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A simple model to study subspace clustering is the high-dimensional$k$-Gaussian mixture model where the cluster means are sparse vectors. Here weprovide an exact asymptotic characterization of the statistically optimalreconstruction error in this model in the high-dimensional regime withextensive sparsity, i.e. when the fraction of non-zero components of thecluster means $\rho$, as well as the ratio $\alpha$ between the number ofsamples and the dimension are fixed, while the dimension diverges. We identifythe information-theoretic threshold below which obtaining a positivecorrelation with the true cluster means is statistically impossible.Additionally, we investigate the performance of the approximate message passing(AMP) algorithm analyzed via its state evolution, which is conjectured to beoptimal among polynomial algorithm for this task. We identify in particular theexistence of a statistical-to-computational gap between the algorithm thatrequire a signal-to-noise ratio $\lambda_{\text{alg}} \ge k / \sqrt{\alpha} $to perform better than random, and the information theoretic threshold at$\lambda_{\text{it}} \approx \sqrt{-k \rho \log{\rho}} / \sqrt{\alpha}$.Finally, we discuss the case of sub-extensive sparsity $\rho$ by comparing theperformance of the AMP with other sparsity-enhancing algorithms, such assparse-PCA and diagonal thresholding.

Subspace clustering in high-dimensions: Phase transitions & Statistical-to-Computational gap