l_inf-approximation of localized distributions

• URL: http://arxiv.org/abs/2410.11771v1
• Date: Tue, 15 Oct 2024 16:47:05 GMT
• Title: l_inf-approximation of localized distributions
• Authors: Tiangang Cui, Shuigen Liu, Xin Tong,
• Abstract summary: In this work, we establish a dimension independent error bound for the marginals of approximate distributions. We show how to use localized likelihood-informed subspace method and localized score matching, which not only avoid dimension dependence in the approximation error, but also significantly reduce the computational cost.
• Score: 11.211950982996784
• License:
• Abstract: Distributions in spatial model often exhibit localized features. Intuitively, this locality implies a low intrinsic dimensionality, which can be exploited for efficient approximation and computation of complex distributions. However, existing approximation theory mainly considers the joint distributions, which does not guarantee that the marginal errors are small. In this work, we establish a dimension independent error bound for the marginals of approximate distributions. This $\ell_\infty$-approximation error is obtained using Stein's method, and we propose a $\delta$-locality condition that quantifies the degree of localization in a distribution. We also show how $\delta$-locality can be derived from different conditions that characterize the distribution's locality. Our $\ell_\infty$ bound motivates the localization of existing approximation methods to respect the locality. As examples, we show how to use localized likelihood-informed subspace method and localized score matching, which not only avoid dimension dependence in the approximation error, but also significantly reduce the computational cost due to the local and parallel implementation based on the localized structure.

Related papers

• Adaptive $k$-nearest neighbor classifier based on the local estimation of the shape operator [49.87315310656657]
  We introduce a new adaptive $k$-nearest neighbours ($kK$-NN) algorithm that explores the local curvature at a sample to adaptively defining the neighborhood size. Results on many real-world datasets indicate that the new $kK$-NN algorithm yields superior balanced accuracy compared to the established $k$-NN method.
  arXiv  Detail & Related papers  (2024-09-08T13:08:45Z)
• MASALA: Model-Agnostic Surrogate Explanations by Locality Adaptation [3.587367153279351]
  Existing local Explainable AI (XAI) methods select a region of the input space in the vicinity of a given input instance, for which they approximate the behaviour of a model using a simpler and more interpretable surrogate model. We propose a novel method, MASALA, for generating explanations, which automatically determines the appropriate local region of impactful model behaviour for each individual instance being explained.
  arXiv  Detail & Related papers  (2024-08-19T15:26:45Z)
• Rejection via Learning Density Ratios [50.91522897152437]
  Classification with rejection emerges as a learning paradigm which allows models to abstain from making predictions. We propose a different distributional perspective, where we seek to find an idealized data distribution which maximizes a pretrained model's performance. Our framework is tested empirically over clean and noisy datasets.
  arXiv  Detail & Related papers  (2024-05-29T01:32:17Z)
• Guarantee Regions for Local Explanations [29.429229877959663]
  We propose an anchor-based algorithm for identifying regions in which local explanations are guaranteed to be correct. Our method produces an interpretable feature-aligned box where the prediction of the local surrogate model is guaranteed to match the predictive model.
  arXiv  Detail & Related papers  (2024-02-20T06:04:44Z)
• Optimal Scaling for Locally Balanced Proposals in Discrete Spaces [65.14092237705476]
  We show that efficiency of Metropolis-Hastings (M-H) algorithms in discrete spaces can be characterized by an acceptance rate that is independent of the target distribution. Knowledge of the optimal acceptance rate allows one to automatically tune the neighborhood size of a proposal distribution in a discrete space, directly analogous to step-size control in continuous spaces.
  arXiv  Detail & Related papers  (2022-09-16T22:09:53Z)
• Arbitrary Conditional Distributions with Energy [11.081460215563633]
  A more general and useful problem is arbitrary conditional density estimation. We propose a novel method, Arbitrary Conditioning with Energy (ACE), that can simultaneously estimate the distribution $p(mathbfx_u mid mathbfx_o)$. We also simplify the learning problem by only learning one-dimensional conditionals, from which more complex distributions can be recovered during inference.
  arXiv  Detail & Related papers  (2021-02-08T18:36:26Z)
• Linear Optimal Transport Embedding: Provable Wasserstein classification for certain rigid transformations and perturbations [79.23797234241471]
  Discriminating between distributions is an important problem in a number of scientific fields. The Linear Optimal Transportation (LOT) embeds the space of distributions into an $L2$-space. We demonstrate the benefits of LOT on a number of distribution classification problems.
  arXiv  Detail & Related papers  (2020-08-20T19:09:33Z)
• Debiasing Distributed Second Order Optimization with Surrogate Sketching and Scaled Regularization [101.5159744660701]
  In distributed second order optimization, a standard strategy is to average many local estimates, each of which is based on a small sketch or batch of the data. Here, we introduce a new technique for debiasing the local estimates, which leads to both theoretical and empirical improvements in the convergence rate of distributed second order methods.
  arXiv  Detail & Related papers  (2020-07-02T18:08:14Z)
• Log-Likelihood Ratio Minimizing Flows: Towards Robust and Quantifiable Neural Distribution Alignment [52.02794488304448]
  We propose a new distribution alignment method based on a log-likelihood ratio statistic and normalizing flows. We experimentally verify that minimizing the resulting objective results in domain alignment that preserves the local structure of input domains.
  arXiv  Detail & Related papers  (2020-03-26T22:10:04Z)

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.