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Sample complexity bounds for learning high-dimensional simplices in noisy regimes

Saberi, A. H ; Sharif University of Technology | 2023

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  1. Type of Document: Article
  2. DOI: 10.48550/arXiv.2209.05953
  3. Publisher: ML Research Press , 2023
  4. Abstract:
  5. 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 RK, 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 ℓ2distance of at most ε from the true simplex (for any ε > 0). Also, we theoretically show that in order to achieve this bound, it is sufficient to have n ≥ (K eΩ(K/SNR2) samples, where SNR stands for the signal-to-noise ratio. This result solves an open problem in this area of research and shows as long as SNR ≥ Ω (K, 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 from (Ashtiani et al., 2018), 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. © 2023 Proceedings of Machine Learning Research. All rights reserved
  6. Keywords:
  7. Machine learning
  8. Source: Proceedings of Machine Learning Research ; Volume 202 , 2023 , Pages 29514-29541 ; 26403498 (ISSN)
  9. URL: https://arxiv.org/abs/2209.05953