Saade, Florent Krzakala, Lenka Zdeborová

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.

Implementations are are available in matlab http://github.com/alaa-saade/macbeth_matlab and in Julia http://github.com/alaa-saade/macbeth_julia. They are also on Alaa's webpage http://alaasaade.wordpress.com/ and on the SPHINX group software page: http://www.lps.ens.fr/~krzakala/WASP.html

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