Projecting to Manifolds via Unsupervised Learning

We present a new mechanism, called adversarial projection, that projects a given signal onto the intrinsically low dimensional manifold of true data. This operator can be used for solving inverse problems, which consists of recovering a signal from a collection of noisy measurements. Rather than attempt to encode prior knowledge via an analytic regularizer, we leverage available data to project signals directly onto the (possibly nonlinear) manifold of true data (i.e., regularize via an indicator function of the manifold). Our approach avoids the difficult task of forming a direct representation of the manifold. Instead, we directly learn the projection operator by solving a sequence of unsupervised learning problems, and we prove our method converges in probability to the desired projection. This operator can then be directly incorporated into optimization algorithms in the same manner as Plug-and-Play methods, but now with robust theoretical guarantees. Numerical examples are provided.