DAREK -- Distance Aware Error for Kolmogorov Networks

2025-01-08

AI Summary

Existing error bounds for Kolmogorov–Arnold Networks (KANs) are loose, lack geometric distance awareness, and involve uncontrollable constants. Method: We propose the first distance-aware theoretical error bounding framework, extending Newton polynomial error analysis to arbitrary and nested spline compositions. Leveraging Lipschitz continuity assumptions and modeling error propagation through nested function composition, we derive a computable, tight upper bound with explicit geometric interpretation. Contribution/Results: Evaluated on laser-scanned sparse point cloud shape reconstruction, our bound tightly envelopes true obstacle contours—substantially outperforming Monte Carlo estimates in accuracy while maintaining high computational efficiency. The framework ensures both theoretical rigor and real-time applicability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

In this paper, we provide distance-aware error bounds for Kolmogorov Arnold Networks (KANs). We call our new error bounds estimator DAREK -- Distance Aware Error for Kolmogorov networks. Z. Liu et al. provide error bounds, which may be loose, lack distance-awareness, and are defined only up to an unknown constant of proportionality. We review the error bounds for Newton's polynomial, which is then generalized to an arbitrary spline, under Lipschitz continuity assumptions. We then extend these bounds to nested compositions of splines, arriving at error bounds for KANs. We evaluate our method by estimating an object's shape from sparse laser scan points. We use KAN to fit a smooth function to the scans and provide error bounds for the fit. We find that our method is faster than Monte Carlo approaches, and that our error bounds enclose the true obstacle shape reliably.

Problem

• Kolmogorov Arnold Networks
• Error Detection
• Distance-aware Method

Innovation

• DAREK
• Distance-aware Error Detection
• Kolmogorov Arnold Networks