Diffusion means in geometric spaces

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We introduce a location statistic for distributions on non-linear geometric spaces, the diffusion mean, serving as an extension and an alternative to the Fréchet mean. The diffusion mean arises as the generalization of Gaussian maximum likelihood analysis to non-linear spaces by maximizing the likelihood of a Brownian motion. The diffusion mean depends on a time parameter t, which admits the interpretation of the allowed variance of the diffusion. The diffusion t-mean of a distribution X is the most likely origin of a Brownian motion at time t, given the end-point distribution X. We give a detailed description of the asymptotic behavior of the diffusion estimator and provide sufficient conditions for the diffusion estimator to be strongly consistent. Particularly, we present a smeary central limit theorem for diffusion means and we show that joint estimation of the mean and diffusion variance rules out smeariness in all directions simultaneously in general situations. Furthermore, we investigate properties of the diffusion mean for distributions on the sphere Sm. Experimentally, we consider simulated data and data from magnetic pole reversals, all indicating similar or improved convergence rate compared to the Fréchet mean. Here, we additionally estimate t and consider its effects on smeariness and uniqueness of the diffusion mean for distributions on the sphere.

Udgave nummer4
Sider (fra-til)3141-3170
Antal sider30
StatusUdgivet - 2023

Bibliografisk note

Funding Information:
P. Hansen and S. Sommer gratefully acknowledge support by the Novo Nordisk Foundation grant NNF18OC0052000 and the Villum Foundation grants 22924 and 40582. B. Eltzner and S. Huckemann gratefully acknowledge funding by the DFG SFB 1456.

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© 2023 ISI/BS.

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