Skeleton Regression: A Dimension-Independent Approach to Manifold-Structured Data Estimation

Belangrijkste concepten

The proposed skeleton regression framework combines graph-based representation of the underlying manifold structure and nonparametric regression techniques to efficiently estimate the regression function for data with complex geometric structure.

Samenvatting

The content introduces a new regression framework called "Skeleton Regression" designed to handle large-scale, complex data that lies around a low-dimensional manifold with noise. The key idea is to first construct a graph representation, referred to as the "skeleton", to capture the underlying geometric structure of the data. The regression function is then estimated on this skeleton graph using various nonparametric regression techniques, such as kernel smoothing, k-nearest neighbor, and linear spline.
The authors first describe the procedure to construct the skeleton graph from the data, which involves identifying representative points (knots) and connecting them based on the 2-nearest neighbor regions. They then define a skeleton-based distance metric and show how to project the original covariates onto the skeleton.
Next, the authors apply different nonparametric regression methods on the skeleton graph. For kernel regression, they analyze the convergence properties separately for edge points, knots with nonzero mass, and knots with zero mass. For k-nearest neighbor regression, they adapt the method to use the skeleton-based distance. For linear spline regression, they provide an elegant parametric representation using the values at the knots.
The authors also discuss the challenges in applying other nonparametric regression techniques like local polynomial regression, higher-order spline, and orthonormal basis on the skeleton graph due to the lack of well-defined orientation and derivatives.
Finally, the authors demonstrate the effectiveness of the skeleton regression framework through simulations and real data examples, showing its advantages in handling data with underlying geometric structures, additive noise, and noisy observations.

Statistieken

The regression response increases polynomially with the angle and the radius of the two-moon shaped covariates.
The skeleton construction procedure divides the covariate space into a given number of disjoint components.

Citaten

"The main goal of this work is to estimate a scalar response with covariates lying around some manifold structures in a way that utilizes the geometric structure and bypasses the curse of dimensionality."
"The proposed regression framework in this work also adapts to the manifold, as the nonparametric regression models fitted on a graph are dimension-independent."

Belangrijkste Inzichten Gedestilleerd Uit

Skeleton Regression: A Graph-Based Approach to Estimation with Manifold Structure

by Zeyu Wei,Yen... om arxiv.org 05-03-2024

https://arxiv.org/pdf/2303.11786.pdf

Diepere vragen

How can the skeleton construction procedure be further improved to better capture the underlying manifold structure?

The skeleton construction procedure can be enhanced by incorporating more advanced techniques from manifold learning and graph theory. One approach is to utilize local manifold learning algorithms to estimate the local structure around each knot more accurately. This can involve techniques such as local principal component analysis or local linear embedding to capture the intrinsic geometry of the data more effectively. Additionally, incorporating information about the density distribution of the data points can help in determining the regions of interest more precisely. By considering the density of the data points, the skeleton construction can focus on areas with higher data concentration, leading to a more accurate representation of the underlying manifold structure.

What are the potential limitations of the skeleton regression framework in handling highly complex or discontinuous regression functions?

While the skeleton regression framework offers a novel approach to estimating regression functions on manifolds, it may face limitations when dealing with highly complex or discontinuous regression functions. One limitation is the assumption of linearity between knots on the skeleton, which may not capture the intricate relationships in highly complex functions. Discontinuous regression functions pose a challenge as the linear interpolation between knots may not accurately represent abrupt changes in the regression function. Additionally, the skeleton regression framework may struggle with functions that exhibit non-linear behavior or have sharp transitions, as the linear spline model may not be flexible enough to capture such complexities. In such cases, more advanced regression techniques that can handle non-linearity and discontinuities may be more suitable.

Can the skeleton regression framework be extended to handle functional responses or multiple response variables defined on manifolds?

Yes, the skeleton regression framework can be extended to handle functional responses or multiple response variables defined on manifolds by modifying the regression models and data projection techniques. For functional responses, the regression models can be adapted to predict a function value at each point on the manifold, allowing for the estimation of a functional relationship between the covariates and the response. This can involve using functional regression techniques or modeling the response as a curve on the manifold. Similarly, for multiple response variables, the regression models can be extended to predict multiple outcomes simultaneously, considering the relationships between the variables on the manifold. By incorporating appropriate transformations and adjustments in the regression framework, it is possible to accommodate functional responses and multiple response variables within the skeleton regression approach.