Invariant kernels on the space of complex covariance matrices

2024/03/28 by Said, Salem, Steinert, Franziskus, Mostajeran, Cyrus · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2404.02169

Abstract

The present work develops certain analytical tools required to construct and compute invariant kernels on the space of complex covariance matrices. The main result is the L¹--Godement theorem, which states that any invariant kernel, which is (in a certain natural sense) also integrable, can be computed by taking the inverse spherical transform of a positive function. General expressions for inverse spherical transforms are then provided, which can be used to explore new families of invariant kernels, at a rather moderate computational cost. A further, alternative approach for constructing new invariant kernels is also introduced, based on Ramanujan's master theorem for symmetric cones. This leads to a novel closed-form invariant kernel, called the Beta-prime kernel. Numerical experiments highlight the computational and performance advantages of this kernel, especially in the context of two-sample hypothesis testing.

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