Approximate joint diagonalization according to the natural. The study of riemannian geometry is rather meaningless without some basic knowledge on gaussian geometry i. Let g and a be the lie algebras of g and a, respectively, and es, a the root system for, a. That is, its metric conformal function flat space metric. From the perspective of information geometry, the gaussians lie on a specific riemannian manifold. Discrete spectrum for nonriemannian locally symmetric. Riemannian gaussian distributions on the space of symmetric. These are defined as riemannian manifolds m with a transitive connected lie group of isometries g and an isometry.
Multivariate normal distributions parametrized as a riemannian symmetric space. 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. We consider the blind source separation bss problem and the closely related approximate joint diagonalization ajd problem of symmetric positive definite spd matrices. However, it can be very challenging to assume useful parametric distributions for. Theory and application to diffusion tensor mri processing, journal of mathematical imaging and vision, 253. The circularly symmetric version of the complex normal distribution has a slightly different form. A multivariate normal distribution is determined by its covariance matrix and its mean vector. In this paper, we propose for the rst time an approximate. 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. We discuss some invariance properties of the riemannian mean and we use differential geometric.
This probability density function for this distribution is given by px exp. The expression of its probability density function, generalising that of a gaussian distribution on the euclidean space rp, is. Approximate joint diagonalization and geometric mean of. New riemannian priors on the univariate normal model. Intrinsic covariance matrix estimation for multivariate. The distance function based on the fisher information metric has been studied by a number of statisticians. Jan 23, 2018 we consider the blind source separation bss problem and the closely related approximate joint diagonalization ajd problem of symmetric positive definite spd matrices. Skovgaard l 1984 a riemannian geometry of the multivariate normal model. A detailed description of these fibrations is also given. Riemannchristoffel tensor, ricci tensor, scalar curvature, sectional curvature, homogeneous space, lie group, riemannian geometry. Multivariate normal distribution parametrized as a. Local boundary data of eigenfunctions on a riemannian. Multivariate normal distribution parametrized as a riemannian.
The local geometry of a riemannian symmetric space is described completely by the riemannian metric and the riemannian curvature tensor of the space. Wasserstein riemannian geometry of positivedefinite matrices. According to information geometry, the space of ddimensional multivariate gaussians is also a specific riemannian manifold and can be embedded. Our metric is based on the fundamental idea of parametrizing this space as the riemannian symmetric space. There are a number of equivalent ways of characterizing these types. For various statistical purposes, it is desirable to have a. Pdf multivariate normal distributions parametrized as a. 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.
This distribution may be used to describe diffusion tensor data. 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. Symmetric submanifolds of riemannian symmetric spaces. M, 1 the normal space at is linearly spanned by the image. Pdf riemannian gaussian distributions on the space of. To classify the possible isometry classes of m, first note that the universal cover of a riemannian symmetric space is again riemannian symmetric, and. The model is essentially semiparametric, since the joint distribution of y, x is. That is, its metric conformal function flatspace metric. Joint metric learning on riemannian manifold of global. 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. Our metric is based on the fundamental idea of parametrizing this space as the riemannian.
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. Sep 25, 2014 using its value of the riemann tensor, one can show that a maximally symmetric metric is conformally flat. Regression models on riemannian symmetric spaces ncbi. Multivariate normal distributions parametrized as a. In 1, when m gk is an inner riemannian symmetric space and g the invariant metric, this zeroset was shown to. Riemannian curvature of maximally symmetric spaces physics. Gaussian mixture regression on symmetric positive definite. Ref di erential geometry, lie groups, and symmetric spaces. Euclidean space but on a riemannian manifold, a rieman nian metric.
Derivation of the fisher metric this paper considers the riemannian geometry of the univariate normal model, as based on the fisher metric 1. Goh a, vidal r 2008 unsupervised riemannian clustering of probability density functions. May 15, 2009 a manifold is said to be maximally symmetric if it has the same number of symmetries as ordinary euclidean space. Reconstructing the geometric structure of a riemannian. These two problems can be reduced to an optimization problem with three key components. The mean associated with the euclidean metric of the ambient space is the usual arithmetic mean. The construction of a distance function between probability distributions is of importance in mathematical statistics and its applications. In particular, a distribution can be represented via the characteristic function. A reconstruction theorem for riemannian symmetric spaces of noncompact type 5 see 8, chapter iv, section 41. Grenoble alpes, grenoble institute of technology, grenoble france 2 ljk, cnrs, univ. Symmetric spaces and their local versions were studied and classi. Riemannian optimization and approximate joint diagonalization.
