In probability theory and statistics, the multivariate normal distribution, multivariate gaussian distribution, or joint. Symmetric spaces are well understood in riemannian geometry, allowing us to compute distance functions and other. Riemannian gaussian distributions on the space of symmetric positive definite matrices article pdf available in ieee transactions on information theory pp99 july 2015 with 109 reads. One can use a manifestly flat metric like a constant metric for the flat space metric, and then find the riemann tensor from the complete metric. Pdf riemannian gaussian distributions on the space of. In the 1950s atle selberg extended cartans definition of symmetric space to that of weakly symmetric riemannian space, or in current terminology weakly symmetric space. Our metric is based on the fundamental idea of parametrizing this space as the riemannian symmetric space.
Apr 28, 2015 goh a, vidal r 2008 unsupervised riemannian clustering of probability density functions. Then gk is a riemannian symmetric space of noncompact type. Riemannian curvature of maximally symmetric spaces. Mixed region covariance discriminative learning for image. There are a number of equivalent ways of characterizing these types. A detailed description of these fibrations is also given. However, it can be very challenging to assume useful parametric distributions for. Skovgaard l 1984 a riemannian geometry of the multivariate normal model. Riemannian curvature of maximally symmetric spaces physics. Information geometry and asymptotic geodesics on the space. A reconstruction theorem for riemannian symmetric spaces of noncompact type 3 variant functions on a. The study of riemannian geometry is rather meaningless without some basic knowledge on gaussian geometry i. Approximate joint diagonalization according to the natural.
Symmetric spaces and their local versions were studied and classi. Arpm lab characteristic function of a multivariate normal. A riemannian symmetric space is a riemannian manifold s with the property that the geodesic re. Globally symmetric riemannian space encyclopedia of mathematics. Riemannian symmetric spaces massachusetts institute of. According to information geometry, the space of ddimensional multivariate gaussians is also a specific riemannian manifold and can be embedded. Multivariate normal distributions parametrized as a.
These are defined as riemannian manifolds m with a transitive connected lie group of isometries g and an isometry. Riemannian geometry for the statistical analysis of. From now on let x gkbe a riemannian symmetric space of the noncom. By definition, a hermitian symmetric space is a riemannian symmetric space m together with an almost complex structure on m such that the metric is hermitian and such that every symmetry s a is. These two problems can be reduced to an optimization problem with three key components. Symmetric submanifolds of riemannian symmetric spaces. M, 1 the normal space at is linearly spanned by the image. That is, its metric conformal function flatspace metric. The circularly symmetric version of the complex normal distribution has a slightly different form. A riemannian framework for tensor computing xavier pennec, pierre fillard, nicholas ayache.
Reconstructing the geometric structure of a riemannian symmetric space from its satake diagram sebastian klein1 january 24, 2008 abstract. The mean associated with the riemannian metric corresponds to the geometric mean. Joint metric learning on riemannian manifold of global. Multivariate normal distributions parametrized as a riemannian symmetric space. Ref foundations of di erentiable manifolds and lie groups by warner part 2. May 15, 2009 a manifold is said to be maximally symmetric if it has the same number of symmetries as ordinary euclidean space. One definition is that a random vector is said to be kvariate normally distributed if every linear combination of its k components has a univariate normal distribution. In particular, a distribution can be represented via the characteristic function. The distance function based on the fisher information metric has been studied by a number of statisticians. The local geometry of a riemannian symmetric space is described completely by the riemannian metric and the riemannian curvature tensor of the space. For various statistical purposes, it is desirable to have a. Geometric analysis on riemannian and metric spaces september 59, 2016 room 420, research institute for mathematical sciences, kyoto university directions. Theory and application to diffusion tensor mri processing, journal of mathematical imaging and vision, 253. Any twistor space with an integrable complex structure will have an image in jm.
Riemannian laplace distribution on the space of symmetric. That is, its metric conformal function flat space metric. Approximate joint diagonalization and geometric mean of. This distribution may be used to describe diffusion tensor data. L2invariants of locally symmetric spaces 221 for any 0, where the. In this paper, we present a joint metric learning jml model on riemannian manifold of gaussian distributions. Multivariate normal distribution parametrized as a riemannian. A compact simply connected riemannian 4symmetric space decomposes as a product m\ x x mr where each irreducible factor is. The mean associated with the euclidean metric of the ambient space is the usual arithmetic mean. Derivation of the fisher metric this paper considers the riemannian geometry of the univariate normal model, as based on the fisher metric 1. To classify the possible isometry classes of m, first note that the universal cover of a riemannian symmetric space is again riemannian symmetric, and. We propose to study a slightly different metric on the space of multivariate normal distributions on r n.
The challenge of using gaussians to model the vision data is that the space of gaussians is not a linear space. Riemannian gaussian distributions on the space of symmetric. In fact, we arrive at the same formalism and we end up with the same riemannian metric from a pure differential geometric point of view 3, 5, 1116, or from an information geometric point of view, assuming the multivariate normal distribution of the data and adopting the fisher information metric 17, 18, dating back to the seminal works. Due to the specific geometry of riemannian manifolds, classical. This space is a riemannian symmetric space of negative curvature, where for. Wasserstein riemannian geometry of positivedefinite matrices. We describe a new covariance descriptor, which could improve the discriminative learning ability of region covariance descriptor by taking into account the mean of feature vectors. Riemannchristoffel tensor, ricci tensor, scalar curvature, sectional curvature, homogeneous space, lie group, riemannian geometry. The random primitive x has a probability density function px real, positive and integrable function if 8x 2 a. Ruh3 university of fribourg, fribourg, switzerland email. A reconstruction theorem for riemannian symmetric spaces of. S can be seen as a differentiable manifold under certain regularity conditions, with a coordinate system. In 1, when m gk is an inner riemannian symmetric space and g the invariant metric, this zeroset was shown to. Goh a, vidal r 2008 unsupervised riemannian clustering of probability density functions.
