Solving linear inverse problems using generative models

Shirin Jalali, Xin Yuan

Abstract: Compressed sensing (CS) algorithms recover a signal from its under-determined linear measurements via exploiting its structure. Starting from sparsity, recovery methods have steadily moved towards more complex structures. Emerging machine learning tools, e.g., generative models that are based on neural nets, potentially learn general complex structures from training data. Inspired by the success of such models in various computer vision tasks, researchers in CS have recently started to employ them to design efficient recovery methods. Consider a generative model defined by function g : U <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> → R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , where U denotes a bounded subset of R. Assume that the function g is trained such that it can describe the class of desired signals Q c Rn. The standard problem in noiseless CS is to recover x ∈ Q from under-determined linear measurements y = Ax, where y ∈ R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> and m <;<; n. A recovery method based on g finds g(u), u ∈ υν, which has the minimum measurement error. In this paper, the performance of such a recovery method is studied and for an L-Lipschitz function g, it is proven that if the number of measurements (m) is larger than twice the dimension of the generative model (k), then x can be recovered from y, with a distortion that is a function of the distortion induced by g in describing x, i.e., min <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u∈Uk</sub> ||g(u) - x||. Finally, using projected gradient descent to solve the aforementioned optimization, some preliminary numerical results are reported.