Core Concepts
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.
Abstract
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.
Stats
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.
Quotes
"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."