The invariant riemannian metric generally used for spd manifolds is. We consider the blind source separation bss problem and the closely related approximate joint diagonalization ajd problem of symmetric positive definite spd matrices. Using its value of the riemann tensor, one can show that a maximallysymmetric metric is conformally flat. New riemannian priors on the univariate normal model. The space of spd matrices forms the interior of a convex cone in the space symd see fig. Sep 25, 2014 using its value of the riemann tensor, one can show that a maximally symmetric metric is conformally flat. A manifold with an affine connection is called a locally symmetric affine space if the torsion tensor and the covariant derivative of the curvature tensor vanish identically.
A reconstruction theorem for riemannian symmetric spaces. A reconstruction theorem for riemannian symmetric spaces of noncompact type 5 see 8, chapter iv, section 41. The expression of its probability density function, generalising that of a gaussian distribution on the euclidean space rp, is. Reconstructing the geometric structure of a riemannian. Approximate joint diagonalization according to the natural riemannian distance florent bouchard 1. A manifold is said to be maximally symmetric if it has the same number of symmetries as ordinary euclidean space.
Our metric is based on the fundamental idea of parametrizing this space as the riemannian. Wrapped gaussian process regression on riemannian manifolds. If is an algebra of compact type, then is an algebra of noncompact type and vice versa each globally symmetric riemannian space generates an orthogonal symmetric lie algebra, where is the lie algebra of the group and is the identity in the group. Discrete spectrum for nonriemannian locally symmetric spaces. Introduction to riemannian symmetric spaces and rspaces. In this paper we introduce metricbased means for the space of positivedefinite matrices. Ref di erential geometry, lie groups, and symmetric spaces.
Wasserstein riemannian geometry of positivedefinite matrices 3 singular. Euclidean space but on a riemannian manifold, a rieman nian metric. The selfinformation, also known as the information content of a signal, random variable, or event is defined as the negative logarithm of the probability of the given outcome occurring when applied to a discrete random variable, the selfinformation can be represented as citation needed. A riemannian gaussian distribution, denoted gy, depends on two parameters, y 2p mand. Intrinsic covariance matrix estimation for multivariate. Statistics on multivariate normal distributions halinria. G is the isometry group of s with the properties sxx x, dsxx. Grenoble alpes, grenoble institute of technology, grenoble france 2 ljk, cnrs, univ. Discrete spectrum for nonriemannian locally symmetric. The complex analogues of the riemannian symmetric spaces are the hermitian symmetric spaces. Covariance matrices, known as symmetric positive definite spd matrices, are usually regarded as points lying on riemannian manifolds. This probability density function for this distribution is given by px exp.
Multivariate normal distributions parametrized as a riemannian symmetric space miroslav lovric 1 and maung minoo2 mcmaster university, hamilton, ontario, canada email. Symmetric positive definite matrices, tensor, riemannian metric, gaussian distribution. Due to the specific geometry of riemannian manifolds, classical learning. Eigenfunctions on a riemannian symmetric space 641 admits an asymptotic expansion along a wall of type av the usual notation for the split component of a standard parabolic. Let m be a complete connected parallel submanifold of a pseudoeuclidean space e satisfying that, for any point pe. Gaussian distributions and thus measure the dissimilarity of two sets with the distance.
The construction of a distance function between probability distributions is of importance in mathematical statistics and its applications. A general name given to various types of spaces in differential geometry. In this paper, firstly, we will show the following. In this paper, we propose for the rst time an approximate. In probability theory and statistics, the multivariate normal distribution, multivariate gaussian distribution, or joint normal distribution is a generalization of the onedimensional normal distribution to higher dimensions. Local boundary data of eigenfunctions on a riemannian. Symmetric submanifolds of riemannian symmetric spaces sankaran viswanath may 11, 2000 1 introduction a symmetric space is a riemannian manifold that is symmetric about each of its points. However, when the response is in a riemannian symmetric space rss and the. We propose to study a slightly different metric on the space of multivariate normal distributions on. Globally symmetric riemannian space encyclopedia of. Each globally symmetric riemannian space generates an orthogonal symmetric lie algebra, where is the lie algebra of the group and is the identity in the group. A manifold with an affine connection is called a locally symmetric affine space if the torsion tensor and the covariant derivative of the curvature tensor vanish identically a pseudo riemannian manifold is called a locally symmetric pseudo riemannian space if the covariant derivative of its curvature tensor.
Discriminant analysis on riemannian manifold of gaussian. Symmetric spaces are well understood in riemannian geometry, allowing us to compute distance functions and other relevant geometric data. A multivariate normal distribution is determined by its covariance matrix and its mean vector. We discuss some invariance properties of the riemannian mean and we use differential geometric. Let g and a be the lie algebras of g and a, respectively, and es, a the root system for, a. Lecture notes on symmetric spaces university of augsburg. A locally symmetric space is a quotient x nx of a reductive symmetric space x gh by a discrete subgroup of g acting properly discontinuously and freely. One can get a metric of a maximallysymmetric space in an interesting way. We propose to study a slightly different metric on the space of multivariate normal distributions on n. Regression models on riemannian symmetric spaces ncbi. According to information geometry, the space of ddimensional multivariate gaussians is also a specific riemannian manifold and can be embedded into the space of symmetric positive definite spd.