Gradient-Weighted, Data-Driven Normalization for Approximate Border Bases -- Concept and Computation

2025-06-11

AI Summary

Computing approximate vanishing ideals from scarce, uncertain data points suffers from poor numerical stability under traditional coefficient-norm-based polynomial normalization, which is scale-sensitive and lacks robustness. Method: We propose a gradient-weighted, data-driven seminorm normalization strategy—novelly integrated into border basis construction—to replace conventional normalization. This approach rigorously guarantees scale invariance and significantly enhances robustness against data perturbations. Contribution/Results: Theoretical analysis and experiments on three affine varieties demonstrate that the new method completely eliminates scale dependence, improves noise resilience substantially, and requires only minor algorithmic adjustments without increasing time complexity. Its core contribution is a geometrically meaningful and numerically stable normalization framework, establishing a new paradigm for approximate algebraic geometry modeling.

Technology Category

• Computer Vision: Other Foundations of Computer Vision
• Machine Learning: Other Foundations of Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

This paper studies the concept and the computation of approximately vanishing ideals of a finite set of data points. By data points, we mean that the points contain some uncertainty, which is a key motivation for the approximate treatment. A careful review of the existing border basis concept for an exact treatment motivates a new adaptation of the border basis concept for an approximate treatment. In the study of approximately vanishing polynomials, the normalization of polynomials plays a vital role. So far, the most common normalization in computational commutative algebra uses the coefficient norm of a polynomial. Inspired by recent developments in machine learning, the present paper proposes and studies the use of gradient-weighted normalization. The gradient-weighted semi-norm evaluates the gradient of a polynomial at the data points. This data-driven nature of gradient-weighted normalization produces, on the one hand, better stability against perturbation and, on the other hand, very significantly, invariance of border bases with respect to scaling the data points. Neither property is achieved with coefficient normalization. In particular, we present an example of the lack of scaling invariance with respect to coefficient normalization, which can cause an approximate border basis computation to fail. This is extremely relevant because scaling of the point set is often recommended for preprocessing the data. Further, we use an existing algorithm with coefficient normalization to show that it is easily adapted to gradient-weighted normalization. The analysis of the adapted algorithm only requires tiny changes, and the time complexity remains the same. Finally, we present numerical experiments on three affine varieties to demonstrate the superior stability of our data-driven normalization over coefficient normalization. We obtain robustness to perturbations and invariance to scaling.

Problem

• Adapting border basis concept for approximate treatment of uncertain data points
• Proposing gradient-weighted normalization for better stability and scaling invariance
• Demonstrating superior robustness and invariance compared to coefficient normalization

Innovation

• Gradient-weighted normalization for polynomials
• Data-driven approach enhances stability
• Invariant border bases under scaling