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We consider the problem of learning high dimensional polynomialtransformations of Gaussians. Given samples of the form $p(x)$, where $x\simN(0, \mathrm{Id}_r)$ is hidden and $p: \mathbb{R}^r \to \mathbb{R}^d$ is afunction where every output coordinate is a low-degree polynomial, the goal isto learn the distribution over $p(x)$. This problem is natural in its ownright, but is also an important special case of learning deep generativemodels, namely pushforwards of Gaussians under two-layer neural networks withpolynomial activations. Understanding the learnability of such generativemodels is crucial to understanding why they perform so well in practice.  Our first main result is a polynomial-time algorithm for learning quadratictransformations of Gaussians in a smoothed setting. Our second main result is apolynomial-time algorithm for learning constant-degree polynomialtransformations of Gaussian in a smoothed setting, when the rank of theassociated tensors is small. In fact our results extend to anyrotation-invariant input distribution, not just Gaussian. These are the firstend-to-end guarantees for learning a pushforward under a neural network withmore than one layer.  Along the way, we also give the first polynomial-time algorithms withprovable guarantees for tensor ring decomposition, a popular generalization oftensor decomposition that is used in practice to implicitly store largetensors.

Learning Polynomial Transformations