
Riemannian Gaussian distributions, random matrix ensembles and diffusion kernels
We show that the Riemannian Gaussian distributions on symmetric spaces, ...
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Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices
Symmetric Positive Definite (SPD) matrices have become popular to encode...
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WassersteinRiemannian Geometry of Positivedefinite Matrices
The Wasserstein distance on multivariate nondegenerate Gaussian densiti...
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Low rank approximation of positive semidefinite symmetric matrices using Gaussian elimination and volume sampling
Positive semidefinite matrices commonly occur as normal matrices of lea...
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Symmetric Positive Semidefinite Riemannian Geometry with Application to Domain Adaptation
In this paper, we present new results on the Riemannian geometry of symm...
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Geometrical and statistical properties of Mestimates of scatter on Grassmann manifolds
We consider data from the Grassmann manifold G(m,r) of all vector subspa...
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Spectral estimates for saddle point matrices arising in weak constraint fourdimensional variational data assimilation
We consider the largesparse symmetric linear systems of equations that ...
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Gaussian distributions on Riemannian symmetric spaces in the large N limit
We consider Gaussian distributions on certain Riemannian symmetric spaces. In contrast to the Euclidean case, it is challenging to compute the normalization factors of such distributions, which we refer to as partition functions. In some cases, such as the space of Hermitian positive definite matrices or hyperbolic space, it is possible to compute them exactly using techniques from random matrix theory. However, in most cases which are important to applications, such as the space of symmetric positive definite (SPD) matrices or the Siegel domain, this is only possible numerically. Moreover, when we consider, for instance, highdimensional SPD matrices, the known algorithms for computing partition functions can become exceedingly slow. Motivated by notions from theoretical physics, we will discuss how to approximate the partition functions in the large N limit: an approximation that gets increasingly better as the dimension of the underlying symmetric space (more precisely, its rank) gets larger. We will give formulas for leading order terms in the case of SPD matrices and related spaces. Furthermore, we will characterize the large N limit of the Siegel domain through a singular integral equation arising as a saddlepoint equation.
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