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Over the past few years, we developed a mathematically rigorous method to study the dynamical processes associated to nonlinear Forchheimer flows for slightly compressible fluids. We have proved the existence of a geometric transformation…
In this paper we use methods from Stochastic Analysis to establish Li-Yau type estimates for positive solutions of the heat equation. In particular, we want to emphasize that Stochastic Analysis provides natural tools to derive local…
We study estimation problems in safety-critical applications with streaming data. Since estimation problems can be posed as optimization problems in the probability space, we devise a stochastic projected Wasserstein gradient flow that…
Despite the widespread use of gradient-based algorithms for optimizing high-dimensional non-convex functions, understanding their ability of finding good minima instead of being trapped in spurious ones remains to a large extent an open…
We investigate a family of approximate multi-step proximal point methods, accelerated by implicit linear discretizations of gradient flow. The resulting methods are multi-step proximal point methods, with similar computational cost in each…
We propose RicciFlowRec, a geometric recommendation framework that performs root cause attribution via Ricci curvature and flow on dynamic financial graphs. By modelling evolving interactions among stocks, macroeconomic indicators, and…
We explore two aspects of geometric approximation via a coupling approach to Stein's method. Firstly, we refine precision and increase scope for applications by convoluting the approximating geometric distribution with a simple translation…
We explicitly construct parameter transformations between gradient flows in metric spaces, called curves of maximal slope, having different exponents when the associated function satisfies a suitable convexity condition. These…
A generic, fast and asymptotically efficient method for parametric estimation is described. It is based on the projected stochastic gradient descent on the log-likelihood function corrected by a single step of the Fisher scoring algorithm.…
Stochastic-gradient sampling methods are often used to perform Bayesian inference on neural networks. It has been observed that the methods in which notions of differential geometry are included tend to have better performances, with the…
This article deals with flow of plane curves driven by the curvature and external force. We make use of such a geometric flow for the purpose of image segmentation. A parametric model for evolving curves with uniform and curvature adjusted…
We analyze convergence rates of stochastic optimization procedures for non-smooth convex optimization problems. By combining randomized smoothing techniques with accelerated gradient methods, we obtain convergence rates of stochastic…
We introduce a general class of algorithms and supply a number of general results useful for analysing these algorithms when applied to regular graphs of large girth. As a result, we can transfer a number of results proved for random…
We investigate the test risk of continuous-time stochastic gradient flow dynamics in learning theory. Using a path integral formulation we provide, in the regime of a small learning rate, a general formula for computing the difference…
In this book chapter, we briefly describe the main components that constitute the gradient descent method and its accelerated and stochastic variants. We aim at explaining these components from a mathematical point of view, including…
A new approach of obtaining stratified random samples from statistically dependent random variables is described. The proposed method can be used to obtain samples from the input space of a computer forward model in estimating expectations…
We introduce data structures for solving robust regression through stochastic gradient descent (SGD) by sampling gradients with probability proportional to their norm, i.e., importance sampling. Although SGD is widely used for large scale…
We study curvature pinching estimates of Ricci flow on complete 3- dimensional manifolds without bounded curvature assumption. We will derive some general curvature conditions which are preserved on any complete solution of 3-dim Ricci…
The evolution by horizontal mean curvature flow (HMCF) is a partial differential equation in a sub-Riemannian setting with application in IT and neurogeometry (see Citti-Franceschiello-Sanguinetti-Sarti, 2016). Unfortunately this equation…
Ricci curvature and Ricci flow have proven to be powerful tools for analyzing the geometry of discrete structures, particularly on undirected graphs, where they have been applied to tasks ranging from community detection to graph…