Universal Kernels on Non-Standard Input Spaces

By Andreas Christmann, Ingo Steinwart. 2010

115 citations recorded by the source corpus.

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In Statistical Learning Theory

Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “For example, Christmann and Steinwart (2010) considers a Gaussian-like kernel”
Kernel-Based Just-In-Time Learning for Passing Expectation Propagation Messages (2015, methodology): “Rather, we use a general characteristic kernel on input distributions [Christmann and Steinwart, 2010, eq.”
K2-ABC: Approximate Bayesian Computation with Infinite Dimensional Summary Statistics via Kernel Embeddings (2015, background): “…are possible on kernel embeddings - one can define a positive definite kernel on probability measures themselves using their representation in a Hilbert space, e.g., κ (Fx, Fy) = exp ( −MMD 2(Fx, Fy) ) , > 0 is an example of a kernel on probability measures (Christmann & Steinwart, 2010).”
Learning Theory for Distribution Regression (2014, methodology): “Methods based on parametric assumption in a RKHS: A heuristic related to the parametric approach is to assume that the training distributions are Gaussians in a reproducing kernel Hilbert space (see for example Jebara et al., 2004; Zhou and Chellappa, 2006, and references therein).”
Kernel Mean Shrinkage Estimators (2014, background): “, κ(Pi,Pj) = exp(‖μ̂Pi − μ̂Pj‖H/2σ), see Christmann and Steinwart (2010). Our goal in this experiment is to investigate if the shrinkage estimators of the kernel mean improve the performance of discriminative learning on distributions.”
Distribution-Free Distribution Regression (2013, background): “In this framework, the role of universal kernels have been studied by Christmann and Steinwart [2010]. Recently, the representer theorem has also been generalized for the space of probability distributions [Muandet et al.”
Domain Generalization via Invariant Feature Representation (2013, background): “Here, k X is the kernel on H X and the kernel on distributions is k P ( P , P ′ ) := κ ( µ P , µ P ′ ) where κ is a positive deﬁnite kernel (Christmann and Steinwart 2010, Muandet et al. 2012).”
Distribution to Distribution Regression (2013, background): “In this framework, the role of universal kernels have been studied by Christmann and Steinwart [2010]. Recently, the representer theorem has also been generalized for the space of probability distributions [Muandet et al.”
Domain Generalization by Marginal Transfer Learning (2017, methodology): “Fortunately, this was studied by Christmann and Steinwart (2010).”
Multi-Task Learning for Contextual Bandits (2017, methodology): “To define a kernel on Z = PX , we use the same construction described in [5], originally introduced by Steinwart and Christmann [8].”
Generalizing from Several Related Classification Tasks to a New Unlabeled Sample (2011, background): “…decomposition: sup f∈Bk(R) ∣∣∣∣∣∣ 1N N∑ i=1 1 ni ni∑ j=1 `(f(X̃ij), Yij)− E(f,∞) ∣∣∣∣∣∣ ≤ sup f∈Bk(R) ∣∣∣∣∣∣ 1N N∑ i=1 1 ni ni∑ j=1 ( `(f(P̂ (i)X , Xij), Yij)− `(f(P (i) X , Xij), Yij) )∣∣∣∣∣∣ + sup f∈Bk(R) ∣∣∣∣∣∣ 1N N∑ i=1 1 ni ni∑ j=1 `(f(P (i)X , Xij), Yij)− E(f,∞) ∣∣∣∣∣∣ =: (I) + (II).”
Measuring Association on Topological Spaces Using Kernels and Geometric Graphs (2020, background): “With the above examples of graph functionals in mind, recall that the next step in computing η̂n is to construct characteristic kernels on Y which is an area of active research in the machine learning community (see [39, 28, 26, 98]).”
Separability of reproducing kernel spaces (2015, background): “7 & 12], [7], [6], Steinwart and Christmann [34], [35], De Vito, Rosasco and Toigo [9], Hable and Christmann [18, Thm.”