Solving inverse problems via auto-encoders

Compressed sensing (CS) is about recovering a structured signal from its under-determined linear measurements. Starting from sparsity, recovery methods have steadily moved towards more complex structures. Emerging machine learning tools such as generative functions that are based on neural networks are able to learn general complex structures from training data. This makes them potentially powerful tools for designing CS algorithms. Consider a desired class of signals $\cal Q$, ${\cal Q}\subset{R}^n$, and a corresponding generative function $g:{\cal U}^k\rightarrow {R}^n$, ${\cal U}\subset {R}$, such that $\sup_{{\bf x}\in {\cal Q}}\min_{{\bf u}\in{\cal U}^k}{1\over \sqrt{n}}\|g({\bf u})-{\bf x}\|\leq δ$. A recovery method based on $g$ seeks $g({\bf u})$ with minimum measurement error. In this paper, the performance of such a recovery method is studied, under both noisy and noiseless measurements. In the noiseless case, roughly speaking, it is proven that, as $k$ and $n$ grow without bound and $δ$ converges to zero, if the number of measurements ($m$) is larger than the input dimension of the generative model ($k$), then asymptotically, almost lossless recovery is possible. Furthermore, the performance of an efficient iterative algorithm based on projected gradient descent is studied. In this case, an auto-encoder is used to define and enforce the source structure at the projection step. The auto-encoder is defined by encoder and decoder (generative) functions $f:{R}^n\to{\cal U}^k$ and $g:{\cal U}^k\to{R}^n$, respectively. We theoretically prove that, roughly, given $m>40k\log{1\over δ}$ measurements, such an algorithm converges to the vicinity of the desired result, even in the presence of additive white Gaussian noise. Numerical results exploring the effectiveness of the proposed method are presented.