Positive Definite Matrices

Bhatia, Rajendra

doi:10.1515/9781400827787

Cited by 805 publications

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“…The function z → z α is operator concave on L + (H) for 0 ≤ α ≤ 1 and operator convex on L + (H) for 1 ≤ α ≤ 2 (see, respectively, theorems 4.2.3 and 1.5.8 in [4]). So the function f α (z) = z α − z /(α − 1) is operator convex for α ∈ [0, 2] and α = 1.…”

Section: Definitions and Notationmentioning

confidence: 99%

Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

Rastegin

2013

Math Phys Anal Geom

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Pinsker's and Fannes' type bounds on the Tsallis relative entropy are derived. The monotonicity property of the quantum f -divergence is used for its estimating from below. For order α ∈ (0, 1), a family of lower bounds of Pinsker type is obtained. For α > 1 and the commutative case, upper continuity bounds on the relative entropy in terms of the minimal probability in its second argument are derived. Both the lower and upper bounds presented are reformulated for the case of Rényi's entropies. The Fano inequality is extended to Tsallis' entropies for all α > 0. The deduced bounds on the Tsallis conditional entropy are used for obtaining inequalities of Fannes' type.

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Section: Definitions and Notationmentioning

confidence: 99%

Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

Rastegin

2013

Math Phys Anal Geom

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“…The mathematics of Riemannian manifolds, Cartan-Hadamard manifolds, and P(n) is well-understood; the book by Bridson and Haefliger [8] is an invaluable reference on metric spaces of nonpositive curvature, and Bhatia [5] provides a detailed study of P(n) in particular. However, there are very few algorithmic results for problems in these spaces.…”

Section: Related Workmentioning

confidence: 99%

“…. , X d , where X d is the convex hull of X 0 [5]. This is the convex combination of X 0 , and the boundary is partitioned into a finite number of faces uniquely determined by painting segments among d-point subsets of X 0 .…”

Section: Introductionmentioning

confidence: 99%

Horoball Hulls and Extents in Positive Definite Space

Fletcher

Moeller

Phillips

et al. 2011

Lecture Notes in Computer Science

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Abstract. The space of positive definite matrices P(n) is a Riemannian manifold with variable nonpositive curvature. It includes Euclidean space and hyperbolic space as submanifolds, and poses significant challenges for the design of algorithms for data analysis. In this paper, we develop foundational geometric structures and algorithms for analyzing collections of such matrices. A key technical contribution of this work is the use of horoballs, a natural generalization of halfspaces for non-positively curved Riemannian manifolds. We propose generalizations of the notion of a convex hull and a centerpoint and approximations of these structures using horoballs and based on novel decompositions of P(n). This leads to an algorithm for approximate hulls using a generalization of extents.

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“…Since (1) ⇒ (2) is trivial and (3) ⇒ (1) is a well-known result (see e.g. Chapter 4 in [2]), it is enough to verify that (2) ⇒ (3) holds. Consider the continuous map ψ :…”

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confidence: 95%

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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Positive Definite Matrices

Cited by 805 publications

References 0 publications

Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

Horoball Hulls and Extents in Positive Definite Space

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

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