I am interested in probability theory and in its interplay with analysis and geometry.  Particularly I am interested in:

• the Schrödinger Bridge Problem (SP) and its relationship with  (Entropic) Optimal Transport,

• Optimal Transport point of view in (Diffusion) Generative Modelling

• stochastic optimal control problems: long-time behaviour and exponential ergodicity,

• large deviations theory,

• high-dimensional probability.

During the first year of my PhD I have worked on a generalization of SP in a hypocoercitivity setting where the underlying reference measure is the underdamped Langevin dynamics (whose density solves the kinetic Fokker-Planck equation). We have named it the kinetic Schrödinger problem and we have investigated the exponential ergodicity and long-time behaviour of this hypocoercive stochastic control problem. These results are described in the paper "Entropic turnpike estimates for the kinetic Schrödinger problem".

During my second year we have worked on the relation between SP and Optimal Transport in the small-time limit and on quantitative stability bounds that measure how sensitive SP is when varying the marginals . These two problems can be both addressed relying on gradient estimates for the Schrödinger potentials, known as corrector estimates (the name comes from the long-time behaviour of SP and its control interpretation). We highlight for the first time their relevance in the short-time regime and more precisely we prove:

• convergence of the gradients of the Schrödinger  potentials towards the Brenier map in the small-time limit;

• quantitative stability estimates for the optimal values and optimal couplings for SP, that we express in terms of a negative order weighted homogeneous Sobolev norm (which encodes the linearized behaviour of the 2-Wasserstein distance)

These results are described in the preprint "Gradient estimates for the Schrödinger potentials: convergence to the Brenier map and quantitative stability".

In my third year (maybe the best one during a 4-years PhD) I have started working on Sinkhorn algorithm, a very fast and efficient iterative algorithm that coverges to the solution of SP (or equivalently EOT). The study of its convergence is tightly related to quantitative stability estimates for SP. On discrete/compact/bounded settings it is very well known the exponential convergence of the algorithm. In the paper "Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach" we have further investigated the algorithm on the compact setting, namely on the torus, by showing an interesting link between updates of Sinkhorn algorithm and Stochastich Optimal Control problems and then deducing from that the convergence of the gradients of Sinkhorn iterates.  Inspired by the results obtained there via coupling techniques, in the follow-up paper "Quantitative contraction rates for Sinkhorn algorithm: beyond bounded costs and compact marginals", we have investigated the convergence of the algorithm on the Euclidean space and we have provided the very first exponential convergence result for quadratic cost and unbounded marginals.

During this last year of PhD I have been working on my PhD thesis, made of the above explained  contributions, which I will defend in TU/e on the 23rd of May 2024. Lately, I have been interested in the study of generative modelling with Schrödinger bridges and optimal transport tools. 

• G. Conforti, A. Durmus, G. Greco. Quantitative contraction rates for Sinkhorn algorithm: beyond bounded costs and compact marginals. arXiv:2304.04451 (04/2023)

• G. Greco, M. Noble, G. Conforti, A. Durmus. Non-asymptotic convergence bounds for Sinkhorn iterates and their gradients: a coupling approach. Accepted in 36th Annual Conference on Learning Theory (COLT 2023)  PMLR 195:716-746 (2023)

• A. Chiarini, G. Conforti, G. Greco, L. Tamanini. Gradient estimates for the Schrödinger potentials: convergence to the Brenier map and quantitative stability. Communications in Partial Differential Equations 48:6, 895-943 (2023) 

• A. Chiarini, G. Conforti, G. Greco, Z. Ren. Entropic turnpike estimates for the kinetic Schrödinger problem. Electron. J. Probab. 27: 1-32 (2022)