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Mode and Ridge Estimation in Euclidean and Directional Product Spaces: A Mean Shift Approach

Yikun Zhang, Yen-Chi Chen

TL;DR

This work extends the (subspace constrained) mean shift algorithm to such product spaces, addressing potential challenges in the generalization process, and establishes the algorithmic convergence of the proposed methods, along with practical implementation guidelines.

Abstract

The set of local modes and density ridge lines are important summary characteristics of the data-generating distribution. In this work, we focus on estimating local modes and density ridges from point cloud data in a product space combining two or more Euclidean and/or directional metric spaces. Specifically, our approach extends the (subspace constrained) mean shift algorithm to such product spaces, addressing potential challenges in the generalization process. We establish the algorithmic convergence of the proposed methods, along with practical implementation guidelines. Experiments on simulated and real-world datasets demonstrate the effectiveness of our proposed methods.

Mode and Ridge Estimation in Euclidean and Directional Product Spaces: A Mean Shift Approach

TL;DR

This work extends the (subspace constrained) mean shift algorithm to such product spaces, addressing potential challenges in the generalization process, and establishes the algorithmic convergence of the proposed methods, along with practical implementation guidelines.

Abstract

The set of local modes and density ridge lines are important summary characteristics of the data-generating distribution. In this work, we focus on estimating local modes and density ridges from point cloud data in a product space combining two or more Euclidean and/or directional metric spaces. Specifically, our approach extends the (subspace constrained) mean shift algorithm to such product spaces, addressing potential challenges in the generalization process. We establish the algorithmic convergence of the proposed methods, along with practical implementation guidelines. Experiments on simulated and real-world datasets demonstrate the effectiveness of our proposed methods.

Paper Structure

This paper contains 18 sections, 10 theorems, 90 equations, 6 figures, 5 tables.

Table of Contents

  1. Experimental Setup and Additional Results
  2. Bandwidth Selection
  3. Manifold-Recovering Error Measure
  4. Simulation 1: Mode-Seeking on $\Omega_1 \times \mathbb{R}$
  5. Simulation 2: Mode-Seeking on $\Omega_1\times \Omega_1$
  6. Simulation 3: Ridge-Finding on $\Omega_2 \times \mathbb{R}$
  7. Simulation 4: Surface-Recovering on $\Omega_2\times \mathbb{R}$
  8. Statistical Consistency
  9. Assumptions and Consistency of KDE
  10. Mode Consistency
  11. Ridge Consistency
  12. Linear Convergence of the Mean Shift and SCMS Algorithms on $\mathcal{S}_1\times \mathcal{S}_2$
  13. (Subspace Constrained) Gradient Ascent Framework on $\mathcal{S}_1\times \mathcal{S}_2$
  14. Intrinsic Step Size of the Proposed Mean Shift Algorithms Under the Gradient Ascent Framework on $\mathcal{S}_1\times \mathcal{S}_2$
  15. Intrinsic Step Size of the Proposed SCMS Algorithm Under the Subspace Constrained Gradient Ascent Framework on $\mathcal{S}_1\times \mathcal{S}_2$
  16. ...and 3 more sections

Key Result

proposition 1

Let $\hat{f}_{\bm{h}}(\bm{x},\bm{y})$ in KDE_prod be defined on the product space $\mathcal{S}_1\times \mathcal{S}_2$ with $\bm{h}=(h_1,h_2)$. If the von Mises kernel profile $L(r)=e^{-r}$ is used for directional components and Gaussian kernel profile $k(s) = e^{-s/2}$ is applied to Euclidean compon (b) When $\mathcal{S}_1\times \mathcal{S}_2 = \Omega_{D_1}\times \Omega_{D_2}$, where $C_q(\kappa)

Figures (6)

  • Figure 1: Local modes obtained by our mean shift algorithm on the simulated DirLin and DirDir data. In each panel, the red dots are estimated local modes while the blue dots are simulated points.
  • Figure 2: Estimated ridges obtained by various SCMS algorithms on the spherical cone data. In each panel, the red surface is the hidden manifold structure while the blue dots are final convergent points of the corresponding SCMS algorithm.
  • Figure 3: One-step iteration of the (simultaneous) mean shift algorithm on $\Omega_{D_1}\times \mathbb{R}^{D_2}$ projected onto $\Omega_{D_1}$.
  • Figure 4: One-step iteration of the proposed SCMS algorithm on $\Omega_{D_1}\times \mathbb{R}^{D_2}$ projected onto $\Omega_{D_1}$.
  • Figure 5: Proposed SCMS algorithm with different choices of the step size $\eta$ on the simulated directional-linear dataset. In Panels (b-e), the blue points are the final convergent points of the SCMS algorithm under a certain step size $\eta$, while the green curve indicates the underlying curve structure.
  • ...and 1 more figures

Theorems & Definitions (16)

  • proposition 1: Explicit LSCV loss under von Mises and/or Gaussian kernels
  • remark 1
  • lemma 1
  • lemma 2
  • theorem 1
  • theorem 2
  • lemma 3
  • lemma 4
  • lemma 5
  • proof : Proof of Theorem \ref{['Thm:MS_lin_conv']}
  • ...and 6 more