Using auto-encoders for solving ill-posed linear inverse problems
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Compressed sensing algorithms recover a signal from its under-determined linear measurements by exploiting its structure. Starting from sparsity, recovery methods have steadily moved towards more complex structures. Recently, the emerging machine learning techniques, especially the generative models based on neural nets, potentially, can learn general complex structures. Inspired by the success of such models in various computer vision tasks, researchers in compressed sensing have recently started to employ them to design efficient recovery methods. Consider a generative model defined as function g: U^k→R^n, U⊂R. Assume that the function g is trained such that it can describe a class of desired signals Q⊂R^n. The standard problem in noiseless compressed sensing is to recover x∈ Q from under-determined linear measurements y=A x, where y∈R^m and m≪ n. A recovery method based on g finds g( u), ∈ U^k, which has the minimum measurement error. In this paper, the performance of such a recovery method is studied and 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. _ u∈ U^kg( u)- x. To derive an efficient method, an algorithm based on projected gradient descent is proposed. It is proven that, given enough measurements, the algorithm converges to the optimal solution and is robust to measurement noise. Numerical results showing the effectiveness of the proposed method are presented.
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