Joint AMS-ECE Seminar 

Gromov’s method for particle flow filters

Fred Daum

Raytheon

Tuesday, 4/24/18, 2:00pm
Light Engineering 250

Abstract: We derive a new exact stochastic particle flow filter, using a theorem of Gromov. Our filter is many orders of magnitude faster than standard particle filters for high dimensional problems, and our filter beats the extended Kalman filter accuracy by orders of magnitude for difficult nonlinear problems. Our theory uses particle flow to compute Bayes’ rule, rather than a pointwise multiply. We do not use resampling of particles or proposal densities or any MCMC method. But rather, we design the particle flow with the solution of a linear first order highly underdetermined PDE. We solve this PDE as an exact formula which is valid for arbitrary smooth nowhere vanishing densities. Gromov proves that there exists a “nice” solution to a linear constant coefficient PDE for smooth functions if and only if the number of unknowns is sufficiently large (at least the number of linearly independent equations plus the dimension of the state vector). A “nice” solution of the PDE means that we do not need any integration, and hence it is very fast. To dispel the mystery of Gromov’s theorem we show the simplest non-trivial example. We also show several generalizations of Gromov’s theorem. Particle flow is similar to optimal transport, but it is much simpler and faster because we avoid solving a variational problem. Optimal transport is (almost always) deterministic whereas our particle flow is stochastic, like all such algorithms that actually work robustly for difficult high dimensional problems.

Bio: Fred Daum is an IEEE Fellow, a Principal Fellow at Raytheon, a Distinguished Lecturer for the IEEE and a graduate of Harvard University. Fred's exact fixed finite dimensional nonlinear filter theory generalizes the Kalman and Beneš filters. Fred’s particle flow nonlinear filter is many orders of magnitude faster than standard particle filters for the same accuracy. He has published more than one hundred technical papers, and he has given invited lectures at MIT, Harvard, Yale, Caltech, the Technion, Ecole Normale Superieure de Paris, Brown, Georgia Tech., Duke, Melbourne Univ., Univ. of Toulouse, Univ. of New South Wales, Univ. of Canterbury, Liverpool Univ., Xidian Univ., Univ. of Illinois at Chicago, Washington Univ. at St Louis, etc.