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We consider the problem of Gaussian mixture clustering in the high-dimensional limit where the data consists of $m$ points in $n$ dimensions, $n,m rightarrow infty$ and $alpha = m/n$ stays finite. Using exact but non-rigorous methods from statistical physics, we determine the critical value of $alpha$ and the distance between the clusters at which it becomes information-theoretically possible to reconstruct the membership into clusters better than chance. We also determine the accuracy achievable by the Bayes-optimal estimation algorithm. In particular, we find that when the number of clusters is sufficiently large, $r > 4 + 2 sqrt{alpha}$, there is a gap between the threshold for information-theoretically optimal performance and the threshold at which known algorithms succeed.
We consider the phase retrieval problem of reconstructing a $n$-dimensional real or complex signal $mathbf{X}^{star}$ from $m$ (possibly noisy) observations $Y_mu = | sum_{i=1}^n Phi_{mu i} X^{star}_i/sqrt{n}|$, for a large class of correlated real a
We present a novel framework exploiting the cascade of phase transitions occurring during a simulated annealing of the Expectation-Maximisation algorithm to cluster datasets with multi-scale structures. Using the weighted local covariance, we can ext
We analyze the connection between minimizers with good generalizing properties and high local entropy regions of a threshold-linear classifier in Gaussian mixtures with the mean squared error loss function. We show that there exist configurations tha
We study the problem of detecting a structured, low-rank signal matrix corrupted with additive Gaussian noise. This includes clustering in a Gaussian mixture model, sparse PCA, and submatrix localization. Each of these problems is conjectured to exhi
Generalised linear models for multi-class classification problems are one of the fundamental building blocks of modern machine learning tasks. In this manuscript, we characterise the learning of a mixture of $K$ Gaussians with generic means and covar