ResearchTrend.AI
  • Papers
  • Communities
  • Events
  • Blog
  • Pricing
Papers
Communities
Social Events
Terms and Conditions
Pricing
Parameter LabParameter LabTwitterGitHubLinkedInBlueskyYoutube

© 2025 ResearchTrend.AI, All rights reserved.

  1. Home
  2. Papers
  3. 2306.05722
21
0

Ridge Estimation with Nonlinear Transformations

9 June 2023
Zheng Zhai
Hengchao Chen
Zhigang Yao
ArXivPDFHTML
Abstract

Ridge estimation is an important manifold learning technique. The goal of this paper is to examine the effects of nonlinear transformations on the ridge sets. The main result proves the inclusion relationship between ridges: \cR(f∘p)⊆\cR(p)\cR(f\circ p)\subseteq \cR(p)\cR(f∘p)⊆\cR(p), provided that the transformation fff is strictly increasing and concave on the range of the function ppp. Additionally, given an underlying true manifold \cM\cM\cM, we show that the Hausdorff distance between \cR(f∘p)\cR(f\circ p)\cR(f∘p) and its projection onto \cM\cM\cM is smaller than the Hausdorff distance between \cR(p)\cR(p)\cR(p) and the corresponding projection. This motivates us to apply an increasing and concave transformation before the ridge estimation. In specific, we show that the power transformations fq(y)=yq/q,−∞<q≤1f^{q}(y)=y^q/q,-\infty<q\leq 1fq(y)=yq/q,−∞<q≤1 are increasing and concave on \RR+\RR_+\RR+​, and thus we can use such power transformations when ppp is strictly positive. Numerical experiments demonstrate the advantages of the proposed methods.

View on arXiv
Comments on this paper