Sample Complexity Bounds for Learning High-dimensional Simplices in Noisy Regimes

• URL: http://arxiv.org/abs/2209.05953v2
• Date: Sat, 29 Apr 2023 02:17:39 GMT
• Title: Sample Complexity Bounds for Learning High-dimensional Simplices in Noisy Regimes
• Authors: Amir Hossein Saberi, Amir Najafi, Seyed Abolfazl Motahari and Babak H. Khalaj
• Abstract summary: We find a sample complexity bound for learning a simplex from noisy samples. We show that as long as $mathrmSNRgeOmegaleft(K1/2right)$, the sample complexity of the noisy regime has the same order to that of the noiseless case.
• Score: 5.526935605535376
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size $n$ is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in $\mathbb{R}^K$, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a $\ell_2$ distance of at most $\varepsilon$ from the true simplex (for any $\varepsilon>0$). Also, we theoretically show that in order to achieve this bound, it is sufficient to have $n\ge\left(K^2/\varepsilon^2\right)e^{\Omega\left(K/\mathrm{SNR}^2\right)}$ samples, where $\mathrm{SNR}$ stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as $\mathrm{SNR}\ge\Omega\left(K^{1/2}\right)$, the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in \citep{ashtiani2018nearly}, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.

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