Sample Complexity Bounds for Learning High-dimensional Simplices in Noisy Regimes
This solves an important open problem in high-dimensional statistics, with potential applications in machine learning and data analysis where simplex models are used.
The paper tackles the problem of learning a high-dimensional simplex from noisy samples, proving that an algorithm can output a simplex within ε ℓ₂ distance of the true one with high probability, requiring n ≥ (K²/ε²)e^{Ω(K/SNR²)} samples, and showing that sample complexity matches the noiseless case when SNR ≥ Ω(K^{1/2}).
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 $n\ge\left(K^2/\varepsilon^2\right)e^{Ω\left(K/\mathrm{SNR}^2\right)}$ samples, where $\mathrm{SNR}$ stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as $\mathrm{SNR}\geΩ\left(K^{1/2}\right)$, 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.