## Archive for optimal transport

## PR[AI]RIE colloquium [12/12]

Posted in Statistics with tags colloquium, optimal transport, Paris, Paris Artificial Intelligence Research Institute, Schrödinger bridge, Sinkhorn algorithm, The Prairie Chair, Wasserstein distance on December 2, 2022 by xi'an## coupling for the Gibbs sampler

Posted in Books, Mountains, pictures, Running, Statistics, Travel, University life with tags AISTATS 2022, BNP13, Hamming distance, label switching, Lago Llanquihue, maximal coupling, optimal transport, Orlin's algorithm, Orsono volcano, partition, single chain on November 27, 2022 by xi'an**A**t BNP13, Brian Trippe presented the AISTAT 2022 paper he recently wrote with Tin D. Nguyen and Tamara Broderick. Which made me read their 2021 paper on the topic. There, they note that coupling may prove challenging, which they blame on label switching. Considering a naïve Gibbs sampler on the space of partitions, meaning allocating each data-point to one of the existing partitions or to a singleton, they construct an optimal transport coupling under Hamming distance. Which appears to be achievable in O(NK³log{K}), if K is the maximal number of partitions among both chains. The paper does not goes deeply into the implementation, which involves [to quote] (a) *computing* the* distances between each pair of partitions in the Cartesian product of supports of the Gibbs conditionals* and (b) *solving the optimal transport problem*. Except in the appendix where the book-keeping necessary to achieve O(K²) for pairwise distances and the remaining complexity follows from the standard Orlin’s algorithm. What remains unclear from the paper is that, while the chains couple faster (fastest?), the resulting estimators do not necessarily improve upon budget-equivalent alternatives. (The reason for the failure of the single chain in Figure 2 is hard to fathom.)

## sampling, transport, and diffusions

Posted in pictures, Running, Statistics, Travel, University life with tags causality, delayed rejection sampling, Flatiron building, Flatiron Institute, HMC, Hyvärinnen score, Madison Square Garden, normalising flow, NYC, optimal transport, Restore, Simmons Foundation, simulation, Sinkhorn algorithm, WABC, Wasserstein distance on November 18, 2022 by xi'an

**T**his week, I am attending a very cool workshop at the Flatiron Institute (not in the Flatiron building!, but close enough) on Sampling, Transport, and Diffusions, organised by Bob Carpenter and Michael Albergo. It is quite exciting as I do not know most participants or their work! The Flatiron Institute is a private institute focussed on fundamental science funded by the Simons Foundation (in such working conditions universities cannot compete with!).

Eric Vanden-Eijden gave an introductory lecture on using optimal transport notion to improve sampling, with a PDE/ODE approach of continuously turning a base distribution into a target (formalised by the distribution at time one). This amounts to solving a velocity solution to an KL optimisation objective whose target value is zero. Velocity parameterised as a deep neural network density estimator. Using a score function in a reverse SDE inspired by Hyvärinnen (2005), with a surprising occurrence of Stein’s unbiased estimator, there for the same reasons of getting rid of an unknown element. In a lot of environments, simulating from the target is the goal and this can be achieved by MCMC sampling by normalising flows, learning the transform / pushforward map.

At the break, Yuling Yao made a very smart remark that testing between two models could also be seen as an optimal transport, trying to figure an optimal transform from one model to the next, rather than the bland mixture model we used in our mixtestin paper. At this point I have no idea about the practical difficulty of using / inferring the parameters of this continuum but one could start from normalising flows. Because of time continuity, one would need some driving principle.

Esteban Tabak gave another interest talk on simulating from a conditional distribution, which sounds like a no-problem when the conditional density is known but a challenge when only pairs are observed. The problem is seen as a transport problem to a barycentre obtained as a distribution independent from the conditioning z and then inverting. Constructing maps through flows. Very cool, even possibly providing an answer for causality questions.

Many of the transport talks involved normalizing flows. One by [Simons Fellow] Christopher Jazynski about adding to the Hamiltonian (in HMC) an artificial flow field (Vaikuntanathan and Jarzynski, 2009) to make up for the Hamiltonian moving too fast for the simulation to keep track. Connected with Eric Vanden-Eijden’s talk in the end.

An interesting extension of delayed rejection for HMC by Chirag Modi, with a manageable correction *à la* Antonietta Mira. Johnatan Niles-Weed provided a nonparametric perspective on optimal transport following Hütter+Rigollet, 21 AoS. With forays into the Sinkhorn algorithm, mentioning Aude Genevay’s (Dauphine graduate) regularisation.

Michael Lindsey gave a great presentation on the estimation of the trace of a matrix by the Hutchinson estimator for sdp matrices using only matrix multiplication. Solution surprisingly relying on Gibbs sampling called thermal sampling.

And while it did not involve optimal transport, I gave a short (lightning) talk on our recent adaptive restore paper: although in retrospect a presentation of Wasserstein ABC could have been more suited to the audience.