# A Universal Reproducing Kernel Hilbert Space from Polynomial Alignment and IMQ Distance

arXiv: https://arxiv.org/abs/2605.03262 · HTML: https://mlsanity.com/paper/2605.03262/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 48 references that are in the map.
- First posted on arXiv 5 May 2026.

## Builds on (most cited first, 18 of 18)

- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Compressive Transformers for Long-Range Sequence Modelling](https://mlsanity.com/paper/1911.05507/) (2019)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [Deep Kernel Learning](https://mlsanity.com/paper/1511.02222/) (2015)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [A PAC-Bayesian Approach to Spectrally-Normalized Margin Bounds for Neural Networks](https://mlsanity.com/paper/1707.09564/) (2017)
- [The Power of Depth for Feedforward Neural Networks](https://mlsanity.com/paper/1512.03965/) (2015)
- [Linearized two-layers neural networks in high dimension](https://mlsanity.com/paper/1904.12191/) (2019)
- [Toward Deeper Understanding of Neural Networks: The Power of Initialization and a Dual View on Expressivity](https://mlsanity.com/paper/1602.05897/) (2016)
- [Gaussian Process Kernels for Pattern Discovery and Extrapolation](https://mlsanity.com/paper/1302.4245/) (2013)
- [A Priori Estimates of the Population Risk for Two-layer Neural Networks](https://mlsanity.com/paper/1810.06397/) (2018)
- [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020)
- [Pad'e Activation Units: End-to-end Learning of Flexible Activation Functions in Deep Networks](https://mlsanity.com/paper/1907.06732/) (2019)
- [Convolutional Kernel Networks](https://mlsanity.com/paper/1406.3332/) (2014)
- [Neural networks and rational functions](https://mlsanity.com/paper/1706.03301/) (2017)
- [A representer theorem for deep kernel learning](https://mlsanity.com/paper/1709.10441/) (2017)

## Built on by (most cited first, 1 of 1)

- [Bernstein-Schur Kernels: Random Features by Sketched Modulation and Radial Randomization](https://mlsanity.com/paper/2606.11255/) (2026)

## Abstract (arXiv)

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-√ 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.