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arXiv · 2601.21717

On sample complexity for covariance estimation via the unadjusted Langevin algorithm

Shogo Nakakita
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Abstract

We establish sample complexity guarantees for estimating the covariance matrix of a strongly log-concave smooth distribution using the unadjusted Langevin algorithm (ULA). We quantitatively compare our complexity estimates on single-chain ULA with embarrassingly parallel ULA and derive that the sample complexity of the single-chain approach is smaller than that of embarrassingly parallel ULA by a logarithmic factor in the dimension and the reciprocal of the prescribed precision, with the difference arising from effective bias reduction through burn-in. The key technical contribution is a concentration bound for the sample covariance matrix around its expectation, derived via a log-Sobolev inequality for the joint distribution of ULA iterates.

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Shogo Nakakita. 2026-01-29. On sample complexity for covariance estimation via the unadjusted Langevin algorithm. https://arxiv.org/abs/2601.21717

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