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

Gaussian processes are arguably the most important class of spatiotemporal models within machine learning. They encode prior information about the modeled function and can be used for exact or approximate Bayesian learning. In many applications, particularly in physical sciences and engineering, but also in areas such as geostatistics and neuroscience, invariance to symmetries is one of the most fundamental forms of prior information one can consider. The invariance of a Gaussian process' covariance to such symmetries gives rise to the most natural generalization of the concept of stationarity to such spaces. In this work, we develop constructive and practical techniques for building stationary Gaussian processes on a very large class of non-Euclidean spaces arising in the context of symmetries. Our techniques make it possible to (i) calculate covariance kernels and (ii) sample from prior and posterior Gaussian processes defined on such spaces, both in a practical manner. This work is split into two parts, each involving different technical considerations: part I studies compact spaces, while part II studies non-compact spaces possessing certain structure. Our contributions make the non-Euclidean Gaussian process models we study compatible with well-understood computational techniques available in standard Gaussian process software packages, thereby making them accessible to practitioners

Kernel density estimation on the Siegel space applied to radar processing

Main techniques of probability density estimation on Riemannian manifolds are reviewed in the case of the Siegel space. For computational reasons we chose to focus on the kernel density estimation. The main result of the paper is the expression of Pelletier's kernel density estimator. The method is applied to density estimation of reflection coefficients from radar observations

Iwasawa N=8 Attractors

Starting from the symplectic construction of the Lie algebra e_7(7) due to Adams, we consider an Iwasawa parametrization of the coset E_7(7)/SU(8), which is the scalar manifold of N=8, d=4 supergravity. Our approach, and the manifest off-shell symmetry of the resulting symplectic frame, is determined by a non-compact Cartan subalgebra of the maximal subgroup SL(8,R) of E_7(7). In absence of gauging, we utilize the explicit expression of the Lie algebra to study the origin of E_7(7)/SU(8) as scalar configuration of a 1/8-BPS extremal black hole attractor. In such a framework, we highlight the action of a U(1) symmetry spanning the dyonic 1/8-BPS attractors. Within a suitable supersymmetry truncation allowing for the embedding of the Reissner-Nordstrom black hole, this U(1) is interpreted as nothing but the global R-symmetry of pure N=2 supergravity. Moreover, we find that the above mentioned U(1) symmetry is broken down to a discrete subgroup Z_4, implying that all 1/8-BPS Iwasawa attractors are non-dyonic near the origin of the scalar manifold. We can trace this phenomenon back to the fact that the Cartan subalgebra of SL(8,R) used in our construction endows the symplectic frame with a manifest off-shell covariance which is smaller than SL(8,R) itself. Thus, the consistence of the Adams-Iwasawa symplectic basis with the action of the U(1) symmetry gives rise to the observed Z_4 residual non-dyonic symmetry.Comment: 1+26 page

Peripheral separability and cusps of arithmetic hyperbolic orbifolds

For X = R, C, or H it is well known that cusp cross-sections of finite volume X-hyperbolic (n+1)-orbifolds are flat n-orbifolds or almost flat orbifolds modelled on the (2n+1)-dimensional Heisenberg group N_{2n+1} or the (4n+3)-dimensional quaternionic Heisenberg group N_{4n+3}(H). We give a necessary and sufficient condition for such manifolds to be diffeomorphic to a cusp cross-section of an arithmetic X-hyperbolic (n+1)-orbifold. A principal tool in the proof of this classification theorem is a subgroup separability result which may be of independent interest.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-32.abs.htm

Spherical representations of Lie supergroups

The classical Cartan-Helgason theorem characterises finite-dimensional spherical representations of reductive Lie groups in terms of their highest weights. We generalise the theorem to the case of a reductive symmetric supergroup pair $(G,K)$ of even type. Along the way, we compute the Harish-Chandra $c$-function of the symmetric superspace $G/K$. By way of an application, we show that all spherical representations are self-dual in type AIII|AIII.Comment: 37 pages; title changed; substantially revised version; accepted for publication, J. Func. Anal. (2014

Cusps of arithmetic orbifolds

This thesis investigates cusp cross-sections of arithmetic real, complex, and quaternionic hyperbolic $n$--orbifolds. We give a smooth classification of these submanifolds and analyze their induced geometry. One of the primary tools is a new subgroup separability result for general arithmetic lattices.Comment: 76 pages; Ph.D. thesi