A direct approach for function approximation on data defined manifolds.

Area of Science:

• Mathematics
• Computer Science
• Machine Learning

Background:

• Traditional function approximation methods often assume known domains (e.g., cubes, spheres), leading to conservative results when data is sparse.
• Manifold learning assumes data defines an unknown manifold, requiring multi-stage procedures like approximating the Laplace-Beltrami operator or creating manifold atlases for function approximation.

Purpose of the Study:

• To propose a direct function approximation method for unknown, data-defined manifolds.
• To develop universal constructions that do not require prior knowledge of the target function beyond continuity.
• To enhance function approximation capabilities in deep networks operating on manifolds.

Main Methods:

• A novel, direct approach to function approximation on manifolds defined by data.
• Avoidance of computing the eigen-decomposition of operators or constructing manifold atlases.
• Development of constructions that are universal and do not require classical training phases.

Main Results:

• The proposed method provides direct function approximation without complex intermediate steps.
• Approximation error degree estimation is provided, avoiding the saturation phenomenon for smooth functions.
• Demonstration of how results can be applied to deep networks via good error propagation.

Conclusions:

• The study presents a more efficient and direct method for function approximation on data-defined manifolds.
• The universal constructions offer flexibility and reduce reliance on prior assumptions about the target function.
• The findings have implications for improving deep network performance in manifold learning scenarios.