Data-intrinsic approximation in metric spaces

• URL: http://arxiv.org/abs/2510.13496v1
• Date: Wed, 15 Oct 2025 12:45:29 GMT
• Title: Data-intrinsic approximation in metric spaces
• Authors: Jürgen Dölz, Michael Multerer,
• Abstract summary: We consider the approximation of labeled data samples, mathematically described as site-to-value maps between finite metric spaces.<n>We propose an algorithm for its efficient computation and present a sample based approximation theory for labeled data.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Analysis and processing of data is a vital part of our modern society and requires vast amounts of computational resources. To reduce the computational burden, compressing and approximating data has become a central topic. We consider the approximation of labeled data samples, mathematically described as site-to-value maps between finite metric spaces. Within this setting, we identify the discrete modulus of continuity as an effective data-intrinsic quantity to measure regularity of site-to-value maps without imposing further structural assumptions. We investigate the consistency of the discrete modulus of continuity in the infinite data limit and propose an algorithm for its efficient computation. Building on these results, we present a sample based approximation theory for labeled data. For data subject to statistical uncertainty we consider multilevel approximation spaces and a variant of the multilevel Monte Carlo method to compute statistical quantities of interest. Our considerations connect approximation theory for labeled data in metric spaces to the covering problem for (random) balls on the one hand and the efficient evaluation of the discrete modulus of continuity to combinatorial optimization on the other hand. We provide extensive numerical studies to illustrate the feasibility of the approach and to validate our theoretical results.

Related papers

• Efficient Covariance Estimation for Sparsified Functional Data [51.69796254617083]
  proposed Random-knots (Random-knots-Spatial) and B-spline (Bspline-Spatial) estimators of the covariance function are computationally efficient.<n>Asymptotic pointwise of the covariance are obtained for sparsified individual trajectories under some regularity conditions.
  arXiv  Detail & Related papers  (2025-11-23T00:50:33Z)
• Statistical Inference in Tensor Completion: Optimal Uncertainty Quantification and Statistical-to-Computational Gaps [7.174572371800217]
  This paper presents a simple yet efficient method for statistical inference of tensor linear forms using incomplete and noisy observations. It is suitable for various statistical inference tasks, including constructing confidence intervals, inference under heteroskedastic and sub-exponential noise, and simultaneous testing.
  arXiv  Detail & Related papers  (2024-10-15T03:09:52Z)
• Density Estimation via Binless Multidimensional Integration [45.21975243399607]
  We introduce the Binless Multidimensional Thermodynamic Integration (BMTI) method for nonparametric, robust, and data-efficient density estimation. BMTI estimates the logarithm of the density by initially computing log-density differences between neighbouring data points. The method is tested on a variety of complex synthetic high-dimensional datasets, and is benchmarked on realistic datasets from the chemical physics literature.
  arXiv  Detail & Related papers  (2024-07-10T23:45:20Z)
• Synthetic Tabular Data Validation: A Divergence-Based Approach [8.062368743143388]
  Divergences quantify discrepancies between data distributions. Traditional approaches calculate divergences independently for each feature. We propose a novel approach that uses divergence estimation to overcome the limitations of marginal comparisons.
  arXiv  Detail & Related papers  (2024-05-13T15:07:52Z)
• Scalable Bayesian inference for the generalized linear mixed model [2.45365913654612]
  We introduce a statistical inference algorithm at the intersection of AI and Bayesian inference. Our algorithm is an extension of gradient MCMC with novel contributions that address the treatment of correlated data. We apply our algorithm to a large electronic health records database.
  arXiv  Detail & Related papers  (2024-03-05T14:35:34Z)
• Minimally Supervised Learning using Topological Projections in Self-Organizing Maps [55.31182147885694]
  We introduce a semi-supervised learning approach based on topological projections in self-organizing maps (SOMs) Our proposed method first trains SOMs on unlabeled data and then a minimal number of available labeled data points are assigned to key best matching units (BMU) Our results indicate that the proposed minimally supervised model significantly outperforms traditional regression techniques.
  arXiv  Detail & Related papers  (2024-01-12T22:51:48Z)
• Learning to Bound Counterfactual Inference in Structural Causal Models from Observational and Randomised Data [64.96984404868411]
  We derive a likelihood characterisation for the overall data that leads us to extend a previous EM-based algorithm. The new algorithm learns to approximate the (unidentifiability) region of model parameters from such mixed data sources. It delivers interval approximations to counterfactual results, which collapse to points in the identifiable case.
  arXiv  Detail & Related papers  (2022-12-06T12:42:11Z)
• Partial Counterfactual Identification from Observational and Experimental Data [83.798237968683]
  We develop effective Monte Carlo algorithms to approximate the optimal bounds from an arbitrary combination of observational and experimental data. Our algorithms are validated extensively on synthetic and real-world datasets.
  arXiv  Detail & Related papers  (2021-10-12T02:21:30Z)
• Distributed Learning of Finite Gaussian Mixtures [21.652015112462]
  We study split-and-conquer approaches for the distributed learning of finite Gaussian mixtures. New estimator is shown to be consistent and retains root-n consistency under some general conditions. Experiments based on simulated and real-world data show that the proposed split-and-conquer approach has comparable statistical performance with the global estimator.
  arXiv  Detail & Related papers  (2020-10-20T16:17:47Z)
• Instability, Computational Efficiency and Statistical Accuracy [101.32305022521024]
  We develop a framework that yields statistical accuracy based on interplay between the deterministic convergence rate of the algorithm at the population level, and its degree of (instability) when applied to an empirical object based on $n$ samples. We provide applications of our general results to several concrete classes of models, including Gaussian mixture estimation, non-linear regression models, and informative non-response models.
  arXiv  Detail & Related papers  (2020-05-22T22:30:52Z)
• 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)
• A Robust Functional EM Algorithm for Incomplete Panel Count Data [66.07942227228014]
  We propose a functional EM algorithm to estimate the counting process mean function under a missing completely at random assumption (MCAR) The proposed algorithm wraps several popular panel count inference methods, seamlessly deals with incomplete counts and is robust to misspecification of the Poisson process assumption. We illustrate the utility of the proposed algorithm through numerical experiments and an analysis of smoking cessation data.
  arXiv  Detail & Related papers  (2020-03-02T20:04:38Z)

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.