Hilbert space embeddings of conditional distributions with applications to dynamical systems

By Le Song, Jonathan Huang, Alex Smola, Kenji Fukumizu. 2009

415 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Statistical and Nonlinear Physics, Probabilistic & Bayesian Methods

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Citations in the corpus

Discriminative Embeddings of Latent Variable Models for Structured Data (2016, methodology): “Our idea is to model each structured data point as a latent variable model, then embed the graphical model into feature spaces (Smola et al., 2007; Song et al., 2009), and use inner product in the embedding space to define kernels.”
Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Such a condition, however, is often impractically strong: as noted in Song et al. (2009) and”
Consistent Kernel Mean Estimation for Functions of Random Variables (2016, background): “An important notion in probability theory is that of a conditional distribution, which can also be represented using KME (Song et al., 2009).”
Computing functions of random variables via reproducing kernel Hilbert space representations (2015, neutral): “ on random variables. Such approximations typically have coefﬁcients 2Rn0and 2Rm0that are not uniform, that may not sum to one, and that may take negative values [34], e.g., for conditional mean maps [40, 9]. We propose to approximate the kernel mean m[f(X;Y)] by the estimator mˆ[f(X ;Y)]:= 1 åm 0 i”
Kernel Mean Estimation via Spectral Filtering (2014, background): “Moreover, the solution to (7) is also in a similar form to the regularized conditional emb eddingμY |X = CYX(Ck + λI) [9].”
Characteristic Kernels and Infinitely Divisible Distributions (2014, methodology): “Note, however, that the kernel mean estimators in applications (Song et al., 2009; Fukumizu et al., 2013) are given by a form of the weighted sum m̂P = ∑n i=1 wik(·, Xi) with data {Xi} n i=1 and associated weights w ∈ Rn. Weights w are allowed to take negative values, and the kernel mean estimator…”
Filtering with State-Observation Examples via Kernel Monte Carlo Filter (2013, methodology): “kX (·,Xi), t = 1,...,T. 5 Related Work Fully nonparametric SSMs also have been considered, where not only the observation model but the state-transition model are learned nonparametrically with data. Song et al. (2009) and Fukumizu et al. (2011) proposed ﬁlteri ng methods based on RKHS embeddings. K”
Hilbert Space Embeddings of POMDPs (2012, methodology): “According to [Song et al., 2009], a mapping from kX (x, ·) ∈ HX to µY |x ∈ HY for all x ∈ X can be characterized by conditional embedding operator UY”
Modelling transition dynamics in MDPs with RKHS embeddings (2012, methodology): “These approaches have been particularly successful in performing inference in graphical models, where the model parameters are learned nonparametrically from data (Song et al., 2010b; 2009; 2011).”
Conditional mean embeddings as regressors (2012, methodology): “Following (Song et al., 2009; 2010), we deﬁne a second kernel K : X × X → R with associated RKHS H K , and consider the estimate where , and where =1 , and λ is a chosen regularization parameter.”
Kernel Belief Propagation (2011, methodology): “An expression for the conditional distribution embedding was proposed by Song et al. (2009).”
Kernel Bayes' Rule (2010, methodology): “The proposed kernel realization of Bayes’ rule is an extensi on of the approach used in [20] for state-space models.”
Smooth Operators (2013, methodology): “Kernel-based approximations to conditional expectations have been widely studied, and their links with vector-valued regression are established (Song et al., 2009; Grünewälder et al., 2012a).”
Conditional Adversarial Domain Adaptation (2017, methodology): “…feature maps has been successfully applied to embed joint distribution or conditional distribution into reproducing kernel Hilbert spaces (RKHSs) (Song et al., 2009; 2010; Song & Dai, 2013; Long et al., 2017) to enable learning from distributions, lending its great capability of modeling…”
Gradient-based kernel method for feature extraction and variable selection (2012, methodology): “This reproducing property combined with the following kernel estimator of the conditional expectation (see [8, 9, 19] for details) will provide a method for dimension reduction.”
Recycling Privileged Learning and Distribution Matching for Fairness (2017, methodology): “To generalize this to any type and multiple dependence of protected characteristics, we can use the Hilbert Space embedding of conditional distributions framework of [44, 45].”