Local regression on path spaces with signature metrics

• URL: http://arxiv.org/abs/2510.16728v1
• Date: Sun, 19 Oct 2025 06:40:51 GMT
• Title: Local regression on path spaces with signature metrics
• Abstract summary: We study non-parametric regression and classification for path-valued data.<n>We introduce a functional Nadaraya-Watson estimator that combines the signature transform from rough path theory with local kernel regression.
• Score: 0.669087470775851
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study nonparametric regression and classification for path-valued data. We introduce a functional Nadaraya-Watson estimator that combines the signature transform from rough path theory with local kernel regression. The signature transform provides a principled way to encode sequential data through iterated integrals, enabling direct comparison of paths in a natural metric space. Our approach leverages signature-induced distances within the classical kernel regression framework, achieving computational efficiency while avoiding the scalability bottlenecks of large-scale kernel matrix operations. We establish finite-sample convergence bounds demonstrating favorable statistical properties of signature-based distances compared to traditional metrics in infinite-dimensional settings. We propose robust signature variants that provide stability against outliers, enhancing practical performance. Applications to both synthetic and real-world data - including stochastic differential equation learning and time series classification - demonstrate competitive accuracy while offering significant computational advantages over existing methods.

Related papers

• Inversion-Free Natural Gradient Descent on Riemannian Manifolds [5.556612138252781]
  This paper proposes an inversion-free natural gradient method for probability distributions with convex parameters.<n>The manifold setting offers several advantages: one can implicitly enforce parameter constraints.<n>A limited-memory variant of the algorithm with sub-quadratic storage complexity is proposed.
  arXiv  Detail & Related papers  (2026-04-03T11:08:59Z)
• No-rank Tensor Decomposition Using Metric Learning [0.0]
  This paper introduces a no-rank tensor decomposition framework grounded in metric learning.<n>We provide theoretical guarantees for the framework's convergence and establish bounds on its metric properties.<n>Our approach achieves superior performance with smaller training datasets compared to transformer-based methods.
  arXiv  Detail & Related papers  (2025-11-03T18:21:53Z)
• Kernel Representation and Similarity Measure for Incomplete Data [55.62595187178638]
  Measuring similarity between incomplete data is a fundamental challenge in web mining, recommendation systems, and user behavior analysis.<n>Traditional approaches either discard incomplete data or perform imputation as a preprocessing step, leading to information loss and biased similarity estimates.<n>This paper presents a new similarity measure that directly computes similarity between incomplete data in kernel feature space without explicit imputation in the original space.
  arXiv  Detail & Related papers  (2025-10-15T09:41:23Z)
• Semi-parametric Functional Classification via Path Signatures Logistic Regression [1.210026603224224]
  We propose Path Signatures Logistic Regression, a semi-parametric framework for classifying vector-valued functional data.<n>Our results highlight the practical and theoretical benefits of integrating rough path theory into modern functional data analysis.
  arXiv  Detail & Related papers  (2025-07-09T08:06:50Z)
• Generalization Bound of Gradient Flow through Training Trajectory and Data-dependent Kernel [55.82768375605861]
  We establish a generalization bound for gradient flow that aligns with the classical Rademacher complexity for kernel methods.<n>Unlike static kernels such as NTK, the LPK captures the entire training trajectory, adapting to both data and optimization dynamics.
  arXiv  Detail & Related papers  (2025-06-12T23:17:09Z)
• Distances for Markov chains from sample streams [16.443304244634767]
  Bisimulation metrics are powerful tools for measuring similarities between processes, and specifically Markov chains.<n>Recent advances have uncovered that bisimulation metrics are, in fact, optimal-transport distances, which has enabled the development of fast algorithms for computing such metrics with provable accuracy and runtime guarantees.<n>This is often an impractical assumption in most real-world scenarios, where typically only sample trajectories are available.<n>We propose a new optimization method that addresses this limitation and estimates bisimulation metrics based on sample access, without requiring explicit transition models.
  arXiv  Detail & Related papers  (2025-05-23T15:09:04Z)
• Amortised inference of fractional Brownian motion with linear computational complexity [0.0]
  We introduce a simulation-based, amortised Bayesian inference scheme to infer the parameters of random walks. Our approach learns the posterior distribution of the walks' parameters with a likelihood-free method. We adapt this scheme to show that a finite decorrelation time in the environment can furthermore be inferred from individual trajectories.
  arXiv  Detail & Related papers  (2022-03-15T14:43:16Z)
• Scalable Variational Gaussian Processes via Harmonic Kernel Decomposition [54.07797071198249]
  We introduce a new scalable variational Gaussian process approximation which provides a high fidelity approximation while retaining general applicability. We demonstrate that, on a range of regression and classification problems, our approach can exploit input space symmetries such as translations and reflections. Notably, our approach achieves state-of-the-art results on CIFAR-10 among pure GP models.
  arXiv  Detail & Related papers  (2021-06-10T18:17:57Z)
• Fast and Robust Online Inference with Stochastic Gradient Descent via Random Scaling [0.9806910643086042]
  We develop a new method of online inference for a vector of parameters estimated by the Polyak-Rtupper averaging procedure of gradient descent algorithms. Our approach is fully operational with online data and is rigorously underpinned by a functional central limit theorem.
  arXiv  Detail & Related papers  (2021-06-06T15:38:37Z)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)
• Random extrapolation for primal-dual coordinate descent [61.55967255151027]
  We introduce a randomly extrapolated primal-dual coordinate descent method that adapts to sparsity of the data matrix and the favorable structures of the objective function. We show almost sure convergence of the sequence and optimal sublinear convergence rates for the primal-dual gap and objective values, in the general convex-concave case.
  arXiv  Detail & Related papers  (2020-07-13T17:39:35Z)
• Asymptotic Analysis of an Ensemble of Randomly Projected Linear Discriminants [94.46276668068327]
  In [1], an ensemble of randomly projected linear discriminants is used to classify datasets. We develop a consistent estimator of the misclassification probability as an alternative to the computationally-costly cross-validation estimator. We also demonstrate the use of our estimator for tuning the projection dimension on both real and synthetic data.
  arXiv  Detail & Related papers  (2020-04-17T12:47:04Z)
• Learning Flat Latent Manifolds with VAEs [16.725880610265378]
  We propose an extension to the framework of variational auto-encoders, where the Euclidean metric is a proxy for the similarity between data points. We replace the compact prior typically used in variational auto-encoders with a recently presented, more expressive hierarchical one. We evaluate our method on a range of data-sets, including a video-tracking benchmark.
  arXiv  Detail & Related papers  (2020-02-12T09:54:52Z)

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.