Globally symmetric riemannian space encyclopedia of mathematics. A manifold is said to be maximally symmetric if it has the same number of symmetries as ordinary euclidean space. A general name given to various types of spaces in differential geometry. 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. Riemannian symmetric spaces massachusetts institute of. We propose to study a slightly different metric on the space of multivariate normal distributions on r n. Riemannian laplace distribution on the space of symmetric. Then gk is a riemannian symmetric space of noncompact type. However, when the response is in a riemannian symmetric space rss and the. 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. Apr 28, 2015 goh a, vidal r 2008 unsupervised riemannian clustering of probability density functions. Lecture notes on symmetric spaces university of augsburg. 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.
A riemannian framework for tensor computing xavier pennec, pierre fillard, nicholas ayache. In this paper we introduce metricbased means for the space of positivedefinite matrices. 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. Discrete spectrum for nonriemannian locally symmetric spaces. Ref foundations of di erentiable manifolds and lie groups by warner part 2. S can be seen as a differentiable manifold under certain regularity conditions, with a coordinate system. In probability theory and statistics, the multivariate normal distribution, multivariate gaussian distribution, or joint.
We propose to study a slightly different metric on the space of multivariate normal distributions on. Riemannian geometry for the statistical analysis of diffusion. Symmetric positive definite matrices, tensor, riemannian metric, gaussian distribution. The complex analogues of the riemannian symmetric spaces are the hermitian symmetric spaces. L2invariants of locally symmetric spaces 221 for any 0, where the. A reconstruction theorem for riemannian symmetric spaces of. Reconstructing the geometric structure of a riemannian symmetric space from its satake diagram sebastian klein1 january 24, 2008 abstract. 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.
Ruh3 university of fribourg, fribourg, switzerland email. Statistics on multivariate normal distributions halinria. From now on let x gkbe a riemannian symmetric space of the noncom. 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. This space is a riemannian symmetric space of negative curvature, where for. A reconstruction theorem for riemannian symmetric spaces of noncompact type 3 variant functions on a. The invariant riemannian metric generally used for spd manifolds is. Riemannian geometry for the statistical analysis of. In this paper, we present a joint metric learning jml model on riemannian manifold of gaussian distributions.
Covariance matrices, known as symmetric positive definite spd matrices, are usually regarded as points lying on riemannian manifolds. Due to the specific geometry of riemannian manifolds, classical learning. Any twistor space with an integrable complex structure will have an image in jm. G is the isometry group of s with the properties sxx x, dsxx. The space of spd matrices forms the interior of a convex cone in the space symd see fig.
Due to the specific geometry of riemannian manifolds, classical. Symmetric spaces are well understood in riemannian geometry, allowing us to compute distance functions and other relevant geometric data. A riemannian symmetric space is a riemannian manifold s with the property that the geodesic re. 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. Approximate joint diagonalization according to the natural riemannian distance florent bouchard 1. The random primitive x has a probability density function px real, positive and integrable function if 8x 2 a. Wasserstein riemannian geometry of positivedefinite matrices 3 singular. A compact simply connected riemannian 4symmetric space decomposes as a product m\ x x mr where each irreducible factor is.
A riemannian gaussian distribution, denoted gy, depends on two parameters, y 2p mand. Gaussian distributions and thus measure the dissimilarity of two sets with the distance. Using its value of the riemann tensor, one can show that a maximallysymmetric metric is conformally flat. Globally symmetric riemannian space encyclopedia of.
Mixed region covariance discriminative learning for image. A reconstruction theorem for riemannian symmetric spaces. 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. Arpm lab characteristic function of a multivariate normal. Introduction to riemannian symmetric spaces and rspaces. Geometric analysis on riemannian and metric spaces september 59, 2016 room 420, research institute for mathematical sciences, kyoto university directions. Discriminant analysis on riemannian manifold of gaussian. The mean associated with the riemannian metric corresponds to the geometric mean. In this paper, firstly, we will show the following. Riemannian curvature of maximally symmetric spaces. Arpm lab characteristic function of a multivariate. Multivariate normal distributions parametrized as a riemannian symmetric space miroslav lovric 1 and maung minoo2 mcmaster university, hamilton, ontario, canada email. We propose to study a slightly different metric on the space of multivariate normal distributions on n.