Hopkins, Samuel B. Mean estimation with sub-Gaussian rates in polynomial time. (English) Zbl 1454.62162 Ann. Stat. 48, No. 2, 1193-1213 (2020). The author studies polynomial time algorithms for estimating the mean of a heavy-tailed multivariate random vector. The only assumption is that the random vector \(X\) has finite mean and covariance. In this setting, the radius of confidence intervals achieved by the empirical mean are large compared to the case that \(X\) is Gaussian or sub-Gaussian. A polynomial time algorithm is proposed to estimate the mean with sub-Gaussian-size confidence intervals under these assumptions. The algorithm is based on a new semidefinite programming relaxation of a high-dimensional median. Previous estimators which assumed only existence of finitely many moments of \(X\) either sacrifice sub-Gaussian performance or are only known to be computable via brute-force search procedures requiring time exponential in the dimension. The algorithm runs in polynomial time, but it is not close to practical for any substantially high-dimensional data set. Reviewer: Pavel Stoynov (Sofia) Cited in 23 Documents MSC: 62H12 Estimation in multivariate analysis 62G32 Statistics of extreme values; tail inference 68W01 General topics in the theory of algorithms 90C22 Semidefinite programming Keywords:multivariate estimation; heavy tails; confidence intervals; sub-Gaussian rates; semidefinite programming; sum of squares method × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid References: [1] Abbe, E., Bandeira, A. S. and Hall, G. (2016). Exact recovery in the stochastic block model. IEEE Trans. Inform. Theory 62 471-487. · Zbl 1359.94047 · doi:10.1109/TIT.2015.2490670 [2] Alon, N., Matias, Y. and Szegedy, M. (1999). The space complexity of approximating the frequency moments. J. Comput. System Sci. 58 137-147. · Zbl 0938.68153 · doi:10.1006/jcss.1997.1545 [3] Alon, N. and Naor, A. (2006). 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