Dimensionality Reduction for Supervised Learning with Reproducing Kernel Hilbert Spaces

By Kenji Fukumizu, Francis R. Bach, Michael I. Jordan. 2004

816 citations recorded by the source corpus.

Subfields: Representation & Self-Supervised Learning, Statistical Learning Theory

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “= ·] is an element of H for all g ∈ G may not hold in general (Fukumizu et al. 2004, Song et al. 2009).2 This technical issue can be circumvented by resorting to a regularized version of (4.5), i.e., CYX (CXX + εI)−1k(x, ·) where ε > 0 denotes a regularization parameter.”
Domain Adaptation with Conditional Transferable Components (2016, methodology): “We adopt the kernel dimensionality reduction framework (Fukumizu et al., 2004) to achieve so.”
Kernel-based Conditional Independence Test and Application in Causal Discovery (2011, methodology): “First, X and Y are not necessarily disjoint, and H X and H Y can be any RKHS’s.”
Learning Theory for Distribution Regression (2014, methodology): “When bounded kernels are used, mean embeddings exist for all probability measures (Fukumizu et al., 2004).”
Kernel Mean Shrinkage Estimators (2014, background): “That is, it preserves all information about the distribution (Fukumizu et al., 2004; Sriperumbudur et al., 2008).”
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, background): “l k: XX! Ris called positive deﬁnite (p.d.), if for all n2N, c 1;:::;c n 2R, and X 1;:::;X n 2X, we have Xn i=1 Xn j=1 c ic jk(X i;X j) 0: 2 kis bounded on Xif sup x2X k(x;x) <1. 7 is one-to-one (Fukumizu et al., 2004, 2008; Sriperumbudur et al., 2010). This means that the RKHS is rich enough to ”
Density Estimation in Infinite Dimensional Exponential Families (2013, methodology): “The mean and covariance operators have found many applications in nonparametric hypothesis testing [20, 18] and dimensionality reduction [16, 17].”
Kernel Adaptive Metropolis-Hastings (2013, methodology): “We will work with the Gaussian measure on the RKHSk with meank(·, y) and covarianceν2Cz, wherez = {zi}ni=1 is the subset of the chain history.”
Kernel Mean Estimation and Stein's Effect (2013, positive): “Representing the distribution as a mean function in the RKHS has several advantages: 1) the representation with appropriate choice of kernel k has been shown to preserve all information about the distribution [5, 6, 7]; 2) basic operations on the distribution can be carried out by means of inner pro”
Testing Hypotheses by Regularized Maximum Mean Discrepancy (2013, background): “Characteristic kernels [6], [21], [8] injectively map the probability distribution onto its mean element in the corresponding RKHSs.”
One-Class Support Measure Machines for Group Anomaly Detection (2013, methodology): “The following theorem due to Fukumizu et al. (2004) and Sriperumbudur et al. (2010) gives a promising property of representing distributions as mean elements in the RKHS.”
Domain Generalization via Invariant Feature Representation (2013, background): “…or is closely related to many wellknown dimension reduction algorithms including kernel principal component analysis (KPCA) (Schölkopf et al. 1998, Fukumizu et al. 2004a), transfer component analysis (TCA) (Pan et al. 2011), and covariance operator inverse regression (COIR) (Kim and Pavlovic…”
Kernel Belief Propagation (2011, methodology): “Following Fukumizu et al. (2004), we define the covariance operator CXsXt which allows us to compute the expectation of the product of function f(Xs) and g(Xt), i.”
Kernel Bayes' Rule (2010, methodology): “We begin with a review of some basic concepts and tools concerning statistics on RKHS [1, 3, 6, 7].”
Universality, Characteristic Kernels and RKHS Embedding of Measures (2010, methodology): “…signed Radon measures generalizes the concept of RKHS embedding of Radon probability measures, the latter being related to characteristic kernels (Fukumizu et al., 2004, 2008; Sriperumbudur et al., 2008), in the following section, we relate the notion of universality to characteristic kernels.”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, background): “If k is translation invariant on M = Rd, the rate is independent of d (Gretton et al., 2007; Sriperumbudur et al., 2009b), an important property when dealing with high dimensions. Moreover, γF is not straightforward to compute when F is Cb(M), Cbu(M), FW or Fβ (Weaver, 1999, Section 2.3): by contras”