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Journal Article

Statistical Analysis of Random Objects Via Metric Measure Laplacians


Munk,  Axel
Research Group of Statistical Inverse Problems in Biophysics, Max Planck Institute for Multidisciplinary Sciences, Max Planck Society;

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Mordant, G., & Munk, A. (2023). Statistical Analysis of Random Objects Via Metric Measure Laplacians. SIAM Journal on Mathematics of Data Science, 5(2), 528-557. doi:10.1137/22M1491022.

Cite as: https://hdl.handle.net/21.11116/0000-000E-351A-A
In this paper, we consider a certain convolutional Laplacian for metric measure spaces and investigate its potential for the statistical analysis of complex objects. The spectrum of that Laplacian serves as a signature of the space under consideration and the eigenvectors provide the principal directions of the shape, its harmonics. These concepts are used to assess the similarity of objects or understand their most important features in a principled way which is illustrated in various examples. Adopting a statistical point of view, we define a mean spectral measure and its empirical counterpart. The corresponding limiting process of interest is derived and statistical applications are discussed.