This 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.
While the Sampling, Transport, Diffusion workshop at the Flatiron Institute kept me happily busy, and while I did not stay any longer, spending a few days in New York City was a treat and I took advantage of my early hours to go running along the river sides, first south of the Flatiron building, then north to the Queensboro Bridge and over it, and last north along the Hudson River. The East River side is much less convenient for running as the path is repeatedly blocked by construction / storage sites and Xing the Queensboro Bridge gave a great view of Manhattan, albeit at the risk of being hit by a bike / scooter / moppet, as the path was shared with [an endless flow of] speeding electric bicycles. As I had never been to this part of the city, I was unaware of the cable car / gondola to Roosevelt Island (surprisingly called tram), which I would have taken given an extra day. Came by uponchance over a Trump Tower, which I ignored was so inappropriately close to the UN Headquarters! Running on the uninterrupted Hudson River trail was much nicer (and busier) despite the freezing wind that day.
For once (!) I stayed in an hotel, reserved by the Flatiron, and for the three nights I was there it was most tolerable, except for the usual background noise found in hotels, both from heating fans and patrons discussing in the corridors after hours. But the staff was helpful to the point of purchasing a kettle for my early morning tea. As the workshop provided an enormous amount of food at all times (and there was a true matcha tea provider around the corner!), it did not matter in the least.
