@article{20040,
  abstract     = {Contractive coupling rates have been recently introduced by Conforti as a tool to establish convex Sobolev inequalities (including modified log-Sobolev and Poincaré inequality) for some classes of Markov chains. In this work, for most of the examples discussed by Conforti, we use contractive coupling rates to prove stronger inequalities, in the form of curvature lower bounds (in entropic and discrete Bakry–Émery sense) and geodesic convexity of some entropic functionals. In addition, we recall and give straightforward generalizations of some notions of coarse Ricci curvature, and we discuss some of their properties and relations with the concepts of couplings and coupling rates: as an application, we show exponential contraction of the p-Wasserstein distance for the heat flow in the aforementioned examples.},
  author       = {Pedrotti, Francesco},
  issn         = {1050-5164},
  journal      = {The Annals of Applied Probability},
  number       = {1},
  pages        = {196 -- 250},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Contractive coupling rates and curvature lower bounds for Markov chains}},
  doi          = {10.1214/24-aap2113},
  volume       = {35},
  year         = {2025},
}

@article{20050,
  abstract     = {We prove upper bounds on the L∞-Wasserstein distance from optimal transport between strongly log-concave probability densities and log-Lipschitz perturbations. In the simplest setting, such a bound amounts to a transport-information inequality involving the L∞-Wasserstein metric and the relative L∞-Fisher information. We show that this inequality can be sharpened significantly in situations where the involved densities are anisotropic. Our proof is based on probabilistic techniques using Langevin dynamics. As an application of these results, we obtain sharp exponential rates of convergence in Fisher’s infinitesimal model from quantitative genetics, generalising recent results by Calvez, Poyato, and Santambrogio in dimension 1 to arbitrary dimensions.},
  author       = {Khudiakova, Kseniia and Maas, Jan and Pedrotti, Francesco},
  issn         = {1050-5164},
  journal      = {The Annals of Applied Probability},
  number       = {3},
  pages        = {1913--1940},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher’s infinitesimal model}},
  doi          = {10.1214/25-aap2162},
  volume       = {35},
  year         = {2025},
}

@article{20591,
  abstract     = {In this paper we derive estimates for the Hessian of the logarithm (log-Hessian) for solutions to the heat equation. For initial data in the form of log-Lipschitz perturbation of strongly log-concave measures, the log-Hessian admits an explicit, uniform (in space) lower bound. This yields a new estimate for the Lipschitz constant of a transport map pushing forward the standard Gaussian to a measure in this class. On the other hand, we show that assuming only fast decay of the tails of the initial datum does not suffice to guarantee uniform log-Hessian upper bounds.},
  author       = {Brigati, Giovanni and Pedrotti, Francesco},
  issn         = {1083-589X},
  journal      = {Electronic Communications in Probability},
  publisher    = {Institute of Mathematical Statistics},
  title        = {{Heat flow, log-concavity, and Lipschitz transport maps}},
  doi          = {10.1214/25-ECP717},
  volume       = {30},
  year         = {2025},
}

@inproceedings{18897,
  abstract     = {Score-based generative models (SGMs) are powerful tools to sample from complex data distributions. Their underlying idea is to (i) run a forward process for time T1 by adding noise to the data, (ii) estimate its score function, and (iii) use such estimate to run a reverse process. As the reverse process is initialized with the stationary distribution of the forward one, the existing analysis paradigm requires T1→∞. This is however problematic: from a theoretical viewpoint, for a given precision of the score approximation, the convergence guarantee fails as T1 diverges; from a practical viewpoint, a large T1 increases computational costs and leads to error propagation. This paper addresses the issue by considering a version of the popular predictor-corrector scheme: after running the forward process, we first estimate the final distribution via an inexact Langevin dynamics and then revert the process. Our key technical contribution is to provide convergence guarantees which require to run the forward process only for a fixed finite time T1. Our bounds exhibit a mild logarithmic dependence on the input dimension and the subgaussian norm of the target distribution, have minimal assumptions on the data, and require only to control the L2 loss on the score approximation, which is the quantity minimized in practice.},
  author       = {Pedrotti, Francesco and Maas, Jan and Mondelli, Marco},
  booktitle    = {Transactions on Machine Learning Research},
  issn         = {2835-8856},
  title        = {{Improved convergence of score-based diffusion models via prediction-correction}},
  year         = {2024},
}

