A Fourier analytique approach to Gaussian mixture learning
Suppose that we are given independent, identically distributed random samples $x_1,\cdots,x_n$ from a mixture at most $k$ many $d$-dimensional spherical Gaussian distributions $μ_1,\cdots,μ_{k_0}$ of identical and known variance $σ^2$ in each coordinate, such that the minimum $\ell^2$ distance between two distinct centers $y_l$ and $y_j$ is greater than $2Δσ\min\{\sqrt{d},\sqrt k\}$, where $Δ>C_0$, and $C_0$ is a sufficiently large universal constant. We develop a randomized algorithm that learns the centers $y_l$'s of the Gaussian components to within an $\ell^2$ distance of $k^{-\tilde C_0}$ -- in presence of arbitrarily large number of components and in arbitrary dimension, when the weights are known to be uniform. Furthermore, if the number of components is $k= Ω(2^d)$, then for arbitrary universal constant $c>0$, even for unknown weights, the algorithm learns the centers to within an $\ell^2$ distance of $d^{-\tilde C_0}$ and the weights up to an accuracy of $cw_{min}$, with probability greater than $1 - \exp(-k/c)$, provided that the weights lie in $[c/k,1/ck]$, and the minimum separation is just $2c\sqrt d$. The number of samples and the computational time is bounded above by $\mathrm{poly}(k, d)$ in either case. Such a bound on the sample and computational complexity was previously unknown in the regime of non-constant dimension, and in particular, when $d$ is not $O(1)$. When $d = O(1)$, this complexity bound follows from work of Regev and Vijayaraghavan, where it has also been shown that the sample complexity of learning a random mixture of Gaussians in a ball of radius $o(\sqrt{d})$ in $d$ dimensions, when $d$ is $Θ( \log k)$, is at least super-polynomial in $k, d$, showing that our result is tight in this case.