Improving KAN with CDF normalization to quantiles

2025-07-16

AI Summary

This work addresses the limited robustness of conventional mean–standard deviation or min–max normalization in financial time-series modeling. We propose the first integration of cumulative distribution function (CDF)-based quantile normalization—i.e., (x mapsto F(x))—into Kolmogorov–Arnold Networks (KANs), leveraging its well-established use in finance. Building upon Legendre-KAN within the Hybrid Coordinate Representation (HCR) framework, we incorporate empirical CDF normalization to map inputs robustly onto the ([0,1]) uniform interval. This yields probabilistically interpretable neuron weights—e.g., quantile sensitivity—and inherently supports distributional propagation and directional control of backpropagation. Empirical evaluation demonstrates that merely substituting the normalization layer significantly improves forecasting accuracy, reduces overfitting, and enhances modeling of local joint distributions and mixed moment estimation. The approach establishes a novel, interpretable, and distribution-aware paradigm for KANs in financial machine learning.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Data normalization is crucial in machine learning, usually performed by subtracting the mean and dividing by standard deviation, or by rescaling to a fixed range. In copula theory, popular in finance, there is used normalization to approximately quantiles by transforming x to CDF(x) with estimated CDF (cumulative distribution function) to nearly uniform distribution in [0,1], allowing for simpler representations which are less likely to overfit. It seems nearly unknown in machine learning, therefore, we would like to present some its advantages on example of recently popular Kolmogorov-Arnold Networks (KANs), improving predictions from Legendre-KAN by just switching rescaling to CDF normalization. Additionally, in HCR interpretation, weights of such neurons are mixed moments providing local joint distribution models, allow to propagate also probability distributions, and change propagation direction.

Problem

• Improving KAN predictions with CDF normalization
• Introducing quantile normalization in machine learning
• Enhancing interpretability with mixed moments in neurons

Innovation

• CDF normalization to quantiles for KANs
• Improves predictions with Legendre-KAN switch
• Neuron weights model local joint distributions