The problem with the Monge-Ampere approach is that no one really knows a good method for solving it (things are better than they were ~10 years ago, where I’m not sure there were methods to solve it) even in two dimensions. Apparently if you add a pseudo-time variable, you can convert the transport problem into a set of ‘fluid dynamics’ equations, but in high dimensions. But again, methods for solving PDEs numerically in >10 (realistically >3) dimensions don’t really exist, even if the PDE is linear.

]]>Uh?! More details maybe…?

]]>Cool: a lot of people had the same reaction, indeed!!!

Markov: there are Markovian ways of solving fixed point equations. It is however used for real vectors. I think your resolution is what I was hinting at… Thanks!

One question: I’m not really sure what you mean by Markovian in this context. You can certainly (if everything is regular enough) use the multivariate ‘change of variables’ formula to end up with a Monge-Ampere equation (at least for regular optimal transport), which is a local, fully nonlinear, second order PDE. But I have a feeling you’re talking about something else…

Other semi-random comments:

– I suspect that there are actually some decent approximation theory results for these polynomial expansion, but the resulting bounds are probably too pessimistic for a practical stopping criterion )but still good enough to justify the procedure).

– I wonder what happens when some of the random variables are infinite dimensional. Optimal transport theory covers these cases (and the maps are often monotone iirc), but it’s always interesting to try to turn this into a numerical method. (In particular, can it bypass some of the problems that MCMC, INLA and VB have to differing extents with REALLY big problems [i.e. can we avoid determinants?])

– I really wonder how these calculations simplify for certain classes of models. It would be nice if the computations could be simplified based on known features of common models.

– Your point on monotonicity is interesting, although you can write it as the gradient of a convex function, I’m not convinced that that’s easier…

– One of my favourite things about this paper is that it seems to bridge the gap between the ‘Uncertainly Quantification’ work that’s done in applied maths / engineering / numerical analysis and something more statistical. I hope they develop this further!

Regardless of all of these things (and even if this method actually works on real problems), this is MASSIVELY COOL!

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