Manifold Fitting under Unbounded Noise

• URL: http://arxiv.org/abs/1909.10228v3
• Date: Sun, 9 Jun 2024 06:22:48 GMT
• Title: Manifold Fitting under Unbounded Noise
• Authors: Zhigang Yao, Yuqing Xia,
• Abstract summary: We introduce a new manifold-fitting method, by which the output manifold is constructed by directly estimating the tangent spaces at the projected points on the underlying manifold. Our new method provides theoretical convergence in high probability, in terms of the upper bound of the distance between the estimated and underlying manifold.
• Score: 4.54773250519101
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: There has been an emerging trend in non-Euclidean statistical analysis of aiming to recover a low dimensional structure, namely a manifold, underlying the high dimensional data. Recovering the manifold requires the noise to be of certain concentration. Existing methods address this problem by constructing an approximated manifold based on the tangent space estimation at each sample point. Although theoretical convergence for these methods is guaranteed, either the samples are noiseless or the noise is bounded. However, if the noise is unbounded, which is a common scenario, the tangent space estimation at the noisy samples will be blurred. Fitting a manifold from the blurred tangent space might increase the inaccuracy. In this paper, we introduce a new manifold-fitting method, by which the output manifold is constructed by directly estimating the tangent spaces at the projected points on the underlying manifold, rather than at the sample points, to decrease the error caused by the noise. Assuming the noise is unbounded, our new method provides theoretical convergence in high probability, in terms of the upper bound of the distance between the estimated and underlying manifold. The smoothness of the estimated manifold is also evaluated by bounding the supremum of twice difference above. Numerical simulations are provided to validate our theoretical findings and demonstrate the advantages of our method over other relevant manifold fitting methods. Finally, our method is applied to real data examples.

Related papers

• Sampling and estimation on manifolds using the Langevin diffusion [45.57801520690309]
  Two estimators of linear functionals of $mu_phi $ based on the discretized Markov process are considered. Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion.
  arXiv  Detail & Related papers  (2023-12-22T18:01:11Z)
• Robust Inference of Manifold Density and Geometry by Doubly Stochastic Scaling [8.271859911016719]
  We develop tools for robust inference under high-dimensional noise. We show that our approach is robust to variability in technical noise levels across cell types.
  arXiv  Detail & Related papers  (2022-09-16T15:39:11Z)
• Improving Diffusion Models for Inverse Problems using Manifold Constraints [55.91148172752894]
  We show that current solvers throw the sample path off the data manifold, and hence the error accumulates. To address this, we propose an additional correction term inspired by the manifold constraint. We show that our method is superior to the previous methods both theoretically and empirically.
  arXiv  Detail & Related papers  (2022-06-02T09:06:10Z)
• Inferring Manifolds From Noisy Data Using Gaussian Processes [17.166283428199634]
  Most existing manifold learning algorithms replace the original data with lower dimensional coordinates. This article proposes a new methodology for addressing these problems, allowing the estimated manifold between fitted data points.
  arXiv  Detail & Related papers  (2021-10-14T15:50:38Z)
• Residual Overfit Method of Exploration [78.07532520582313]
  We propose an approximate exploration methodology based on fitting only two point estimates, one tuned and one overfit. The approach drives exploration towards actions where the overfit model exhibits the most overfitting compared to the tuned model. We compare ROME against a set of established contextual bandit methods on three datasets and find it to be one of the best performing.
  arXiv  Detail & Related papers  (2021-10-06T17:05:33Z)
• Manifold Hypothesis in Data Analysis: Double Geometrically-Probabilistic Approach to Manifold Dimension Estimation [92.81218653234669]
  We present new approach to manifold hypothesis checking and underlying manifold dimension estimation. Our geometrical method is a modification for sparse data of a well-known box-counting algorithm for Minkowski dimension calculation. Experiments on real datasets show that the suggested approach based on two methods combination is powerful and effective.
  arXiv  Detail & Related papers  (2021-07-08T15:35:54Z)
• Hard-label Manifolds: Unexpected Advantages of Query Efficiency for Finding On-manifold Adversarial Examples [67.23103682776049]
  Recent zeroth order hard-label attacks on image classification models have shown comparable performance to their first-order, gradient-level alternatives. It was recently shown in the gradient-level setting that regular adversarial examples leave the data manifold, while their on-manifold counterparts are in fact generalization errors. We propose an information-theoretic argument based on a noisy manifold distance oracle, which leaks manifold information through the adversary's gradient estimate.
  arXiv  Detail & Related papers  (2021-03-04T20:53:06Z)
• LAAT: Locally Aligned Ant Technique for discovering multiple faint low dimensional structures of varying density [0.0]
  In manifold learning, several studies indicate solutions for removing background noise or noise close to the structure when the density is substantially higher than that exhibited by the noise. We propose a novel method to extract manifold points in the presence of noise based on the idea of Ant colony optimization. In contrast to the existing random walk solutions, our technique captures points that are locally aligned with major directions of the manifold.
  arXiv  Detail & Related papers  (2020-09-17T14:22:50Z)
• Manifold Learning via Manifold Deflation [105.7418091051558]
  dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. Many popular methods can fail dramatically, even on simple two-dimensional Manifolds. This paper presents an embedding method for a novel, incremental tangent space estimator that incorporates global structure as coordinates. Empirically, we show our algorithm recovers novel and interesting embeddings on real-world and synthetic datasets.
  arXiv  Detail & Related papers  (2020-07-07T10:04:28Z)

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.