Global universality via discrete-time signatures

• URL: http://arxiv.org/abs/2603.09773v1
• Date: Tue, 10 Mar 2026 15:12:03 GMT
• Title: Global universality via discrete-time signatures
• Abstract summary: We show that piecewise linears of Brownian motion satisfies this integrability condition.<n>We obtain $Lp$-approximation results for path-dependent functionals, random ordinary differential equations, and differential equations driven by Brownian motion.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We establish global universal approximation theorems on spaces of piecewise linear paths, stating that linear functionals of the corresponding signatures are dense with respect to $L^p$- and weighted norms, under an integrability condition on the underlying weight function. As an application, we show that piecewise linear interpolations of Brownian motion satisfies this integrability condition. Consequently, we obtain $L^p$-approximation results for path-dependent functionals, random ordinary differential equations, and stochastic differential equations driven by Brownian motion.

Related papers

• Weighted universal approximation of differentiable maps on infinite-dimensional manifolds [1.552674656890411]
  We generalize the universal approximation theorem for functional input neural networks to differentiable maps.<n>A FNN maps the input from a possibly infinite-dimensional weighted manifold to the real-valued hidden layer.<n>We show that linear functions of the signature are able to approximate path space functionals including their directional derivatives.
  arXiv  Detail & Related papers  (2026-06-08T17:57:40Z)
• GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains [56.121725064621295]
  Score-based diffusion models in function spaces offer a principled framework for modelling function-valued data.<n>However, practical implementations have struggled to fully realise these benefits.<n>We propose a novel architecture that represents generalised convolutional kernels as finite graph kernels.
  arXiv  Detail & Related papers  (2026-05-05T08:33:52Z)
• Stability and Generalization of Push-Sum Based Decentralized Optimization over Directed Graphs [55.77845440440496]
  Push-based decentralized communication enables optimization over communication networks, where information exchange may be asymmetric.<n>We develop a unified uniform-stability framework for the Gradient Push (SGP) algorithm.<n>A key technical ingredient is an imbalance-aware generalization bound through two quantities.
  arXiv  Detail & Related papers  (2026-02-24T05:32:03Z)
• Universal approximation with signatures of non-geometric rough paths [0.0]
  We prove that linear functionals of the signature of the resulting rough paths approximate continuous functionals on rough path spaces uniformly on compact sets.<n>We construct the signature of a path extended by its pathwise quadratic variation terms based on general pathwise integration la Fllmer.
  arXiv  Detail & Related papers  (2026-02-05T17:16:25Z)
• Global universal approximation with Brownian signatures [0.0]
  We show that linear functionals acting on signatures of time-extended rough paths are dense with respect to an $Lp$-distance.<n>We demonstrate that these $Lp$-type universal approximation theorems apply in particular to Brownian motion.
  arXiv  Detail & Related papers  (2025-12-18T10:49:20Z)
• Generalized Gradient Norm Clipping & Non-Euclidean $(L_0,L_1)$-Smoothness [51.302674884611335]
  This work introduces a hybrid non-Euclidean optimization method which generalizes norm clipping by combining steepest descent and conditional gradient approaches.<n>We discuss how to instantiate the algorithms for deep learning and demonstrate their properties on image classification and language modeling.
  arXiv  Detail & Related papers  (2025-06-02T17:34:29Z)
• Entropic Mirror Descent for Linear Systems: Polyak's Stepsize and Implicit Bias [55.72269695392027]
  This paper focuses on applying entropic mirror descent to solve linear systems.<n>The main challenge for the convergence analysis stems from the unboundedness of the domain.<n>To overcome this without imposing restrictive assumptions, we introduce a variant of Polyak-type stepsizes.
  arXiv  Detail & Related papers  (2025-05-05T12:33:18Z)
• Global universal approximation of functional input maps on weighted spaces [3.386401892906348]
  We introduce so-called functional input neural networks defined on a possibly infinite dimensional weighted space with values also in a possibly infinite dimensional output space.<n>We prove a global universal approximation result on weighted spaces for continuous functions going beyond the usual approximation on compact sets.<n>We emphasize that the reproducing Hilbert kernel space of the signature kernels are Cameron-Martin spaces of certain Gaussian processes.
  arXiv  Detail & Related papers  (2023-06-05T23:06:32Z)
• Reinforcement Learning from Partial Observation: Linear Function Approximation with Provable Sample Efficiency [111.83670279016599]
  We study reinforcement learning for partially observed decision processes (POMDPs) with infinite observation and state spaces. We make the first attempt at partial observability and function approximation for a class of POMDPs with a linear structure.
  arXiv  Detail & Related papers  (2022-04-20T21:15:38Z)
• General Solution and Canonical Quantization of the Conic Path Constrained Second-Class System [0.0]
  We consider the problem of constrained motion along a conic path under a given external potential function. We perform the canonical quantization in a consistent way in terms of the corresponding Dirac brackets. The complete Dirac brackets algebra in phase space as well as its physical realization in terms of differential operators is explicitly obtained.
  arXiv  Detail & Related papers  (2022-02-15T13:46:56Z)
• The Connection between Discrete- and Continuous-Time Descriptions of Gaussian Continuous Processes [60.35125735474386]
  We show that discretizations yielding consistent estimators have the property of invariance under coarse-graining' This result explains why combining differencing schemes for derivatives reconstruction and local-in-time inference approaches does not work for time series analysis of second or higher order differential equations.
  arXiv  Detail & Related papers  (2021-01-16T17:11:02Z)
• Optimal oracle inequalities for solving projected fixed-point equations [53.31620399640334]
  We study methods that use a collection of random observations to compute approximate solutions by searching over a known low-dimensional subspace of the Hilbert space. We show how our results precisely characterize the error of a class of temporal difference learning methods for the policy evaluation problem with linear function approximation.
  arXiv  Detail & Related papers  (2020-12-09T20:19:32Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.