Injective Hilbert Space Embeddings of Probability Measures

By Bharath K. Sriperumbudur, Arthur Gretton, Kenji Fukumizu, Gert R. G. Lanckriet, Bernhard Schölkopf. 2008

180 citations recorded by the source corpus.

Subfields: Statistics and Probability, Statistical Learning Theory

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Citations in the corpus

Discriminative Embeddings of Latent Variable Models for Structured Data (2016, neutral): “…model to the entire dataset, and then uses the empirical Fisher information matrix and the Fisher score of each data point to define the kernel; The latter method instead fits a different generative model for each data point, and then uses inner products between distributions to define the kernel.”
Training generative neural networks via Maximum Mean Discrepancy optimization (2015, background): “(For more details, see Sriperumbudur et al. [12].)”
Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “Hence, if the Λ is supported on the entire Rd, we have by the uniqueness of the characteristic function that μP uniquely determine P (Sriperumbudur et al. 2008; 2010; 2011a).”
Kernel‐based tests for joint independence (2016, background): “ mean embedding is injective. A kernel is called characteristic if the mean embedding Π is injective (see Fukumizu et al., 2007). The Gaussian kernel (2.1) on Rm, for example, is characteristic (e.g. Sriperumbudur et al., 2008, Theorem 7). 2.2 Deﬁnition of dHSIC and independence property Our goal is”
Computing functions of random variables via reproducing kernel Hilbert space representations (2015, positive): “One of the advantages of this approach is that it allows us to apply RKHS methods to probability distributions, often with strong theoretical guarantees (Sriperumbudur et al. 2008, 2010).”
Kernel Mean Shrinkage Estimators (2014, background): “That is, it preserves all information about the distribution (Fukumizu et al., 2004; Sriperumbudur et al., 2008).”
A Kernel Independence Test for Random Processes (2014, methodology): “Hilbert Schmidt Independence Criterion Let k, l be positive definite characteristic kernels (Sriperumbudur et al., 2008): this is to say that mappings of probability measures to the corresponding RKHSs are injective.”
Hilbert Space Embeddings of Predictive State Representations (2013, methodology): “In this paper, we fully generalize PSRs to continuous observations and actions using a recent concept called Hilbert space embeddings of distributions [7, 8].”
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, methodology): “Embedding probability distributions into Reproducing Kernel Hilbert Spaces (RKHSs) yields a linear method that takes information of higher order statistics into account [9], [20], [21].”
Equivalence of distance-based and RKHS-based statistics in hypothesis testing (2012, methodology): “In this section, we establish the equivalence between the distance-based approach and the RKHS-based approach to two-sample and independence testing from Sections 2 and 3, respectively.”
Hypothesis testing using pairwise distances and associated kernels (2012, background): “There are many other kernels that generate ρ, however; for example, the distance kernel induced by ρ and “centered at zero” is k̃(z, z′) = e−σ‖z−z ′‖2 + 1− e−σ‖z‖ 2 − e−σ‖z ′‖2 .”
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.”
On the empirical estimation of integral probability metrics (2012, background): “[21, 43] have shown that when k is measurable and bounded,”
Hilbert Space Embeddings and Metrics on Probability Measures (2009, background): “Therefore, by Corollary 11 of Steinwart (2001), k is universal.”
Kernel dimension reduction in regression (2009, background): “R m or on a compact subset of Rm with respect to the Borel σ-field [2, 15, 28].”
On integral probability metrics, φ-divergences and binary classification (2009, methodology): “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”
A Kernel Two-Sample Test (2012, methodology): “For = 1 the distanceDr(p,q) is known as the Ĺevy distance (Feller, 1971), and forr = 2 we encounter a distance measure derived from the Renyi entropy (Gokcay and Principe, 2002).”
Learning predictive models of a depth camera & manipulator from raw execution traces (2014, background): “A characteristic RKHS is one for which the mean map is injective: that is, each distribution P has a unique embedding [33].”