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On the Dynamical System of Principal Curves in $\mathbb R^d$

Robert Beinert, Arian Bërdëllima, Manuel Gräf, Gabriele Steidl

TL;DR

This work generalizes principal curves from the plane to $\mathbb{R}^d$ with $d\ge3$, formulating a differential-equation description that characterizes self-consistent curves as critical points of the energy $D_{\bm X}^2(\Gamma)$. Building on stochastic and variational viewpoints, it introduces a moving-frame construction and transverse moments to express self-consistency as an explicit ODE system for $\Gamma(s)$ and its tangent directions. The authors then compute principal curves for uniformly distributed data on several 3D domains, showing symmetry-induced planar reductions, and derive concrete examples including rectangular triangles, triangular prisms, and helices on cylinders; these illustrate the method's versatility in higher dimensions. The results open avenues for extending principal-curve computations to other measures and manifolds, with potential applications in nonlinear dimensionality reduction and geometric data analysis, while also suggesting directions for robustness and generalizations.

Abstract

Principal curves are natural generalizations of principal lines arising as first principal components in the Principal Component Analysis. They can be characterized from a stochastic point of view as so-called self-consistent curves based on the conditional expectation and from the variational-calculus point of view as saddle points of the expected difference of a random variable and its projection onto some curve, where the current curve acts as argument of the energy functional. Beyond that, Duchamp and Stützle (1993,1996) showed that planar curves can by computed as solutions of a system of ordinary differential equations. The aim of this paper is to generalize this characterization of principal curves to $\mathbb R^d$ with $d \ge 3$. Having derived such a dynamical system, we provide several examples for principal curves related to uniform distribution on certain domains in $\mathbb R^3$.

On the Dynamical System of Principal Curves in $\mathbb R^d$

TL;DR

This work generalizes principal curves from the plane to with , formulating a differential-equation description that characterizes self-consistent curves as critical points of the energy . Building on stochastic and variational viewpoints, it introduces a moving-frame construction and transverse moments to express self-consistency as an explicit ODE system for and its tangent directions. The authors then compute principal curves for uniformly distributed data on several 3D domains, showing symmetry-induced planar reductions, and derive concrete examples including rectangular triangles, triangular prisms, and helices on cylinders; these illustrate the method's versatility in higher dimensions. The results open avenues for extending principal-curve computations to other measures and manifolds, with potential applications in nonlinear dimensionality reduction and geometric data analysis, while also suggesting directions for robustness and generalizations.

Abstract

Principal curves are natural generalizations of principal lines arising as first principal components in the Principal Component Analysis. They can be characterized from a stochastic point of view as so-called self-consistent curves based on the conditional expectation and from the variational-calculus point of view as saddle points of the expected difference of a random variable and its projection onto some curve, where the current curve acts as argument of the energy functional. Beyond that, Duchamp and Stützle (1993,1996) showed that planar curves can by computed as solutions of a system of ordinary differential equations. The aim of this paper is to generalize this characterization of principal curves to with . Having derived such a dynamical system, we provide several examples for principal curves related to uniform distribution on certain domains in .

Paper Structure

This paper contains 10 sections, 7 theorems, 66 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries in Probability Theory
  3. Principal Curves
  4. Characterization via Differential Equations
  5. Principal Curves of Uniformly Distributed Random Variables
  6. Symmetric and Rotation-Invariant Domains
  7. Rectangular Triangles and Squares
  8. Triangular-Based Prism
  9. Infinite Cylinder
  10. Conclusion

Key Result

Theorem 2.1

Let $(\Omega, \mathcal{A}, P)$ be a probability space, and let $\bm X:\Omega \to \mathbb R^d$ be a random vector with $\bm X \in L_1(\Omega,P)$. For any sub-$\sigma$-algebra $\mathcal{F} \subset \mathcal{A}$, there exists a random variable $\bm Z:\Omega \to \mathbb R^d$ with the following properties If $\tilde{\bm Z}:\Omega \to \mathbb R^d$ is another random vector satisfying (i) and (ii), then I

Figures (5)

  • Figure 1: Two examples for curves and uniformly distributed random variable $\bm X$. Left: $\bm X$ corresponds to the uniform distribution on the quarter circle of radius 1. Its principal curve is just the quarter circle of radius $\frac{2}{3}$. The barycenters of the Voronoi cells converge to the curve if the curve is sampled denser. Right: $\bm X$ corresponds to the uniform distribution on the square of side length 1. For a quarter parabolic curve (blue dots), we calculate the barycenters (red dots) of the corresponding Voronoi cells. These centers do not converge to the curve even if the regions become arbitrary thin; so the shown curve is not principal.
  • Figure 2: For simplicity, the region $\mathbb X$ (gray) is shown as cuboid. If the curve $\Gamma$ (blue) leaves the plane $\mathcal{H}_3$, the section $\lambda_\Gamma^{-1}(\bar{I})$ becomes non-symmetric. The additional region in $\mathcal{H}_3^-$, which is here schematically shown by the green, dashed line, pulls the conditional expectation into $\mathcal{H}_3^-$, whereas $\Gamma(I)$ is contained in $\mathcal{H}_3^+$.
  • Figure 3: Left: A solution to \ref{['eq:PCdgl_triangle_extended']} with $x_1(0)=1$, $x_2(0)=0$, $\zeta(0)=\frac{\pi}{2}$ (blue) and for comparison the diagonal (orange). Right: The derived non-trivial closed principal curve for the square (red) by composition of 8 identical principal curves on rectangular triangles. We show the normals (black) and the border of the projection domain.
  • Figure 4: Principal curve in the triangular prism whose base corresponds to the points $(0,1,0)$, $(\sqrt3/2,-1/2,0)$, $(-2,-1/2,0)$. The length of the prism is chosen such that both bases are parallel. The intersection of the normal planes $\mathbb X(s)$ with the surface of the prism are shown as red triangles. Since they do not intersect each other, the curve is admissible. The barycenters of the related Voronoi cells (green dots) numerically coincide with the curve. Note the scaling with respect to the $x_3$-axis.
  • Figure 5: Helices for several parameters $(a,b) \in \{(0.2, 0.5), (0.6, 0.35), (0.66, 0.1)\}$ (from top to bottom). The curves are sampled equidistantly (blue dots). For some sampling points the Voronoi regions intersected with the cylinder of radius $r=1$ are depicted in golden color. Note that for the second and third set of parameters the helices are not admissible curves for the infinite cylinder.

Theorems & Definitions (12)

  • Theorem 2.1
  • Theorem 3.1: Hastie & Stuetzle HS89
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • Theorem 5.1
  • proof
  • Corollary 5.2
  • proof
  • Example 5.3
  • ...and 2 more