###### 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.

Overview: In this note, I log some basic observations about diffusion-based generative models.

8/14/2023Overview: In this note, I describe some aspects of hierarchical structure in MCMC algorithms, and how they can be of theoretical and practical relevance.

8/9/2023Overview: In this note, I discuss a recurrent question which can be used to generate research questions about methods of all sorts. I then discuss a specific instance of how this question has proved fruitful in the theory of optimisation algorithms. Methods and Approximations A nice story is that when Brad Efron derived the bootstrap, it was done in service of the question “What is the jackknife an approximation to?”. I can't help but agree that there's something quite exciting about research questions which have this same character of ''What is (this existing thing) an approximation to?''. One bonus tilt on this which I appreciate is that there can be multiple levels of approximation, and hence many answers to the same question. One well-known example is gradient descent, which can be viewed as an approximation to the proximal point method, which can then itself be viewed as an approximation to a gradient flow. There are probably even more stops along the way here. In this case, there is even the perspective that from the perspective of mathematical theory, there may be at least as much to be gained by stopping off at the proximal point interpretation, as there is from the gradient flow perspective. My experience is that generalist applied mathematicians get to grips with the gradient flow quickly, but optimisation theorists can squeeze more out of the PPM formulation. There is thus some hint that using this 'intermediate' approximation can be particularly insightful in its own right. It would be interesting to collect more examples with this character.

5/22/2023Overview: In this note, I prove Hoeffding's inequality from the perspectives of martingales and convex ordering. The Basic Construction Let $-\infty<a<x<b<\infty$, and define a random variable $M$ with law $M\left(x;a,b\right)$ by \begin{align} M=\begin{cases} a & \text{w.p. }\frac{b-x}{b-a}\ b & \text{w.p. }\frac{x-a}{b-a}. \end{cases}

5/22/2023
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