@phdthesis{17336,
  abstract     = {This thesis deals with the study of stochastic processes and their ergodicity properties. The
variety of problems encountered calls for a set of different approaches, ranging from classical to
modern ones: a special place is held by probabilistic methods based on couplings, by functional
inequalities, and by the theory of gradient flows in the space of measures.

The material is organized as follows. Chapter 1 contains the introduction to this thesis, starting
with a general presentation of some of the relevant topics. Section 1.1 is dedicated to the
theory of gradient flows in metric spaces, and introduces the first contribution of this thesis
[DSMP24], which is presented in detail in Chapter 2. Section 1.2 moves to the topic of
curvature of Markov chains, concluding with a brief description of our second contribution
[Ped23], which is included in Chapter 3. Section 1.3 discusses applications of stochastic
processes to the theory of sampling, in particular the recent framework of score-based diffusion
models, and our contribution [PMM24], which is contained in Chapter 4. Section 1.4 discusses
some related problems, concerning the regularization properties of the heat flow. It serves
as a motivation for the work [BP24], which we report in Chapter 5. Finally, Section 1.5
discusses the last contribution of this thesis, which can be found in Chapter 6. It deals with
the convergence to equilibrium of a particular stochastic model from quantitative genetics:
this is established via some functional inequalities, which we prove with probabilistic arguments
based on couplings.
},
  author       = {Pedrotti, Francesco},
  issn         = {2663-337X},
  pages        = {183},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Functional inequalities and convergence of stochastic processes}},
  doi          = {10.15479/at:ista:17336},
  year         = {2024},
}

@article{17143,
  abstract     = {This paper deals with local criteria for the convergence to a global minimiser for gradient flow trajectories and their discretisations. To obtain quantitative estimates on the speed of convergence, we consider variations on the classical Kurdyka–Łojasiewicz inequality for a large class of parameter functions. Our assumptions are given in terms of the initial data, without any reference to an equilibrium point. The main results are convergence statements for gradient flow curves and proximal point sequences to a global minimiser, together with sharp quantitative estimates on the speed of convergence. These convergence results apply in the general setting of lower semicontinuous functionals on complete metric spaces, generalising recent results for smooth functionals on Rn. While the non-smooth setting covers very general spaces, it is also useful for (non)-smooth functionals on Rn.
.},
  author       = {Dello Schiavo, Lorenzo and Maas, Jan and Pedrotti, Francesco},
  issn         = {1088-6850},
  journal      = {Transactions of the American Mathematical Society},
  number       = {6},
  pages        = {3779--3804},
  publisher    = {American Mathematical Society},
  title        = {{Local conditions for global convergence of gradient flows and proximal point sequences in metric spaces}},
  doi          = {10.1090/tran/9156},
  volume       = {377},
  year         = {2024},
}

