Learning Entangled Single-Sample Gaussians in the Subset-of-Signals Model
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In the setting of entangled single-sample distributions, the goal is to estimate some common parameter shared by a family of n distributions, given one single sample from each distribution. This paper studies mean estimation for entangled single-sample Gaussians that have a common mean but different unknown variances. We propose the subset-of-signals model where an unknown subset of m variances are bounded by 1 while there are no assumptions on the other variances. In this model, we analyze a simple and natural method based on iteratively averaging the truncated samples, and show that the method achieves error O (√(nln n)/m) with high probability when m=Ω(√(nln n)), matching existing bounds for this range of m. We further prove lower bounds, showing that the error is Ω((n/m^4)^1/2) when m is between Ω(ln n) and O(n^1/4), and the error is Ω((n/m^4)^1/6) when m is between Ω(n^1/4) and O(n^1 - ϵ) for an arbitrarily small ϵ>0, improving existing lower bounds and extending to a wider range of m.
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