Kernels, Associated Structures and Generalizations

By Matthias Hein, Olivier Bousquet. 2004

80 citations recorded by the source corpus.

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “Hence, they have the same eigenvalues (Hein and Bousquet 2004, Rosasco et al. 2010).”
Scalable Kernel Methods via Doubly Stochastic Gradients (2014, positive): “More speciﬁcally, Theorem 1 ( e.g. , Devinatz [16]; Hein & Bousquet [17]) If k ( x, x (cid:48) ) is a PD kernel, then there exists a set Ω , a measure P on Ω , and random function Essentially, the above integral representation relates the kernel function to a random process ω with measure P ( ω ) .”
Kernel Exponential Family Estimation via Doubly Dual Embedding (2018, background): “For arbitrary positive deﬁnite kernel, k ( x, x ), there exists a measure P on X , such that k ( x, x (cid:48) ) = φ w ( x ) φ w ( x (cid:48) ) d P ( w ) Devinatz (1953); Hein and Bousquet (2004), where φ w ( x ) : X → R from L 2 ( X , P ).”
Multi-output learning via spectral filtering (2012, background): “Towards this end, we first recall some examples of kernels defining vector-valued RKHS and their connection to the regularizer choice, while a more complete list of kernels is reported in Appendix A.”
Kernel Methods for Measuring Independence (2005, background): “Our presentation follows and extends the work of Zwald et al. (2004); Hein and Bousquet (2004), who deal with covariance operators froma space to itself rather than from one space to another, and Fukumizu et al. (2004), who use covariance operators as a means of defining conditional covariance…”
Approximate Kernel-Based Conditional Independence Tests for Fast Non-Parametric Causal Discovery (2017, background): “Hein and Bousquet [20] showed that any continuous kernel on a separable space X (e.”
Mercer’s Theorem on General Domains: On the Interaction between Measures, Kernels, and RKHSs (2012, background): “The first sufficient condition, which is a straightforward adaptation from [16], is usually easy to check.”
Matrix-valued kernels for shape deformation analysis (2013, background): “Such theory can be extended to vector-valued functions [15, 30, 53] and has recently been adapted to the study of shape and deformation [58].”
Efficient Output Kernel Learning for Multiple Tasks (2015, methodology): “However, in this paper we use the one-to-one correspondence between real-valued and matrix-valued kernels, see [21], in order to limit the technical overhead.”