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Deep generative modeling has led to new and state of the art approaches forenforcing structural priors in a variety of inverse problems. In contrast topriors given by sparsity, deep models can provide direct low-dimensionalparameterizations of the manifold of images or signals belonging to aparticular natural class, allowing for recovery algorithms to be posed in alow-dimensional space. This dimensionality may even be lower than the sparsitylevel of the same signals when viewed in a fixed basis. What is not known aboutthese methods is whether there are computationally efficient algorithms whosesample complexity is optimal in the dimensionality of the representation givenby the generative model. In this paper, we present such an algorithm andanalysis. Under the assumption that the generative model is a neural networkthat is sufficiently expansive at each layer and has Gaussian weights, weprovide a gradient descent scheme and prove that for noisy compressivemeasurements of a signal in the range of the model, the algorithm converges tothat signal, up to the noise level. The scaling of the sample complexity withrespect to the input dimensionality of the generative prior is linear, and thuscan not be improved except for constants and factors of other variables. To thebest of the authors' knowledge, this is the first recovery guarantee forcompressive sensing under generative priors by a computationally efficientalgorithm.

A Provably Convergent Scheme for Compressive Sensing under Random Generative Priors