Characteristic and Universal Tensor Product Kernels

Zoltán Szabó (1) , Bharath K Sriperumbudur (2)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 Department of Statistics [PennState]

Maximum mean discrepancy (MMD), also called energy distance or N-distance in statistics and Hilbert-Schmidt independence criterion (HSIC), specifically distance covariance in statistics, are among the most popular and successful approaches to quantify the difference and independence of random variables, respectively. Thanks to their kernel-based foundations, MMD and HSIC are applicable on a wide variety of domains. Despite their tremendous success, quite little is known about when HSIC characterizes independence and when MMD with tensor product kernel can discriminate probability distributions. In this paper, we answer these questions by studying various notions of characteristic property of the tensor product kernel.

Mots clés: tensor product kernel; kernel mean embedding; characteristic kernel; I-characteristic kernel; universality; maximum mean discrepancy; Hilbert-Schmidt independence criterion

Domaines: Autres [stat.ML]; Théorie de l'information et codage [math.IT]; Méthodologie [stat.ME]; Théorie [stat.TH]

Zoltán Szabó, Bharath K Sriperumbudur. Characteristic and Universal Tensor Product Kernels. Journal of Machine Learning Research, 2018, 18, pp.233. ⟨hal-01585727v3⟩