Characteristic Kernels on Groups and Semigroups

By Kenji Fukumizu, Bharath K. Sriperumbudur, Arthur Gretton, Bernhard Schölkopf. 2008

111 citations recorded by the source corpus.

Subfields: Statistical Learning Theory, Pure Mathematics

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “…and bounded kernel k is said to be strictly positive definite on a topological space M if and only ifs M k(x,y) dν(x) dν(y) > 0 for all finite non-zero signed Borel measures ν on M. Fukumizu et al. (2009b) also provides numerous examples of characteristic kernels on groups and semi-groups.”
Characteristic Kernels and Infinitely Divisible Distributions (2014, methodology): “The conditions are explored according to the type of domain X and the class of kernel (Gretton et al., 2007; Fukumizu et al., 2008, 2009a,b; Sriperumbudur et al., 2010).”
Filtering with State-Observation Examples via Kernel Monte Carlo Filter (2013, methodology): “Corollary 3 follows from theorem 5 of Fukumizu et al. (2013) and corollary 2.”
On the optimal estimation of probability measures in weak and strong topologies (2013, background): “…R X By deﬁning any kernel for which D k is a metric as the characteristic kernel , it is easy to see that if any k ∈ K is characteristic, then · k F H is a metric on P K . and k k The question of when is D k a metric on M 1+ ( addressed in [13, 14, 21, x d P ( x ) < ∞} (and hence on M 1+ ( R d )).”
A Kernel Test for Three-Variable Interactions (2013, background): “, for all topological spaces where it is possible to construct a valid positive definite function; see [23] for details.”
Testing Hypotheses by Regularized Maximum Mean Discrepancy (2013, background): “The reader is referred to [4], [8] for a detailed study on these kernels.”
Hypothesis testing using pairwise distances and associated kernels (2012, background): “L G ] 2 1 M ay”
Kernel Belief Propagation (2011, methodology): “Our approach applies not only to continuous-valued non-Gaussian variables, but also generalizes to strings and graphs (Schölkopf et al., 2004), groups (Fukumizu et al., 2009), compact manifolds (Wendland, 2005, Chapter 17), and other domains on which kernels may be defined.”
Kernel Bayes' Rule (2010, background): “mplies that the RKHS deﬁned by a characteristic 4 kernel is rich enough to be dense in L2 space up to the constant functions. Other useful conditions for a kernel to be characteristic can be found in [12, 35, 36]. Throughout this paper, when positive deﬁnite kernels on a measurable space are discuss”
Universality, Characteristic Kernels and RKHS Embedding of Measures (2010, methodology): “See Carmeli et al. (2009, Corollary 5), Steinwart and Christmann (2008, Proposition 4.54, Example 4.11) and Sriperumbudur et al. (2009b, Footnote 4).”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, methodology): “Fukumizu et al. (2008, 2009a) extended this characterization to non-compact M and showed that H is characteristic if and only if the direct sum of H and R is dense in the Banach space of r-integrable (for some r ≥ 1) functions.”
On integral probability metrics, φ-divergences and binary classification (2009, background): “e γF(P,Q) = 0 ⇔ φP = φQ ⇔ P = Q, where φP and φQ represent the characteristic functions of P and Q, respectively.4 Therefore, the function class F= {eihω,.i : ω ∈ Rd} induces a metric on P. Recently, [14, 12, 27, 13] considered F to be a unit ball in a reproducing kernel Hilbert space [2, 25] and de”
Characteristic Kernels on Structured Domains Excel in Robotics and Human Action Recognition (2010, methodology): “Theorems 13 and 14 of [7] obtain necessary and sufficient conditions for tailored kernels for histogram-based information.”
Kernels Based Tests with Non-asymptotic Bootstrap Approaches for Two-sample Problems (2012, background): “From Lemma 4 in Gretton et al. (2008), we know that this quantity corresponds to n2 times the squared Maximum Mean Discrepancy on the unit ball in the RKHS HK .”
Convergence rates for a class of estimators based on Stein’s method (2016, result): “, Defn. 4.1). Chwialkowski et al. (2016, Thm. 2.1) proved that when His cc-universal, H 0 is characteristic, meaning that T( 0) = 0 if and only if 0= , where T( 0) := sup k k H + 1 0[ ]: Separately, Fukumizu et al. (2009, Lem. 1) showed that, when a reproducing kernel Hilbert space H 0 is characteri”
A Fast, Consistent Kernel Two-Sample Test (2009, background): “Whe n the embedding is injective, the RKHS is said to becharacteristic[11, 29, 12], and the distance between feature mappings cons titute a metric on distributions.”