Riemannian geometry and matrix geometric means

Bhatia, Rajendra; Holbrook, John

doi:10.1016/j.laa.2005.08.025

Cited by 224 publications

(179 citation statements)

References 9 publications

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“…The GM (4.1) has numerous attractive properties-see for instance [1]-among which, the following variational characterization is very important [11]:…”

Section: Geometric Meanmentioning

confidence: 99%

Positive definite matrices and the S-divergence

Sra

2015

Proc. Amer. Math. Soc.

110

139

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Abstract. Positive definite matrices abound in a dazzling variety of applications. This ubiquity can be in part attributed to their rich geometric structure: positive definite matrices form a selfdual convex cone whose strict interior is a Riemannian manifold. The manifold view is endowed with a "natural" distance function while the conic view is not. Nevertheless, drawing motivation from the conic view, we introduce the S-Divergence as a "natural" distance-like function on the open cone of positive definite matrices. We motivate the S-divergence via a sequence of results that connect it to the Riemannian distance. In particular, we show that (a) this divergence is the square of a distance; and (b) that it has several geometric properties similar to those of the Riemannian distance, though without being computationally as demanding. The S-Divergence is even more intriguing: although nonconvex, we can still compute matrix means and medians using it to global optimality. We complement our results with some numerical experiments illustrating our theorems and our optimization algorithm for computing matrix medians.Key words. Bregman matrix divergence; Log Determinant; Stein Divergence; Jensen-Bregman divergence; matrix geometric mean; matrix median; nonpositive curvature 1. Introduction. Hermitian positive definite (HPD) matrices are a noncommutative generalization of positive reals. They abound in a multitude of applications and exhibit attractive geometric properties-e.g., they form a differentiable Riemannian (also Finslerian) manifold [10,33] that is a well-studied example of a manifold of nonpositive curvature [17, Ch.10]. HPD matrices possess even more structure: (i) they embody a canonical higher-rank symmetric space [51]; and (ii) their closure forms a closed, self-dual convex cone.The convex conic view enjoys great importance in convex optimization [6,43,44] and in nonlinear Perron-Frobenius theory [40]; symmetric spaces are important in algebra, analysis [32,39,51], and optimization [43,52]; while the manifold view (Riemannian or Finslerian) plays diverse roles-see [10, Ch.6] and [46].The manifold view is equipped with a with a "natural" distance function while the conic view is not. Nevertheless, drawing motivation from the convex conic view, we introduce the S-Divergence as a "natural" distance-like function on the open cone of positive definite matrices. Indeed, we prove a sequence of results connecting the S-Divergence to the Riemannian distance. Most importantly, we show that (a) this divergence is the square of a distance; and (b) that it has several geometric properties in common with the Riemannian distance, without being numerically as demanding. This builds an informal link between the manifold and conic views of HPD matrices.1.1. Background and notation. We begin by fixing notation. The letter H denotes some Hilbert space, usually just C n . The inner product between two vectors x and y in H is x, y := x * y (x * denotes 'conjugate transpose'). The set of n × n Hermitian matrices is denoted as H...

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“…The GM (4.1) has numerous attractive properties-see for instance [1]-among which, the following variational characterization is very important [11]:…”

Section: Geometric Meanmentioning

confidence: 99%

Positive definite matrices and the S-divergence

Sra

2015

Proc. Amer. Math. Soc.

110

139

View full text Add to dashboard Cite

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“…This problem has been introduced by many authors. In [3], the authors studied the map φ(A) = SAS * for any invertible matrix S. They have proved that L g 1 (ρ) is invariant under this map. In [5,Proposition 2.3], the authors have shown that A → A −1 also preserves the length with respect to the metric K g 1 .…”

Section: Geodesic Distance Isometries and Geodesic-affine Mapsmentioning

confidence: 99%

“…It leads to the socalled Fisher-Rao metric which is defined by K ϕ D (H, K) = tr D −1 HD −1 K. We note, that this metric plays a significant role in the recent development of the geometric mean of matrices. By [3,10,12] it is known that the geodesic in this Riemannian manifold between A, B ∈ P n is given by…”

Section: Introductionmentioning

confidence: 99%

Preserving problems of geodesic-affine maps and related topics on positive definite matrices

Szokol

Tsai

Zhang

2015

Linear Algebra and its Applications

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Abstract. Based on affine maps in geometry, we study the geodesic-affine maps on Riemannian manifolds Pn of complex positive definite matrices that are induced by different so-called kernel functions. In this article, we are going to describe the structure of all continuous bijective geodesic-affine maps on these manifolds. We also prove that geodesic distance isometries are geodesic-affine maps. Moreover, the forms of all bijective maps which preserve norms of geodesic correspondence are characterized. Indeed, these maps are special examples of geodesic-affine maps.

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“…The Ando-Li-Mathias paper was also important for listing, and deriving for their mean, 10 desirable properties for multivariable geometric means. Moaker and Bhatia and Holbrook were able to establish a number of these important properties for the least-squares mean, but the important question of the monotonicity of this mean, conjectured by Bhatia and Holbrook (5), was left open. However, the authors were recently able to show (9) that all of the properties, in particular the monotonicity, are satisfied in the more general setting of weighted means for any weight ω = ðw 1 ; .…”

Section: Introductionmentioning

confidence: 99%

Weighted means and Karcher equations of positive operators

Lawson

Lim

2013

Proc. Natl. Acad. Sci. U.S.A.

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The Karcher or least-squares mean has recently become an important tool for the averaging and study of positive definite matrices. In this paper, we show that this mean extends, in its general weighted form, to the infinite-dimensional setting of positive operators on a Hilbert space and retains most of its attractive properties. The primary extension is via its characterization as the unique solution of the corresponding Karcher equation. We also introduce power means P t in the infinite-dimensional setting and show that the Karcher mean is the strong limit of the monotonically decreasing family of power means as t → 0 + . We show each of these characterizations provide important insights about the Karcher mean.

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Riemannian geometry and matrix geometric means

Cited by 224 publications

References 9 publications

Positive definite matrices and the S-divergence

Positive definite matrices and the S-divergence

Preserving problems of geodesic-affine maps and related topics on positive definite matrices

Weighted means and Karcher equations of positive operators

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