A differential geometric approach to the geometric mean of. Riemannian geometry and matrix geometric means request pdf. The resulting riemannian space has strong geometrical properties. The former restricts attention to submanifolds of euclidean space while the latter studies manifolds equipped with a riemannian metric. It satisfies most part of the ten properties stated by ando, li and mathias. Two applications computing an invariant subspace of a matrix and the mean of subspaces are. The riemannian mean of positive matrices lixpolytechnique. Their main purpose is to introduce the beautiful theory of riemannian geometry, a still very active area of mathematical research. The riemannian mean of positive matrices rajendra bhatia 2. Riemannian geometry and matrix geometric means core. We show that for n 4 a new matrix mean exists which is simpler to compute than the existing ones. Riemannian geometry is the branch of differential geometry that studies riemannian manifolds, smooth manifolds with a riemannian metric, i. Progress was made recently by making a connection with differential geometry. Symmetric positive defined spd matrix has attracted increasing research focus in imagevideo analysis, which merits in capturing the riemannian geometry in its structured 2d feature representation.
Recent work in the study of the geometric mean of positive definite matrices has. It introduces geometry on manifolds, tensor analysis, pseudo riemannian geometry. Positive definite matrix, geometric mean, riemannian metric. Ju n 20 09 constructing matrix geometric means semantic scholar. It contains many interesting results and gives excellent descriptions of many of the constructions and results in di. Riemannian geometry of grassmann manifolds with a view on. One of its important operator theoretic properties, monotonicity in the m arguments, has been established recently by lawson and lim. The geometric mean of two positive definite matrices has been defined in several ways and studied by several authors, including pusz and woronowicz, and ando. Geometric mean, positivedefinite symmetric matrices, riemannian distance.
I present images from the schwarzschild geometry to support this result pictorially and to lend geometric intuition to the abstract notion of ricci curvature for the pseudo riemannian manifolds of general relativity. The characterizations by these authors do not readily extend to three matrices and it has been a longstanding problem to define a natural geometric mean of three positive definite matrices. Geometric means in a novel vector space structure on symmetric positivedefinite matrices. The mean associated with the riemannian metric corresponds to the geometric mean. The matrix geometric mean indian statistical institute. Pdf a new definition is introduced for the matrix geometric mean of a set of k. Moreover, we show that for n 4 the existing strategies of composing matrix means and.
Riemannian geometry of symmetric positive definite matrices via. S i s j f where s i p 1 2 log p p ip 1 2 and p is the riemannian mean of fp ig. Riemannian geometry has emerged as one of the most powerful mathematical and computational frameworks for analyzing such data. Jun 16, 2010 the new mean is obtained from the properties of the centroid of a triangle rephrased in terms of geodesics in a suitable riemannian geometry on the set of positive definite matrices. A note on geometric mean of positive matrices inase. Request pdf riemannian geometry and matrix geometric means the geometric mean of two positive definite matrices has been defined in several ways and studied by several authors, including pusz. Introduction to geometry and geometric analysis oliver knill this is an introduction into geometry and geometric analysis, taught in the fall term 1995 at caltech. The riemannian mean and matrix inequalities related to the andohiai inequality and chaotic order takeakiyamazaki abstract. The mean associated with the euclidean metric of the ambient space is the usual arithmetic mean. We discuss some invariance properties of the riemannian mean and we use di. M is called boundary of m and is a manifold of dimension n. An attractive candidate for the geometric mean of m positive definite matrices a 1.
However, computation in the vector space on spd matrices cannot capture the geometric properties, which corrupts the classification performance. The geometric mean of two positive definite matrices has been defined in several ways and studied by several authors, including pusz and. Riemannian geometry and geometric means of fixedrank. The riemannian mean of a general number of matrices can be computed using an iterative procedure 4, 11. Extension of the furuta inequality and andohiai log. Riemannian geometry and matrix geometric means sciencedirect. Henriques, jorge batista institute for systems and robotics, faculty of science and technology, university of coimbra, 3030 coimbra, portugal. Frechet mean, symmetric positive definite matrix, lie group, bi invariant metric. This gives, in particular, local notions of angle, length of curves, surface area and volume.
A differential geometric approach to the geometric mean cis upenn. They are indeed the key to a good understanding of it and will therefore play a major role throughout. We will follow the textbook riemannian geometry by do carmo. Riemannian geometry and geometric means of fixedrank positive. Anne collard, rodolphe sepulchre ulg mines paristech silvere. The newton method on abstract riemannian manifolds proposed by s. This is a subject with no lack of interesting examples. Probabilities and statistics on a riemannian manifold 11 2. The extrinsic theory is more accessible because we can visualize curves and. Riemannian geometry, also called elliptic geometry, one of the non euclidean geometries that completely rejects the validity of euclid s fifth postulate and modifies his second postulate. Riemannian geometry and matrix geometric means, linear.
A note on computing matrix geometric means springerlink. In this paper we introduce metricbased means for the space of positivedefinite matrices. Mar 01, 2006 riemannian geometry and matrix geometric means riemannian geometry and matrix geometric means bhatia, rajendra. The theory of plane and space curves and surfaces in the threedimensional euclidean space formed the basis for development of differential geometry during. Lecture 1 notes on geometry of manifolds lecture 1 thu. Euclidean metric of the ambient space is the usual arithmetic mean. Riemannian geometric statistics in medical image analysis is a complete reference on statistics on riemannian manifolds and more general nonlinear spaces with applications in medical image analysis. A geometric understanding of ricci curvature in the context. Learning neural bagofmatrixsummarization with riemannian. Two kinds of isometric families of riemannian metrics 4. Riemannian geometry is a multidimensional generalization of the intrinsic geometry cf.
In this paper, we analyze the process of assembling new matrix geometric means from existing ones, and show what new means can be found, and what cannot be done because of grouptheoretical obstructions. General relativity is used as a guiding example in the last part. Lastly, we denote the vector representation of any. Probabilities and statistics on riemannian manifolds. The riemannian mean and matrix inequalities related to the andohiai inequality and chaotic orderj. Riemannian geometry and geometric means of fixedrank positive semidefinite matrices. The riemannian metric is is given by the matrix function g iju which is. In some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of a and b as the midpoint of the geodesic with respect to a natural riemannian metric joining a and b.
We discuss some invariance properties of the riemannian mean and we use differential geometric tools to give a characterization of this mean. In these formulas, pplanes are represented as the column space of n. One can distinguish extrinsic di erential geometry and intrinsic di erential geometry. Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. Pdf a note on computing matrix geometric means researchgate.
You have to spend a lot of time on basics about manifolds, tensors, etc. Interior geometry of twodimensional surfaces in the euclidean space. The metric of a riemannian space coincides with the euclidean metric of the domain under consideration up to the first order of smallness. The proposed metric is derived from a wellchosen riemannian quotient geometry that generalizes the reductive geometry of the positive cone and the associated natural metric. Riemannian geometric statistics in medical image analysis. A nonparametric riemannian framework on tensor field with. This text is fairly classical and is not intended as an introduction to abstract 2dimensional riemannian.
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