# Draining Fictitious Knots: Restoring Distance-Awareness Guarantees for High-Dimensional Spline Networks

arXiv: https://arxiv.org/abs/2609.15274 · HTML: https://mlsanity.com/paper/2609.15274/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 19 references that are in the map.
- First posted on arXiv 14 Sep 2026.

## Builds on (most cited first, 10 of 10)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Efficient Neural Network Robustness Certification with General Activation Functions](https://mlsanity.com/paper/1811.00866/) (2018)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [On Feature Collapse and Deep Kernel Learning for Single Forward Pass Uncertainty](https://mlsanity.com/paper/2102.11409/) (2021)
- [KKANs: Kurkova-Kolmogorov-Arnold Networks and Their Learning Dynamics](https://mlsanity.com/paper/2412.16738/) (2024)
- [Distance-Aware Error for Spline Networks: A Bottom-Up Approach to Uncertainty](https://mlsanity.com/paper/2501.04757/) (2025)

## Abstract (arXiv)

Kolmogorov-Arnold Networks (KANs) with spline activations have recently shown promise for interpretable function approximation. Distance-Aware Error for Kolmogorov Networks (DAREK) introduces a computationally efficient bottom-up approach to uncertainty quantification by equipping KANs with distance-aware error bounds; yet, in high-dimensional settings, the theoretical guarantees can be weakened by the emergence of fictitious knots. Inspired by the Kolmogorov-Arnold representation theorem, DAREK adopts a componentwise formulation in which each input dimension is treated separately; as a result, induced knot locations may appear in the combined input space without corresponding to actual training data. These fictitious knots mislead the DAREK uncertainty estimator into reporting low uncertainty far from any real observation, violating the distance-awareness guarantee. We identify this failure mode precisely, characterize its geometric structure, and propose a drainage uncertainty mechanism that restores distance-awareness by constructing a monotonically decreasing uncertainty path from any fictitious knot region toward the nearest real knot. The proposed drainage method provides a practical heuristic correction that mitigates the fictitious-knot failure mode while restoring theoretical distance-awareness in high-dimensional settings. Experiments on a 2D synthetic benchmark and a 100-dimensional face dataset show that drainage raises sampled distance-awareness (SDA) from 85% to 98-99%, matching Gaussian processes at lower computational cost.