Matrix power means and the Karcher mean

Lim, Yongdo; Pálfia, Miklós

doi:10.1016/j.jfa.2011.11.012

Cited by 155 publications

(158 citation statements)

References 23 publications

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“…Power means for positive definite matrices have recently been introduced by Lim and Pálfia [17]. Their notion and most of their results readily extend to the setting of positive operators on a Hilbert space, as we point out in this section.…”

Section: Power Meansmentioning

confidence: 98%

“…In [17] Lim and Pálfia have shown in the finite-dimensional setting that the Karcher or least-squares mean is the limit as t → 0 + of the (monotonically decreasing) family of power means P t . We take this characterization as the launch point for our approach to the infinite-dimensional Karcher mean.…”

Section: The Power Mean Limitmentioning

confidence: 99%

“…Section 3 on power means introduces an important tool for later developments, but the fact that well-behaved power means exist for the Hilbert operator setting is of independent interest. Lim and Pálfia [17] have recently shown in the finite-dimensional setting that the Karcher or least-squares mean is the limit as t → 0 + of the power means P t . We show additionally that they are monotonically decreasing, which allows us to deduce the existence of their strong limit.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Karcher means and Karcher equations of positive definite operators

Lawson

Lim

2014

Trans. Amer. Math. Soc. Ser. B

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105

116

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Abstract. The Karcher or least-squares mean has recently become an important tool for the averaging and studying 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 that each of these characterizations provide important insights about the Karcher mean.

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Section: Power Meansmentioning

confidence: 98%

Section: The Power Mean Limitmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Karcher means and Karcher equations of positive definite operators

Lawson

Lim

2014

Trans. Amer. Math. Soc. Ser. B

Self Cite

105

116

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show abstract

“…Section 3 introduces the important tool of power means, which we need to establish existence of the Karcher mean, but the fact that wellbehaved power means exist for the Hilbert operator setting is of independent interest. Lim and Pálfia (13) have recently shown in the finite-dimensional setting that the Karcher or least-squares mean is the limit as t → 0 + of the power means P t . We show additionally that they are monotonically decreasing, which allows us (Section 4) to deduce the existence of their limit in the strong topology in the general Hilbert space setting.…”

Section: Significancementioning

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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“…It has found numerous applications in the analysis of linear and nonlinear operators on cones, see for instance [1,13,17,26] and the references therein. Thompson's metric is also used to study the geometry of cones of positive operators [2,8,9,22] and symmetric cones [16,18,19,21], where it provides an alternative to the usual Riemannian metric. It also appears in the analysis of order-isomorphisms on cones, see [24,25].…”

Section: Introductionmentioning

confidence: 99%