arXiv subjects

Gabriele Bocchi

Publications and source records attributed to Gabriele Bocchi.

2 recordsLinked to original sources

arXiv2026

Triangular-Reference Schr\"odinger Bridges for Time Series Generation

Schr\"odinger bridges for time series (SBTS) generate synthetic paths by projecting, in relative entropy, a Brownian reference onto the path laws that match the joint distribution of the data on the observation grid. The Brownian reference, however, fixes the quadratic variation of the generated paths, which is restrictive when stochastic volatility, correlated noise, or rank-deficient covariance structures must be reproduced. We introduce "Triangular-Reference Schr\"odinger Bridges for Time Series" (TR-SBTS), which keeps the entropy-projection backbone of SBTS but replaces the Brownian reference by a triangular, volatility-informed, intervalwise frozen reference on a state augmented with latent covariance descriptors. The construction remains a single entropy projection on the augmented state: the minimiser is the $h$-transform of the reference, and on each frozen interval the optimal drift has the logarithmic-gradient form $b^\star(t,x)=A\,\nabla\log H(t,x)$, intrinsic to the active covariance directions when the frozen covariance $A$ is degenerate. We prove stability of the frozen approximation and consistency of the associated regularised kernel estimators, describe a reference-aware Nadaraya--Watson implementation of the conditional next-increment law, and evaluate the construction on numerical experiments.

stat.ML↗

arXiv2024

Entropy-minimizing dynamical transport on Riemannian manifolds

Given a smooth Riemannian manifold $(M,g)$, compact and without boundary, we analyze the dynamical optimal mass transport problem where the cost is given by the sum of the kinetic energy and the relative entropy with respect to a reference volume measure $e^{-V}dx$. Under the only assumption that the prescribed marginals lie in $L^1(M)$, and a lower bound on the Ricci curvature, we characterize the minimal curves as unique weak solutions of the optimality system coupling the continuity equation with a backward Hamilton-Jacobi equation (with source given by $\log (m)$). We give evidence that the entropic cost enhances diffusive effects in the evolution of the optimal densities, proving $L^1\to L^\infty$ regularization in time for any initial-terminal data, and smoothness of the solutions whenever the marginals are positive and smooth. We use displacement convexity arguments (in the Eulerian approach) and gradient bounds from quasilinear elliptic equations. We also prove the convergence of optimal curves towards the classical Wasserstein geodesics, as the entropic term is multiplied by a vanishing parameter, showing that this kind of functionals can be used to build a smoothing approximation of the standard optimal transport problem.

math.AP↗