Paper: arXiv 2603.28257
Authors: David Breazu
Abstract
KAN-PCA is an autoencoder that uses a KAN as encoder and a linear map as decoder. It generalizes classical PCA by replacing linear projections with learned B-spline functions on each edge. The motivation is to capture more variance than classical PCA, which becomes inefficient during market crises when the linear assumption breaks down and correlations between assets change dramatically. We prove that if the spline activations are forced to be linear, KAN-PCA yields exactly the same results as classical PCA, establishing PCA as a special case. Experiments on 20 S&P 500 stocks (2015-2024) show that KAN-PCA achieves a reconstruction R^2 of 66.57%, compared to 62.99% for classical PCA with the same 3 factors, while matching PCA out-of-sample after correcting for data leakage in the training procedure.
Complexity vs Empirical Score
- Math Complexity: 6.5/10
- Empirical Rigor: 5.0/10
- Quadrant: Holy Grail — high math complexity, high empirical rigor
Why this score: The paper introduces a novel application of Kolmogorov-Arnold Networks to generalize PCA for nonlinear financial dynamics, backed by a theoretical proof of equivalence under linear constraints. However, the empirical evaluation is limited to a small sample of 20 stocks and exhibits modest out-of-sample improvements once data leakage is addressed.
Research Flowchart
flowchart TD
A["Research Goal: Capture nonlinear asset dependencies during market crises via KAN-PCA"] --> B["Data: S&P 500 Daily Returns (20 stocks, 2015-2024)"]
B --> C["KAN-PCA Architecture: KAN Encoder (B-splines) + Linear Decoder"]
C --> D["Theoretical Proof: Constraining spline activations to linear reduces KAN-PCA to classical PCA"]
C --> E["Model Training & Evaluation: Reconstruction R² & Out-of-Sample Validation"]
E --> F["Key Findings: KAN-PCA achieves 66.57% R² vs 62.99% for PCA (3 factors); matches OOS performance"]