The paper proves global universal approximation theorems for non‑anticipative and general path‑dependent functionals on spaces of piecewise linear paths, showing that linear functionals of the corresponding signatures are dense in $L^p$ and weighted norms. It demonstrates that these results apply to piecewise linear interpolations of Gaussian processes, including Brownian motion and fractional Brownian motion with Hurst parameter $H>1/4$, and provides quantitative convergence rates for signatures of such approximations toward their continuous‑time counterparts. Consequently, the authors obtain $L^p$‑approximation results for path‑dependent functionals of Gaussian processes and for random ordinary and stochastic differential equations driven by Brownian motion.
arXiv:2512. 16396v2 Announce Type: replace-cross Abstract: We establish $L^p$-universal approximation theorems for general path-dependent and non-anticipative functionals on suitable rough path spaces, showing that linear functionals acting on signatures of time-extended rough paths are dense with respect to the $L^p$-distance.
arXiv:2608.28446v1 Announce Type: cross Abstract: For finite-dimensional linear inverse problems where the variables are Gaussian, it is well-known that the minimum-mean-square error estimator takes...
arXiv:2609. 39144v2 Announce Type: new Abstract: We prove a sharp Gaussian approximation for the invariant law of constant-stepsize SGD with bounded additive noise generated by an exogenous uniformly ergodic Markov chain.
arXiv:2609.14922v1 Announce Type: cross Abstract: For constant-stepsize stochastic approximation (SA), the iterates converge in distribution to a stationary law that depends on the stepsize $\alpha.$...
The paper introduces the expected signature kernel as a similarity measure for probability laws on path space, directly applicable to signature maximum mean discrepancies. It characterizes this kernel for group-valued inhomogeneous Lévy processes by solving an extended Goursat-type system, providing exact solutions in the continuous case and well‑posed infinite‑dimensional systems with error‑controlled truncations for jump processes. The authors also show that conditioning on sampled quadratic variations yields an unbiased, lower‑variance statistic for conditionally Gaussian martingales, and present computable finite‑grid implementations that improve variance and test power.
The monograph explores the relationships between Gaussian processes and reproducing kernel Hilbert spaces (RKHS), two widely used approaches that rely on positive definite kernels. It examines how these frameworks connect and are equivalent across key topics such as regression, interpolation, numerical integration, distributional discrepancies, statistical dependence, and Gaussian process sample path properties. By establishing a unifying perspective based on the equivalence between the Gaussian Hilbert space and the RKHS, the work aims to bridge methods developed independently by the machine learning, statistics, and numerical analysis communities.
arXiv:2512. 06143v2 Announce Type: replace Abstract: Despite a large corpus of recent work on scaling up Gaussian processes, a stubborn trade-off between computational speed, prediction and uncertainty quantification accuracy, and customizability persists.
arXiv:2602.13960v2 Announce Type: replace Abstract: Constant-stepsize stochastic approximation (SA) is widely used in learning for computational efficiency, yet the distribution of the iterates is ty...
The paper explores the relationship between Gaussian processes and Gaussian random elements within reproducing kernel Banach spaces. It demonstrates that the covariance operator of a weak second‑order Radon probability measure on such a space is uniquely defined by a symmetric positive‑definite function, and in the Gaussian case it identifies which functions correspond to covariance operators via γ‑radonifying operators. Leveraging these findings, the authors extend Driscoll’s theorem to the setting of reproducing kernel Banach spaces.
The article "Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction" presents a unified primer on diffusion models that applies to both continuous Euclidean data and discrete categorical structures. It develops discrete-time forward noising via Markov kernels and learned reverse dynamics, and connects these to continuous-time limits such as stochastic differential equations in ρ^d and continuous-time Markov chains on finite alphabets, deriving the corresponding Fokker–Planck and master equations. The work also shows how different forward corruption choices—Gaussian processes for continuous spaces and structured categorical transition kernels for discrete spaces—affect reverse dynamics and the evidence lower bound used in training, offering a layered exposition for newcomers, practitioners, and experts alike.
arXiv:2602. 13906v2 Announce Type: replace-cross Abstract: Stochastic approximation (SA) is a method for finding the root of an operator perturbed by noise.
arXiv:2610.19925v1 Announce Type: cross Abstract: Efficient global optimization aims to identify a global optimum of an expensive black-box function within a limited evaluation budget $N$. We investi...