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