Fundamental Limits of Learning High-dimensional Simplices in Noisy Regimes

• URL: http://arxiv.org/abs/2506.10101v1
• Date: Wed, 11 Jun 2025 18:35:38 GMT
• Title: Fundamental Limits of Learning High-dimensional Simplices in Noisy Regimes
• Authors: Seyed Amir Hossein Saberi, Amir Najafi, Abolfazl Motahari, Babak H. khalaj,
• Abstract summary: We prove an algorithm exists that, with high probability, outputs a simplex within $ell$ or total variation (TV) distance at most $varepsilon$ from the true simplex.<n>In the noiseless scenario, our lower bound $n ge Omega(K/varepsilon)$ matches known upper bounds up to constant factors.
• Score: 0.5892638927736114
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we establish sample complexity bounds for learning high-dimensional simplices in $\mathbb{R}^K$ from noisy data. Specifically, we consider $n$ i.i.d. samples uniformly drawn from an unknown simplex in $\mathbb{R}^K$, each corrupted by additive Gaussian noise of unknown variance. We prove an algorithm exists that, with high probability, outputs a simplex within $\ell_2$ or total variation (TV) distance at most $\varepsilon$ from the true simplex, provided $n \ge (K^2/\varepsilon^2) e^{\mathcal{O}(K/\mathrm{SNR}^2)}$, where $\mathrm{SNR}$ is the signal-to-noise ratio. Extending our prior work~\citep{saberi2023sample}, we derive new information-theoretic lower bounds, showing that simplex estimation within TV distance $\varepsilon$ requires at least $n \ge \Omega(K^3 \sigma^2/\varepsilon^2 + K/\varepsilon)$ samples, where $\sigma^2$ denotes the noise variance. In the noiseless scenario, our lower bound $n \ge \Omega(K/\varepsilon)$ matches known upper bounds up to constant factors. We resolve an open question by demonstrating that when $\mathrm{SNR} \ge \Omega(K^{1/2})$, noisy-case complexity aligns with the noiseless case. Our analysis leverages sample compression techniques (Ashtiani et al., 2018) and introduces a novel Fourier-based method for recovering distributions from noisy observations, potentially applicable beyond simplex learning.

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