Computation-information gap in high-dimensional clustering - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2024

Computation-information gap in high-dimensional clustering

Résumé

We investigate the existence of a fundamental computation-information gap for the problem of clustering a mixture of isotropic Gaussian in the high-dimensional regime, where the ambient dimension $p$ is larger than the number $n$ of points. The existence of a computation-information gap in a specific Bayesian high-dimensional asymptotic regime has been conjectured by arXiv:1610.02918 based on the replica heuristic from statistical physics. We provide evidence of the existence of such a gap generically in the high-dimensional regime $p \geq n$, by (i) proving a non-asymptotic low-degree polynomials computational barrier for clustering in high-dimension, matching the performance of the best known polynomial time algorithms, and by (ii) establishing that the information barrier for clustering is smaller than the computational barrier, when the number $K$ of clusters is large enough. These results are in contrast with the (moderately) low-dimensional regime $n \geq poly(p, K)$, where there is no computation-information gap for clustering a mixture of isotropic Gaussian. In order to prove our low-degree computational barrier, we develop sophisticated combinatorial arguments to upper-bound the mixed moments of the signal under a Bernoulli Bayesian model.
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Dates et versions

hal-04483306 , version 1 (29-02-2024)

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Bertrand Even, Christophe Giraud, Nicolas Verzelen. Computation-information gap in high-dimensional clustering. 2024. ⟨hal-04483306⟩
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