@unpublished{17350,
  abstract     = {Score-based generative models (SGMs) are powerful tools to sample from
complex data distributions. Their underlying idea is to (i) run a forward
process for time $T_1$ by adding noise to the data, (ii) estimate its score
function, and (iii) use such estimate to run a reverse process. As the reverse
process is initialized with the stationary distribution of the forward one, the
existing analysis paradigm requires $T_1\to\infty$. This is however
problematic: from a theoretical viewpoint, for a given precision of the score
approximation, the convergence guarantee fails as $T_1$ diverges; from a
practical viewpoint, a large $T_1$ increases computational costs and leads to
error propagation. This paper addresses the issue by considering a version of
the popular predictor-corrector scheme: after running the forward process, we
first estimate the final distribution via an inexact Langevin dynamics and then
revert the process. Our key technical contribution is to provide convergence
guarantees which require to run the forward process only for a fixed finite
time $T_1$. Our bounds exhibit a mild logarithmic dependence on the input
dimension and the subgaussian norm of the target distribution, have minimal
assumptions on the data, and require only to control the $L^2$ loss on the
score approximation, which is the quantity minimized in practice.},
  author       = {Pedrotti, Francesco and Maas, Jan and Mondelli, Marco},
  booktitle    = {arXiv},
  title        = {{Improved convergence of score-based diffusion models via prediction-correction}},
  doi          = {10.48550/arXiv.2305.14164},
  year         = {2024},
}

@unpublished{17352,
  abstract     = {We prove upper bounds on the $L^\infty$-Wasserstein distance from optimal
transport between strongly log-concave probability densities and log-Lipschitz
perturbations. In the simplest setting, such a bound amounts to a
transport-information inequality involving the $L^\infty$-Wasserstein metric
and the relative $L^\infty$-Fisher information. We show that this inequality
can be sharpened significantly in situations where the involved densities are
anisotropic. Our proof is based on probabilistic techniques using Langevin
dynamics. As an application of these results, we obtain sharp exponential rates
of convergence in Fisher's infinitesimal model from quantitative genetics,
generalising recent results by Calvez, Poyato, and Santambrogio in dimension 1
to arbitrary dimensions.},
  author       = {Khudiakova, Kseniia and Maas, Jan and Pedrotti, Francesco},
  booktitle    = {arXiv},
  title        = {{L∞-optimal transport of anisotropic log-concave measures and exponential convergence in Fisher's infinitesimal model}},
  doi          = {10.48550/arXiv.2402.04151},
  year         = {2024},
}

@unpublished{17353,
  abstract     = {In this paper we derive estimates for the Hessian of the logarithm
(log-Hessian) for solutions to the heat equation. For initial data in the form
of log-Lipschitz perturbation of strongly log-concave measures, the log-Hessian
admits an explicit, uniform (in space) lower bound. This yields a new estimate
for the Lipschitz constant of a transport map pushing forward the standard
Gaussian to a measure in this class. Further connections are discussed with
score-based diffusion models and improved Gaussian logarithmic Sobolev
inequalities. Finally, we show that assuming only fast decay of the tails of
the initial datum does not suffice to guarantee uniform log-Hessian upper
bounds.},
  author       = {Brigati, Giovanni and Pedrotti, Francesco},
  booktitle    = {arXiv},
  title        = {{Heat flow, log-concavity, and Lipschitz transport maps}},
  doi          = {10.48550/arXiv.2404.15205},
  year         = {2024},
}

@unpublished{17351,
  abstract     = {Contractive coupling rates have been recently introduced by Conforti as a
tool to establish convex Sobolev inequalities (including modified log-Sobolev
and Poincar\'{e} inequality) for some classes of Markov chains. In this work,
we show how contractive coupling rates can also be used to prove stronger
inequalities, in the form of curvature lower bounds for Markov chains and
geodesic convexity of entropic functionals. We illustrate this in several
examples discussed by Conforti, where in particular, after appropriately
choosing a parameter function, we establish positive curvature in the entropic
and (discrete) Bakry--\'{E}mery sense. In addition, we recall and give
straightforward generalizations of some notions of coarse Ricci curvature, and
we discuss some of their properties and relations with the concepts of
couplings and coupling rates: as an application, we show exponential
contraction of the $p$-Wasserstein distance for the heat flow in the
aforementioned examples.},
  author       = {Pedrotti, Francesco},
  booktitle    = {arXiv},
  title        = {{Contractive coupling rates and curvature lower bounds for Markov chains}},
  doi          = {10.48550/arXiv.2308.00516},
  year         = {2023},
}