Deep neural networks (DNN) and Gaussian processes (GP) are two powerful models with several theoretical connections relating them, but the relationship between their training methods is not well understood. In this paper, we show that certain posterior approximations for Bayesian DNNs equivalent to GP posteriors. This enables us relate solutions iterations of a deep-learning algorithm inference. As result, can obtain kernel nonlinear feature map while DNN. Surprisingly, resulting tangent...

We study a generalization of Kantorovich's optimal transportation problem. Given prescribed family time-dependent probability measures $(\mu_t)$, we aim to find, among all path-continuous stochastic processes whose one-dimensional time marginals coincide with $(\mu_t)$ (if there is any), process that minimizes given energy. After discussing sufficient condition for the energy ensure existence minimizer, investigate fractional Sobolev energies. deterministic path on $p$-Wasserstein space...

Given a probability-measure-valued process $(\mu_t)$, we aim to find, among all path-continuous stochastic processes whose one-dimensional time marginals coincide almost surely with $(\mu_t)$ (if there is any), that minimizes given energy in expectation. Building on our recent study (arXiv:2502.12068), where the minimization of fractional Sobolev was investigated for deterministic paths Wasserstein spaces, now extend results setting address some applications originally motivated study. Two...

Feedback particle filters (FPFs) are Monte-Carlo approximations of the solution filtering problem in continuous time. The samples or particles evolve according to a feedback control law order track posterior distribution. However, it is known that by itself, requirement does not lead unique algorithm. Given filter, another one can be constructed applying time-dependent transformation keeps distribution invariant. Here, we characterize this gauge freedom within class FPFs for linear-Gaussian...

Particle filters (PFs), which are successful methods for approximating the solution of filtering problem, can be divided into two types: weighted and unweighted PFs. It is well known that PFs suffer from weight degeneracy curse dimensionality. To sidestep these issues, have been gaining attention, though they their own challenges. The existing literature on types based distinct approaches. In order to establish a connection, we put forward framework unifies in continuous-time problem. We...

We extend the result of Lisini (Calc Var Partial Differ Equ 28:85-120, 2007) on superposition principle for absolutely continuous curves in $p$-Wasserstein spaces to special case $p=1$. In contrast $p>1$, it is not always possible have lifts curves. Therefore, one needs relax notion a lift by considering bounded variation, or shortly BV-curves, and replace metric speed total variation measure. prove that any BV-curve 1-Wasserstein space can be represented probability measure BV-curves...