Fast Mean Estimation with Sub-Gaussian Rates
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We propose an estimator for the mean of a random vector in R^d that can be computed in time O(n^4+n^2d) for n i.i.d. samples and that has error bounds matching the sub-Gaussian case. The only assumptions we make about the data distribution are that it has finite mean and covariance; in particular, we make no assumptions about higher-order moments. Like the polynomial time estimator introduced by Hopkins, 2018, which is based on the sum-of-squares hierarchy, our estimator achieves optimal statistical efficiency in this challenging setting, but it has a significantly faster runtime and a simpler analysis.
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