Near-optimal sample complexity bounds for learning mixtures of Gaussians
Estimating distributions from observed data is a fundamental task in statistics that has been studied for over a century. We consider such a problem where the distribution is a mixture of k Gaussians in R^d. The objective is density estimation: given i.i.d. samples from the (unknown) distribution, produce a distribution whose total variation distance from the unknown distribution is at most epsilon. We prove that Theta(kd^2/epsilon^2) samples are necessary and sufficient for this task, suppressing logarithmic factors. This improves both the known upper bound and lower bound for this problem.