The intent of this webinar is to demonstrate the optimality of splines for the resolution of inverse problems in imaging and the design of deep neural networks. To that end, I first present a representer theorem that states that the extremal points of a broad class of linear inverse problems with a generalized total-variation constraint are adaptive splines whose type is linked to the underlying regularization operator. When the latter is the second-order derivative operator, then the optimal reconstruction is an adaptive piecewise-linear spline with the smallest possible number of breakpoints. These splines are intrinsically sparse, and hence compatible with the kind of formulation (and algorithms) used in compressed sensing. I then show that these results are relevant to the investigation of neural networks as well. In particular, they yield a functional interpretation of shallow, infinite-width ReLU neural nets. Sparse adaptive splines also turn out to be ideally suited for the specification of deep neural networks with free-form activations.
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