We construct intrinsic Lipschitz graphs in Carnot groups with the property that, at every point, there exist infinitely many different blow-up limits, none of which is a homogeneous subgroup. This provides counterexamples to Rademacher theorem for graphs.

Machine learning techniques have recently gained prominence in physics, yielding a host of new results and insights. One key concept is that backpropagation, which computes the exact gradient any output program with respect to input. This achieved efficiently within differentiable programming paradigm, utilizes automatic differentiation (AD) each step computer chain rule. A classic application ...

We discuss a general technique that can be used to form a differentiable bound on the optima of non-differentiable or discrete objective functions. We form a unified description of these methods and consider under which circumstances the bound is concave. In particular we consider two concrete applications of the method, namely sparse learning and support vector classification. 1 Optimization b...

We consider Backward Stochastic Differential Equations (BSDEs) with generators that grow quadratically in the control variable. In a more abstract setting, we first allow both the terminal condition and the generator to depend on a vector parameter x. We give sufficient conditions for the solution pair of the BSDE to be differentiable in x. These results can be applied to systems of forward-bac...