Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces I: the compact case

Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spaces

“…Deferring a formal definition, we will for now say that this is a compact smooth manifold with a sufficiently rich group G of symmetries acting on it. If we fix an arbitrary point x 0 ∈ X, which is analogous to a sphere's pole, and define g 1 , g 2 ∈ G in a way that ensures g 1 x 0 = x and g 2 x 0 = x -where such g 1 , g 2 exist because G is sufficiently rich-the appropriate analog of Bochner's theorem (Azangulov et al, 2022;Yaglom, 1961) says…”

“…To define their analogs in the setting at hand, we adopt the heat-equation-based approach used in the compact case (Azangulov et al, 2022). This works as follows.…”

“…A proof for heat kernel (ν = ∞) can be found in Gangolli (1968, Proposition 3.1), also see Appendix B for a proof sketch. From this, the Matérn kernel's form follows by applying its definition, changing the order of integration, and appropriate algebra, which works the same way as in the compact case-see Azangulov et al (2022).…”

“…This work is the second part of a two-part series of papers that aims at bringing Gaussian process regression into the machine learning toolbox for a large class of Riemannian manifolds admitting analytical descriptions. To do so, in part I (Azangulov et al, 2022) we studied a rich class of compact manifolds, including compact Lie groups such as the special orthogonal group SO(n) and the special unitary group SU(n), and their homogeneous spaces, such as the sphere S n and Stiefel manifold V(k, n). We developed computational techniques based on suitable generalizations of Bochner's Theorem due to Yaglom (1961), which express stationary kernels on a compact homogeneous space X = G/H as series in terms of characters of irreducible unitary representations of the group G, or the respective zonal spherical functions.…”

Subspace Detours Meet Gromov–Wasserstein

“…Motivated by these examples, many works proposed new tools to handle data lying on Riemannian manifolds. To cite a few, Fletcher et al (2004); Huckemann and Ziezold (2006) developed PCA to perform dimension reduction on manifolds while Le Brigant and Puechmorel (2019) studied density approximation, Feragen et al (2015); Jayasumana et al (2015); Fang et al (2021) studied kernel methods and Azangulov et al (2022Azangulov et al ( , 2023 developed Gaussian processes on (homogeneous) manifolds. More recently, there has been many interests into developing new neural networks with architectures taking into account the geometry of the ambient manifold (Bronstein et al, 2017) such as Residual Neural Networks (Katsman et al, 2022), discrete Normalizing Flows (Bose et al, 2020;Rezende et al, 2020;Rezende and Racanière, 2021) or Continuous Normalizing Flows (Mathieu and Nickel, 2020;Lou et al, 2020;Rozen et al, 2021;Yataka and Shiraishi, 2022).…”