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The goal of compressed sensing is to estimate a high dimensional vector from an underdetermined system of noisy linear equations. In analogy to classical compressed sensing, here we assume a generative model as a prior, that is, we assume the vector is represented by a deep generative model G:Rk→Rn. Classical recovery approaches such as empirical risk minimization (ERM) are guaranteed to succeed when the measurement matrix is subGaussian. However, when the measurement matrix and measurements are heavytailed or have outliers, recovery may fail dramatically. In this paper we propose an algorithm inspired by the MedianofMeans (MOM). Our algorithm guarantees recovery for heavytailed data, even in the presence of outliers. Theoretically, our results show our novel MOMbased algorithm enjoys the same sample complexity guarantees as ERM under subGaussian assumptions. Our experiments validate both aspects of our claims: other algorithms are indeed fragile and fail under heavytailed and/or corrupted data, while our approach exhibits the predicted robustness.

The CSGM framework (BoraJalalPriceDimakis'17) has shown that deepgenerative priors can be powerful tools for solving inverse problems.However, to date this framework has been empirically successful only oncertain datasets (for example, human faces and MNIST digits), and itis known to perform poorly on outofdistribution samples. In thispaper, we present the first successful application of the CSGMframework on clinical MRI data. We train a generative prior on brainscans from the fastMRI dataset, and show that posterior sampling viaLangevin dynamics achieves high quality reconstructions. Furthermore,our experiments and theory show that posterior sampling is robust tochanges in the groundtruth distribution and measurement process.Our code and models are available at: \url{https://github.com/utcsilab/csgmmrilangevin}.

We study the problem of inverting a deep generative model with ReLU activations. Inversion corresponds to finding a latent code vector that explains observed measurements as much as possible. In most prior works this is performed by attempting to solve a nonconvex optimization problem involving the generator. In this paper we obtain several novel theoretical results for the inversion problem. We show that for the realizable case, single layer inversion can be performed exactly in polynomial time, by solving a linear program. Further, we show that for multiple layers, inversion is NPhard and the preimage set can be nonconvex. For generative models of arbitrary depth, we show that exact recovery is possible in polynomial time with high probability, if the layers are expanding and the weights are randomly selected. Very recent work analyzed the same problem for gradient descent inversion. Their analysis requires significantly higher expansion (logarithmic in the latent dimension) while our proposed algorithm can provably reconstruct even with constant factor expansion. We also provide provable error bounds for different norms for reconstructing noisy observations. Our empirical validation demonstrates that we obtain better reconstructions when the latent dimension is large.