Learning finite-dimensional coding schemes with nonlinear reconstruction maps
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This paper generalizes the Maurer--Pontil framework of finite-dimensional lossy coding schemes to the setting where a high-dimensional random vector is mapped to an element of a compact set in a lower-dimensional Euclidean space, and the reconstruction map belongs to a given class of nonlinear maps. We establish a connection to approximate generative modeling under structural constraints using the tools from the theory of optimal transportation. Next, we consider the problem of learning a coding scheme on the basis of a finite collection of training samples and present generalization bounds that hold with high probability. We then illustrate the general theory in the setting where the reconstruction maps are implemented by deep neural nets.
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