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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.18378 |
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| _version_ | 1866909122810609664 |
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| author | Even, Bertrand Giraud, Christophe Verzelen, Nicolas |
| author_facet | Even, Bertrand Giraud, Christophe Verzelen, Nicolas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18378 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computation-information gap in high-dimensional clustering Even, Bertrand Giraud, Christophe Verzelen, Nicolas Statistics Theory 62H30 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. |
| title | Computation-information gap in high-dimensional clustering |
| topic | Statistics Theory 62H30 |
| url | https://arxiv.org/abs/2402.18378 |