Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Metric-based Principal Curve Approach for Learning One-dimensional Manifold

Eliuvish Cuicizion

TL;DR

The paper tackles learning a one-dimensional manifold from high-dimensional data by introducing Metric-based Principal Curve (MPC). MPC optimizes a projection-index loss that combines a metric distance to a smoothed curve with a dispersion-penalty on the projection indices, enabling flexible, dimension-wise manifold learning via choices of smoothing functions, distance metrics, and regularization. Through synthetic experiments on generative models including Number Seven, Spiral Curve, and Golden Bridge, and an MNIST-based application after UMAP projection to 3D, MPC demonstrates robust recovery of latent 1D trajectories. The work fuses differential-geometry-inspired concepts with practical smoothing and regression techniques, offering a versatile framework for intrinsic ordering of points in high-dimensional spaces and providing supporting appendix on Riemannian geometry.

Abstract

Principal curve is a well-known statistical method oriented in manifold learning using concepts from differential geometry. In this paper, we propose a novel metric-based principal curve (MPC) method that learns one-dimensional manifold of spatial data. Synthetic datasets Real applications using MNIST dataset show that our method can learn the one-dimensional manifold well in terms of the shape.

A Metric-based Principal Curve Approach for Learning One-dimensional Manifold

TL;DR

The paper tackles learning a one-dimensional manifold from high-dimensional data by introducing Metric-based Principal Curve (MPC). MPC optimizes a projection-index loss that combines a metric distance to a smoothed curve with a dispersion-penalty on the projection indices, enabling flexible, dimension-wise manifold learning via choices of smoothing functions, distance metrics, and regularization. Through synthetic experiments on generative models including Number Seven, Spiral Curve, and Golden Bridge, and an MNIST-based application after UMAP projection to 3D, MPC demonstrates robust recovery of latent 1D trajectories. The work fuses differential-geometry-inspired concepts with practical smoothing and regression techniques, offering a versatile framework for intrinsic ordering of points in high-dimensional spaces and providing supporting appendix on Riemannian geometry.

Abstract

Principal curve is a well-known statistical method oriented in manifold learning using concepts from differential geometry. In this paper, we propose a novel metric-based principal curve (MPC) method that learns one-dimensional manifold of spatial data. Synthetic datasets Real applications using MNIST dataset show that our method can learn the one-dimensional manifold well in terms of the shape.
Paper Structure (9 sections, 1 theorem, 9 equations, 3 figures, 1 table)

This paper contains 9 sections, 1 theorem, 9 equations, 3 figures, 1 table.

Table of Contents

  1. A Brief Review on Differential Geometry in Statistics
  2. Metric-based Principal Curve
  3. Simulation Studies
  4. Number Seven
  5. Spiral Curve
  6. Golden Bridge
  7. Applications to MNIST data
  8. Conclusion
  9. Appendix: Some Preliminaries in Riemann Geometry

Key Result

Theorem 4

At minima of $L(t,c,c')$, the Euler-Lagrange equation must hold, i.e., Hence, geodesic curves embedded in $\mathbb{R}^d$ satisfy the following system of second-order ordinal differential equations (ODE): where $\otimes$ denotes the Kronecker product and $\text{vec}(\cdot)$ stacks the columns of a matrix into a vector.

Figures (3)

  • Figure 1: Principal curves of seven, spiral and bridge in $\mathbb{R}^3$. Red lines are learned principal curves which represent the trajectory of data manifold.
  • Figure 2: Principal curves of seven, spiral and bridge in $\mathbb{R}^2$. Red lines and colorful points are learned principal curves which represent the trajectory of data manifold. Blue points are raw data in $\mathbb{R}^2$.
  • Figure 3: Principal curves of MNIST. Blue lines are learned principal curves which represent the trajectory of data manifold.

Theorems & Definitions (4)

  • Definition 1: Principal curve assumption
  • Definition 2: Riemannian manifold
  • Definition 3: Geodesic
  • Theorem 4: Euler-Lagrange equation for geodesic hauberg2012geometric