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We prove that $\tilde{\Theta}(k d^2 / \varepsilon^2)$ samples are necessaryand sufficient for learning a mixture of $k$ Gaussians in $\mathbb{R}^d$, up toerror $\varepsilon$ in total variation distance. This improves both the knownupper bounds and lower bounds for this problem. For mixtures of axis-alignedGaussians, we show that $\tilde{O}(k d / \varepsilon^2)$ samples suffice,matching a known lower bound. Moreover, these results hold in theagnostic-learning/robust-estimation setting as well, where the targetdistribution is only approximately a mixture of Gaussians.  The upper bound is shown using a novel technique for distribution learningbased on a notion of `compression.' Any class of distributions that allows sucha compression scheme can also be learned with few samples. Moreover, if a classof distributions has such a compression scheme, then so do the classes ofproducts and mixtures of those distributions. The core of our main result isshowing that the class of Gaussians in $\mathbb{R}^d$ admits a small-sizedcompression scheme.

Near-optimal Sample Complexity Bounds for Robust Learning of Gaussians Mixtures via Compression Schemes