Recent works have shown that MFLD globally minimizes an entropy-regularized convex functional in the space of measures.

Stochastic processes are random variables with values in some space of paths. However, reducing a stochastic process to a path-valued random variable ignores its filtration, i.e. the flow of information carried by the process through time. By conditioning the process on its filtration, we introduce a family of higher order kernel mean embeddings (KMEs) that generalizes the notion of KME and captures additional information related to the filtration. We derive empirical estimators for the associated higher order maximum mean discrepancies (MMDs) and prove consistency. We then construct a filtration-sensitive kernel two-sample test able to pick up information that gets missed by the standard MMD test. In addition, leveraging our higher order MMDs we construct a family of universal kernels on stochastic processes that allows to solve real-world calibration and optimal stopping problems in quantitative finance (such as the pricing of American options) via classical kernel-based regression methods. Finally, adapting existing tests for conditional independence to the case of stochastic processes, we design a causaldiscovery algorithm to recover the causal graph of structural dependencies among interacting bodies solely from observations of their multidimensional trajectories.

Towardsthis end,wedesigned apreconditioned gradient solverforkernelmethods exploiting both GPU acceleration and parallelization with multiple GPUs, implementing out-of-core variants of common linear algebra operations to guarantee optimal hardware utilization.

Finally, we extend Min-AM to solve stochastic programming problems.

Paul Erdős was one of the most prolific mathematicians to ever live, known for showing up at the door of others in the field and declaring they should host and feed him while they do maths together. I come to you with something a little different for my latest maths column - a plea to Hollywood to make a comedy biopic about one of the greatest mathematicians of all time, Paul Erdős. Why is Erdős (pronounced "air-dish") deserving of such acclaim? With almost 1500 papers to his name, he is probably the most prolific mathematician that ever lived, and possibly that will ever live. Unsurprisingly, with that many papers, he is known for his work across many areas of maths, from probability to number theory to graph theory.