###### tags: `one-offs` `transport`

# Non-Optimal Transport Maps

**Overview**: In this note, I sketch some rough thoughts about transport maps, and how in certain applications, optimal transport is not the sharpest route to the answers one seeks.

## Discussion

Something which I've experienced recently is that for certain (admittedly more theoretical) tasks involving measure transport and transport maps, the 'optimal' formulation of the problem can be quite far from the conventional minimum-average-cost approach of OT. For example, there are some really fascinating things going on recently in terms of constructing transport maps with favourable Lipschitz regularity. For certain applications (e.g. geometric functional inequalities), this appears to be far more valuable than a distance-minimising property.

Examples of good (but not optimal!) transport maps include the 'reverse heat flow' approach of Kim and Milman (also studied by Neeman, and by Mikulincer and Shenfeld as the 'Langevin Transport Map') and the 'Brownian Transport Map' (also of Mikulincer and Shenfeld). Interestingly, these maps are built out of stochastic processes which are known to converge well to their equilibrium (in quite a strong sense), which are then somehow translated into deterministic maps with good regularity properties. Somehow, the existence of a stochastic process with good contractivity properties is sufficient to establish the existence of a deterministic flow with similarly good properties (though this may not quite be the structure of the argument).

Each of these works notes that the existence of Lipschitz transport maps (from e.g. a Gaussian measure to the measure of interest) enables the straightforward deduction of various geometric functional inequalities for the measure in question. While certain optimal transport maps **are** known to have good Lipschitz properties (consider the implications of e.g. the contraction theorem of Caffarelli), it is far from clear that the minimal-displacement formulation of conventional optimal transport is the most efficient way to deduce favourable Lipschitz properties.

One problem which is quite open to me is to devise some transport problem whose solution is geared towards ensuring the desired Lipschitz regularity (or will encourage it at some level of generality), while also retaining favourable intrinsic properties as a variational problem. This could prove to be insightful in understanding why e.g. the construction of Kim and Milman is successful, and how it might be generalised.

Another more abstract problem is to generalise these notions to discrete spaces, noting that deterministic transport maps are not even always available in this setting. Perhaps it will still be possible to say something related and fun, though. I also don't know whether these statements have equivalents in the manifold setting.