The number of mixture components equals the rank of an integral operator identified from the data, and counting its singular values above a data-driven threshold yields a consistent estimator.
As the Hilbert-Schmidt norm is an inner product norm, we have ‖Th,x−Th,x′‖2 HS =‖Th,x‖2 HS +‖Th,x′‖2 HS− 2⟨Th,x,Th,x′⟩HS, where⟨·,·⟩HS denotes the Hilbert-Schmidt inner product
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Estimation of the Number of Components of Non-Parametric Multivariate Finite Mixture Models
The number of mixture components equals the rank of an integral operator identified from the data, and counting its singular values above a data-driven threshold yields a consistent estimator.