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This research introduces the Yat kernel, which combines inverse-multiquadric decay with polynomial alignment to simultaneously preserve universal approximation and introduce persistent directional responses in high-dimensional spaces. By analyzing activation covariance, the study demonstrates how this construction avoids Gaussian collapse mechanisms while strictly enlarging the global function space compared to standard IMQ kernels. These findings offer a constructive pathway for designing neural network architectures that maintain radial symmetry without sacrificing directional sensitivity.
A Universal Reproducing Kernel Hilbert Space from Polynomial Alignment and IMQ Distance
arXiv:2605.03262v2 Announce Type: replace Abstract: Inverse-multiquadric (IMQ) kernel sections decay at infinity. Can polynomial alignment add persistent directional responses while preserving universal approximation? The \yat{} (Yat) kernel answers this question by multiplying IMQ distance by a biased squared inner product. With shared positive bias and regularization, its RKHS continuously contains the IMQ RKHS and is universal on every compact domain. Alignment strictly enlarges the global function space: Yat sections retain a quadratic directional trace at infinity. We quantify a second distinction through high-dimensional activation covariance. For a fixed number of orthogonal unit centers and fixed kernel parameters, isotropic Gaussian inputs yield a rank-one rescaled limiting covariance for IMQ and a full-rank limit for Yat. On the radius-$\sqrt d$ sphere, both limits are full rank, isolating shared input-norm fluctuations as the Gaussian collapse mechanism. The connection is also constructive: three positive-bias Yat atoms recover one IMQ section exactly, and three are necessary at nonzero centers when regularization is shared and bias varies across atoms. These results identify how polynomial alignment preserves radial approximation while adding directional structure.
Abstract:Inverse-multiquadric (IMQ) kernel sections decay at infinity. Can polynomial alignment add persistent directional responses while preserving universal approximation? The \yat{} (Yat) kernel answers this question by multiplying IMQ distance by a biased squared inner product. With shared positive bias and regularization, its RKHS continuously contains the IMQ RKHS and is universal on every compact domain. Alignment strictly enlarges the global function space: Yat sections retain a quadratic directional trace at infinity. We quantify a second distinction through high-dimensional activation covariance. For a fixed number of orthogonal unit centers and fixed kernel parameters, isotropic Gaussian inputs yield a rank-one rescaled limiting covariance for IMQ and a full-rank limit for Yat. On the radius-$\sqrt d$ sphere, both limits are full rank, isolating shared input-norm fluctuations as the Gaussian collapse mechanism. The connection is also constructive: three positive-bias Yat atoms recover one IMQ section exactly, and three are necessary at nonzero centers when regularization is shared and bias varies across atoms. These results identify how polynomial alignment preserves radial approximation while adding directional structure.
Machine Learning (cs.LG)
arXiv:2605.03262 [cs.LG]
(or arXiv:2605.03262v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2605.03262 Focus to learn more arXiv-issued DOI via DataCite
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