Fundamental Limits of Learning High-dimensional Simplices in Noisy Regimes
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912425625780224 |
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| author | Saberi, Seyed Amir Hossein Najafi, Amir Motahari, Abolfazl khalaj, Babak H. |
| author_facet | Saberi, Seyed Amir Hossein Najafi, Amir Motahari, Abolfazl khalaj, Babak H. |
| contents | 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 $n \ge Ω(K^3 σ^2/\varepsilon^2 + K/\varepsilon)$ samples, where $σ^2$ denotes the noise variance. In the noiseless scenario, our lower bound $n \ge Ω(K/\varepsilon)$ matches known upper bounds up to constant factors. We resolve an open question by demonstrating that when $\mathrm{SNR} \ge Ω(K^{1/2})$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10101 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fundamental Limits of Learning High-dimensional Simplices in Noisy Regimes Saberi, Seyed Amir Hossein Najafi, Amir Motahari, Abolfazl khalaj, Babak H. Machine Learning 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 $n \ge Ω(K^3 σ^2/\varepsilon^2 + K/\varepsilon)$ samples, where $σ^2$ denotes the noise variance. In the noiseless scenario, our lower bound $n \ge Ω(K/\varepsilon)$ matches known upper bounds up to constant factors. We resolve an open question by demonstrating that when $\mathrm{SNR} \ge Ω(K^{1/2})$, 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. |
| title | Fundamental Limits of Learning High-dimensional Simplices in Noisy Regimes |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2506.10101 |