Means in complete manifolds: uniqueness and approximation. (27th March 2014)

• Record Type: Journal Article
• Authors: Arnaudon, Marc; Miclo, Laurent
• Is Part Of: ESAIM. Volume 18(2014)
• Page Start: 185
• Page End: 206
• Publication Date: 2014-03-27
• Subjects: Probabilities -- Periodicals; Mathematical statistics -- Periodicals; 519.2
• DOI: 10.1051/ps/2013033
• Languages: English
• ISSNs: 1292-8100

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>Let <italic>M</italic> be a complete Riemannian manifold, <italic>M</italic> ∈ ℠and <italic>p</italic> ≥ 1. We prove that almost everywhere on <italic>x</italic> = (<italic>x</italic><sub>1</sub>, <italic>...</italic>, <italic>x</italic><sub><italic>N</italic></sub>) ∈ <italic>M</italic><sup><italic>N</italic></sup> for Lebesgue measure in <italic>M</italic><sup><italic>N</italic></sup>, the measure <inline-formula><alternatives><tex-math id="tex_eq1"><![CDATA[\hbox{$\di \mu(x)=\f1N\sum_{k=1}^N\d_{x_k}$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgr3rg1" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq1" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="italic">μ</mml:mi><mml:mo mathvariant="normal">(</mml:mo><mml:mi mathvariant="italic">x</mml:mi><mml:mo mathvariant="normal">)</mml:mo><mml:mo mathvariant="normal">=</mml:mo><mml:mrow><mml:mfrac><mml:mn mathvariant="normal">1</mml:mn><mml:mi mathvariant="italic">N</mml:mi></mml:mfrac></mml:mrow><mml:munderover><mml:mo>âˆ'</mml:mo><mml:mrow><mml:mi mathvariant="italic">k</mml:mi><mml:mo mathvariant="normal">=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mi mathvariant="italic">N</mml:mi></mml:munderover><mml:msub><mml:mi mathvariant="italic">δ</mml:mi><mml:msub><mml:mi mathvariant="italic">x</mml:mi><mml:mi mathvariant="italic">k</mml:mi></mml:msub></mml:msub></mml:math></alternatives></inline-formula> has a unique <italic>p</italic>â€"mean <italic>e</italic><sub><italic>p</italic></sub>(<italic>x</italic>). As a consequence, if <italic>X</italic> = (<italic>X</italic><sub>1</sub>, <italic>...</italic>, <italic>X</italic><sub><italic>N</italic></sub>) is a <italic>M</italic><sup><italic>N</italic></sup>-valued random variable with absolutely continuous law, then almost surely <italic>μ</italic>(<italic>X</italic>(<italic>ω</italic>)) has a unique <italic>p</italic>â€"mean. In particular if (<italic>X</italic><sub><italic>n</italic></sub>)<sub><italic>n</italic> ≥ 1</sub> is an independent sample of an absolutely continuous law in <italic>M</italic>, then the process <italic>e</italic><sub><italic>p, n</italic></sub>(<italic>ω</italic>) = <italic>e</italic><sub><italic>p</italic></sub>(<italic>X</italic><sub>1</sub>(<italic>ω</italic>), <italic>...</italic>, <italic>X</italic><sub><italic>n</italic></sub>(<italic>ω</italic>)) is well-defined. Assume <italic>M</italic> is compact and consider a probability measure <italic>ν</italic> in <italic>M</italic>. Using partial simulated annealing, we define a continuous semimartingale which converges in probability to the set of minimizers of the integral of distance at power <italic>p</italic> with respect to <italic>ν</italic>. When the set is a singleton, it converges to the <italic>p</italic>â€"mean.</p> </abstract>