Upper and lower bounds for the arithmetic-geometric-harmonic means of positive definite matrices

Sagae, Masahiko; Tanabe, Kunio

doi:10.1080/03081089408818331

Cited by 38 publications

(11 citation statements)

References 2 publications

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“…, x k ) = s k . In [28] Sturm has called this mean the inductive mean for NPC spaces, a mean which appeared earlier in [27,1] for positive definite matrices. Its explicit definition is given inductively by…”

Section: The Methodsmentioning

confidence: 99%

Monotonic properties of the least squares mean

2010

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We settle an open problem of several years standing by showing that the least squares mean for positive definite matrices is monotone for the usual (Loewner) order. Indeed we show this is a special case of its appropriate generalization to partially ordered complete metric spaces of nonpositive curvature. Our techniques extend to establish other basic properties of the least squares mean such as continuity and joint concavity. Moreover, we introduce a weighted least squares mean and derive our results in this more general setting.

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Section: The Methodsmentioning

confidence: 99%

Monotonic properties of the least squares mean

2010

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“…To remedy this problem, another family of means was proposed in [8]. These means are based on the inductive mean (see [9]). The inductive mean of a set of SPD matrices Σ 1 , .…”

Section: Inductive Means and Sequencesmentioning

confidence: 99%

Inductive Means and Sequences Applied to Online Classification of EEG

Massart

Chevallier²

2017

Lecture Notes in Computer Science

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Abstract. The translation of brain activity into user command, through Brain-Computer Interfaces (BCI), is a very active topic in machine learning and signal processing. As commercial applications and out-of-the-lab solutions are proposed, there is an increased pressure to provide online algorithms and real-time implementations. Electroencephalography (EEG) systems offer lightweight and wearable solutions, at the expense of signal quality. Approaches based on covariance matrices have demonstrated good robustness to noise and provide a suitable representation for classification tasks, relying on advances in Riemannian geometry. We propose to equip the minimum distance to mean (MDM) classifier with a new family of means, based on the inductive mean, for block-online classification tasks and to embed the inductive mean in an incremental learning algorithm for online classification of EEG.

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“…There is no formal definition of geometric mean of finite number of positive definite matrices and defining multivariable geometric mean is a non-trivial task and is a recent topic of interest in core linear algebra. A simple and direct method of extending the two-variable geometric mean is the convex combination procedure [1,36,26]:…”

Section: Multivariable Geometric Meansmentioning

confidence: 99%

Factorizations and geometric means of positive definite matrices

Lim

2012

Linear Algebra and its Applications

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In this paper we provide a new class of (metric) geometric means of positive definite matrices varying over Hermitian unitary matrices. We show that each Hermitian unitary matrix induces a factorization of the cone P m of m × m positive definite Hermitian matrices into geodesically convex subsets and a Hadamard metric structure on P m . An explicit formula for the corresponding metric midpoint operation is presented in terms of the geometric and spectral geometric means and show that the resulting two-variable mean is different to the standard geometric mean. Some basic properties comparable to those of the geometric mean and its extensions to finite number of positive definite matrices are studied.

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Upper and lower bounds for the arithmetic-geometric-harmonic means of positive definite matrices

Cited by 38 publications

References 2 publications

Monotonic properties of the least squares mean

Monotonic properties of the least squares mean

Inductive Means and Sequences Applied to Online Classification of EEG

Factorizations and geometric means of positive definite matrices

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