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Algorithms for ridge estimation with convergence guarantees

Wanli Qiao, Wolfgang Polonik

TL;DR

This work addresses the problem of extracting filamentary ridge structures from data by modeling ridges as density-based manifolds characterized by eigenstructure of the density Hessian. It introduces two ODE-based ridge estimation algorithms that maximize a ridgeness functional and come with convergence guarantees to recover the full ridge set, addressing theoretical gaps in the SCMS method. The authors develop a rigorous framework using continuous-time flows and their Euler discretizations, establish convergence rates for ridge recovery (in Hausdorff distance), and provide conditions under which Ridge$(\widehat{f})$ converges to Ridge$(f)$. Empirical results on simulated and real data demonstrate robust ridge recovery, including in complex topologies like intersections, and show advantages over SCMS in terms of ridge connectivity and completeness. The work also discusses practical considerations, variants, and the mathematical implications for modeling and analyzing ridge-based filamentary structures.

Abstract

The extraction of filamentary structure from a point cloud is discussed. The filaments are modeled as ridge lines or higher dimensional ridges of an underlying density. We propose two novel algorithms, and provide theoretical guarantees for their convergences, by which we mean that the algorithms can asymptotically recover the full ridge set. We consider the new algorithms as alternatives to the Subspace Constrained Mean Shift (SCMS) algorithm for which no such theoretical guarantees are known.

Algorithms for ridge estimation with convergence guarantees

TL;DR

This work addresses the problem of extracting filamentary ridge structures from data by modeling ridges as density-based manifolds characterized by eigenstructure of the density Hessian. It introduces two ODE-based ridge estimation algorithms that maximize a ridgeness functional and come with convergence guarantees to recover the full ridge set, addressing theoretical gaps in the SCMS method. The authors develop a rigorous framework using continuous-time flows and their Euler discretizations, establish convergence rates for ridge recovery (in Hausdorff distance), and provide conditions under which Ridge converges to Ridge. Empirical results on simulated and real data demonstrate robust ridge recovery, including in complex topologies like intersections, and show advantages over SCMS in terms of ridge connectivity and completeness. The work also discusses practical considerations, variants, and the mathematical implications for modeling and analyzing ridge-based filamentary structures.

Abstract

The extraction of filamentary structure from a point cloud is discussed. The filaments are modeled as ridge lines or higher dimensional ridges of an underlying density. We propose two novel algorithms, and provide theoretical guarantees for their convergences, by which we mean that the algorithms can asymptotically recover the full ridge set. We consider the new algorithms as alternatives to the Subspace Constrained Mean Shift (SCMS) algorithm for which no such theoretical guarantees are known.

Paper Structure

This paper contains 39 sections, 11 theorems, 176 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Extraction of filamentary structures
  3. Definition
  4. Mean shift algorithm and subspace constrained mean shift algorithm
  5. The SCMS algorithm might miss some parts of the ridge
  6. Measuring ridgeness
  7. Notation
  8. Algorithms
  9. Practical implementation and illustration of the algorithms
  10. Pre-processing
  11. Post-processing
  12. Other more practical aspects
  13. Simulation study
  14. Real Data Application
  15. Main results
  16. ...and 24 more sections

Key Result

Lemma 3

Under (A1)$_{f,4}$, (A2)$_f$, and (A3)$_f$, $\text{Ridge}(f)$ is a compact set; we have ${\lambda_{1}^{\eta}(x)} = \cdots = {\lambda_{k}^{\eta}(x)}=0$ for all $x \in \text{Ridge}(f)$, and there exist positive constants $A, \alpha$ and $\delta^\prime \le \delta$, such that for all $x \in \text{Rid Moreover, the columns of $V^\eta_\perp(x)$ span the normal space of $\text{Ridge}(f)$ at $x\in\text

Figures (11)

  • Figure 2.1: X-cross example with 200 data points, where the black solid dots are data points; the blue circles are the points removed by pre-processing; the red dot is the point removed by post-processing; the green dots are the final output of the algorithm. The right panel shows the final result with the red dot and blue circles removed from the left panel.
  • Figure 2.2: Sorted ridgeness values of the output of Algorithm 2 run on the 200 data points with X-cross as ridges without imposing the threshold $\varepsilon_{\eta}$. A clear jump is visible between the ridgeness values close to zero and those of spurious ridge points. This observation then informed our choice of $\varepsilon_{\eta}=-0.1$ (red line). The point below this threshold corresponds to the red dot in the left panel of Figure \ref{['cross-sin']}.
  • Figure 2.3: Circle: 200 data points.
  • Figure 2.4: Spiral: 200 data points.
  • Figure 2.5: Plots of errors for Model 1 (upper panel) and Model 2 (lower panel), each using 200 random samples, where the heights of the red, green and blue triangles represent the errors in the estimation for Algorithm 1, Algorithm 2 and SCMS, respectively.
  • ...and 6 more figures

Theorems & Definitions (19)

  • Definition 1
  • Remark 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Remark 7
  • Theorem 8
  • Remark 9
  • Theorem 10
  • ...and 9 more