# Geometric Learning with Positively Decomposable Kernels

arXiv: https://arxiv.org/abs/2310.13821 · HTML: https://mlsanity.com/paper/2310.13821/

## Key facts

- Cited by 3 papers in the map (data through 7 Oct 2026).
- Builds on 12 of its 60 references that are in the map.
- First posted on arXiv 20 Oct 2023.

## Builds on (most cited first, 12 of 12)

- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Mat'ern Gaussian processes on Riemannian manifolds](https://mlsanity.com/paper/2006.10160/) (2020)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [Large-Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/1806.00437/) (2018)
- [Kernel Distribution Embeddings: Universal Kernels, Characteristic Kernels and Kernel Metrics on Distributions](https://mlsanity.com/paper/1604.05251/) (2016)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case](https://mlsanity.com/paper/2208.14960/) (2022)
- [Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1412.4172/) (2014)
- [Gaussians on Riemannian Manifolds: Applications for Robot Learning and Adaptive Control](https://mlsanity.com/paper/1909.05946/) (2019)
- [Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spaces](https://mlsanity.com/paper/2301.13088/) (2023)
- [The Gaussian kernel on the circle and spaces that admit isometric embeddings of the circle](https://mlsanity.com/paper/2302.10623/) (2023)
- [Gaussian kernels on non-simply-connected closed Riemannian manifolds are never positive definite](https://mlsanity.com/paper/2303.06558/) (2023)
- [Geodesic squared exponential kernel for non-rigid shape registration](https://mlsanity.com/paper/2112.11853/) (2021)

## Built on by (most cited first, 3 of 3)

- [Invariant kernels on Riemannian symmetric spaces: a harmonic-analytic approach](https://mlsanity.com/paper/2310.19270/) (2023)
- [Memory Capacity of Nonlinear Recurrent Networks: Is it Informative?](https://mlsanity.com/paper/2502.04832/) (2025)
- [Universal kernels via harmonic analysis on Riemannian symmetric spaces](https://mlsanity.com/paper/2506.19245/) (2025)

## Abstract (arXiv)

Kernel methods are powerful tools in machine learning. Classical kernel methods are based on positive-definite kernels, which map data spaces into reproducing kernel Hilbert spaces (RKHS). For non-Euclidean data spaces, positive-definite kernels are difficult to come by. In this case, we propose the use of reproducing kernel Krein space (RKKS) based methods, which require only kernels that admit a positive decomposition. We show that one does not need to access this decomposition in order to learn in RKKS. We then investigate the conditions under which a kernel is positively decomposable. We show that invariant kernels admit a positive decomposition on homogeneous spaces under tractable regularity assumptions. This makes them much easier to construct than positive-definite kernels, providing a route for learning with kernels for non-Euclidean data. By the same token, this provides theoretical foundations for RKKS-based methods in general.