Optimal Transport and Uncertainty

Friday 26 Nov 2021, 14:30 → 20:30 Europe/Rome

Dipartimento di Matematica (Pisa)

live streaming

locandina.pdf

recording

14:30 → 17:30 main contributions

14:30 Euclidean Random Assignment Problems, old and new

An Euclidean Random Assignment Problem (ERAP in short) is as follows:

• there are two $n$-sets $\mathcal{B}=(B_i)_{i=1}^n$ (blue points) and $\mathcal{R}=(R_i)_{I=1}^n$ (red points) of i.i.d. random variables valued on a metric space $(\Omega,D)$ according to a prob. measure $\nu$ (disorder);

• for a permutation (or assignment) $\pi : \mathcal{B} \rightarrow \mathcal{R}$, there is an energy $\mathcal{H}(\pi)=\sum_{i=1}^n D(b_i,r_{\pi(i)})^p$, where $p\in \mathbb{R}$;

what can one say about the random variable $H_{\rm opt} = \min_\pi \mathcal{H}(\pi)$, depending on the choice of $(\Omega,D)$, on the disorder $\nu$, and $p$?

ERAPs were pioneered in statistical physics by Mézard and Parisi in the '80s as toy models for finite-dimensional spin glasses; and any ERAP is Monge-Kantorovich optimal transport problem for the empirical measures of blue and red points, in which $W_p^p(\rho_{\mathcal{B}},\rho_{\mathcal{R}})=\frac{1}{n} H_{\rm opt}$, where $W_p$ is p-Wasserstein distance.

Despite these connections, ERAPs have been exceedingly difficult to understand, and surprisingly few results have been proven to date.

In this talk I will review some selected ideas and results on ERAPs focusing on low dimensions of the underlying space $\Omega$. If time allows, I will discuss current work in progress and touch upon a few research perspectives.

Speaker: Prof. Matteo D'Achille (Université Paris-Est Créteil)

ERAP_OTU_Pisa_v8_final.pdf

15:15 Optimal transport, wave functions and single electron densities

I will discuss some properties of the mapping from wave-functions to single particle densities. In particular I will show that in some case this mapping is open.
The tool will be the construction of special transport plans with given marginals. This partially answers an open question of E.H. Lieb. (From a joint work with Ugo Bindini).

Speaker: Prof. Luigi De Pascale (Università degli Studi di Firenze)

16:00 There is no invariant cyclically monotone Poisson matching in 2d

The optimal matching problem is a classical random variational problem that received interest in the last 30 years. We show that there exists no cyclically monotone invariant matching of two independent Poisson processes in the critical dimension $d=2$. Our argument relies on a recent harmonic approximation theorem together with the two-dimensional local asymptotics for the bipartite matching problem, for which we provide a new self-contained proof based on martingale arguments.

Joint work with M. Huesmann (WWU Münster) and F. Otto (MPI Leipzig)

Speaker: Prof. Francesco Mattesini (University of Münster)

2dnonexistence_slides.pdf

16:45 Optimal quantization strategies for vectorial signals

We will discuss and compare two approaches for quantization of vectorial signals on the input to a computational device: quantizing the whole signal and optimizing the input error, or quantizing separately the components
but optimizing the output error.

Speaker: Prof. Eugene Stepanov (Steklov Institute of Mathematics, St. Petersburg)

17:30 → 18:50 round table

17:30 Optimal transport methods in practical bilevel problems

Speaker: Lina Mallozzi (Università di Napoli Federico II)

17:50 Locating segments via optimal transport

Speaker: Serena Guarino Lo Bianco (Università di Napoli Federico II)

18:10 Random Optimal Transport and friends

Speaker: Dario Trevisan (Università di Pisa)

18:30 Some questions on optimal planar clusters

Speaker: Emanuele Paolini (University of Pisa)

19:30 → 20:30 Dinner

Osteria il Domo. Please let us know if you want to participate. The cost will be around 35€. Meet us at the mathematics department.