Entropy and Learning of Lipschitz Functions under Log-Concave Measures
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916947500728320 |
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| author | Bizeul, Pierre Klartag, Boaz |
| author_facet | Bizeul, Pierre Klartag, Boaz |
| contents | We study regression of $1$-Lipschitz functions under a log-concave measure $μ$ on $\mathbb{R}^d$. We focus on the high-dimensional regime where the sample size $n$ is subexponential in $d$, in which distribution-free estimators are ineffective. We analyze two polynomial-based procedures: the projection estimator, which relies on knowledge of an orthogonal polynomial basis of $μ$, and the least-squares estimator over low-degree polynomials, which requires no knowledge of $μ$ whatsoever. Their risk is governed by the rate of polynomial approximation of Lipschitz functions in $L^2(μ)$. When this rate matches the Gaussian one, we show that both estimators achieve minimax bounds over a wide range of parameters. A key ingredient is sharp entropy estimates for the class of $1$-Lipschitz functions in $L^2(μ)$, which are new even in the Gaussian setting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_10355 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Entropy and Learning of Lipschitz Functions under Log-Concave Measures Bizeul, Pierre Klartag, Boaz Probability Functional Analysis We study regression of $1$-Lipschitz functions under a log-concave measure $μ$ on $\mathbb{R}^d$. We focus on the high-dimensional regime where the sample size $n$ is subexponential in $d$, in which distribution-free estimators are ineffective. We analyze two polynomial-based procedures: the projection estimator, which relies on knowledge of an orthogonal polynomial basis of $μ$, and the least-squares estimator over low-degree polynomials, which requires no knowledge of $μ$ whatsoever. Their risk is governed by the rate of polynomial approximation of Lipschitz functions in $L^2(μ)$. When this rate matches the Gaussian one, we show that both estimators achieve minimax bounds over a wide range of parameters. A key ingredient is sharp entropy estimates for the class of $1$-Lipschitz functions in $L^2(μ)$, which are new even in the Gaussian setting. |
| title | Entropy and Learning of Lipschitz Functions under Log-Concave Measures |
| topic | Probability Functional Analysis |
| url | https://arxiv.org/abs/2509.10355 |