Learning Entangled Single-Sample Gaussians in the Subset-of-Signals Model

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 left(frac√nln nmright) with high probability when m=Omega(√nln n), matching existing bounds for this range of m. We further prove lower bounds, showing that the error is Omegaleft(left(fracnm4right)1/2right) when m is between Omega(ln n) and O(n1/4), and the error is Omegaleft(left(fracnm4right)1/6right) when m is between Omega(n1/4) and O(n1 - ε) for an arbitrarily small ε>0, improving existing lower bounds and extending to a wider range of m.