Structural interpretability in SVMs with truncated orthogonal polynomial kernels

arXiv stat.ML / 4/17/2026

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Key Points

The paper proposes a post-training interpretability method for Support Vector Machines using truncated orthogonal polynomial kernels, leveraging the fact that the RKHS is finite-dimensional with an explicit orthonormal basis.
It expresses the learned decision function exactly in intrinsic RKHS coordinates and introduces ORCA (Orthogonal Representation Contribution Analysis) to diagnose how the classifier’s RKHS norm is distributed.
ORCA uses normalized OKC (Orthogonal Kernel Contribution) indices to quantify contributions across interaction orders, total polynomial degrees, marginal coordinate effects, and pairwise effects.
The approach is fully post-training and does not require retraining or surrogate models, and is demonstrated on both a synthetic double-spiral dataset and a real echocardiogram dataset.
Experiments show that the indices uncover structural complexity properties of the model that predictive accuracy alone fails to reveal.

Abstract

We study post-training interpretability for Support Vector Machines (SVMs) built from truncated orthogonal polynomial kernels. Since the associated reproducing kernel Hilbert space is finite-dimensional and admits an explicit tensor-product orthonormal basis, the fitted decision function can be expanded exactly in intrinsic RKHS coordinates. This leads to Orthogonal Representation Contribution Analysis (ORCA), a diagnostic framework based on normalized Orthogonal Kernel Contribution (OKC) indices. These indices quantify how the squared RKHS norm of the classifier is distributed across interaction orders, total polynomial degrees, marginal coordinate effects, and pairwise contributions. The methodology is fully post-training and requires neither surrogate models nor retraining. We illustrate its diagnostic value on a synthetic double-spiral problem and on a real five-dimensional echocardiogram dataset. The results show that the proposed indices reveal structural aspects of model complexity that are not captured by predictive accuracy alone.