The authors introduce sliced and Lloyd-based dispersion regularizers for hyperspherical embeddings, connect kernel dispersion objectives to maximum mean discrepancy, and show downstream gains in prototype classification and neural machine translation.
Positive definite singular kernels on two-point homogeneous spaces
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abstract
We study positive definiteness of kernels $K(x,y)$ on two-point homogeneous spaces. As opposed to the classical case, which has been developed and studied in the existing literature, we allow the kernel to have an (integrable) singularity for $x=y$. Specifically, the Riesz kernel $d(x,y)^{-s}$ (where $d$ denotes some distance on the space) is a prominent example. We derive results analogous to Schoenberg's characterization of positive definite functions on the sphere, Schur's lemma on the positive definiteness of the product of positive definite functions, and Schoenberg's characterization of functions positive definite on all spheres. We use these results to better understand the behavior of the Riesz kernels for the geodesic and chordal distances on projective spaces.
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cs.LG 1years
2025 1verdicts
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Keep your distance: learning dispersed embeddings on $\mathbb{S}_m$
The authors introduce sliced and Lloyd-based dispersion regularizers for hyperspherical embeddings, connect kernel dispersion objectives to maximum mean discrepancy, and show downstream gains in prototype classification and neural machine translation.