Hilbert Space Embeddings of Hidden Markov Models

By Le Song, Byron Boots, Sajid M. Siddiqi, Geoffrey J. Gordon, Alex Smola. 2010

244 citations recorded by the source corpus.

Subfields: Representation & Self-Supervised Learning, Probabilistic & Bayesian Methods

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Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Song et al. (2010a) relies on the embeddings of the distributions over observations and latent states, and then constructs an operator that represents the joint probabilities in the feature space.”
Computing functions of random variables via reproducing kernel Hilbert space representations (2015, neutral): “It has been applied successfully in many domains such as graphical models [38, 39], two-sample testing [11], domain adaptation [16, 13, 23], and supervised learning on probability distributions [24, 48].”
Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels (2014, background): “f problems in machine learning. These include nonlinear classiﬁcatio n, regression, clustering, semi-supervised learning [Belkin et al., 2006], time-series analysis [Parzen, 1970], sequence modeling [Song et al., 2013], dynamical systems [Boots et al., 2013], hypothesis testing [Harchaoui et al., 20”
Kernel Mean Shrinkage Estimators (2014, methodology): “…appealing to many algorithms in modern kernel methods, namely, two-sample testing via maximum mean discrepancy (MMD) (Gretton et al., 2007, 2012), kernel independence tests (Gretton et al., 2008), Hilbert space embedding of HMMs (Song et al., 2010), and kernel Bayes rule (Fukumizu et al., 2011).”
Characteristic Kernels and Infinitely Divisible Distributions (2014, methodology): “sity estimations (Smola et al., 2007; Song et al., 2008; McCalman et al.,2013), hypothesistests (Gretton et al. 2012, Gretton et al. 2008, Fukumizu et al. 2008), Bayesian inference (Song et al. 2009, Song et al. 2010, Song et al. 2011, Fukumizu et al. 2013, Song et al. 2013, Kanagawa et al. 2016, Ni”
Hilbert Space Embeddings of Predictive State Representations (2013, methodology): “[10] proposed a nonparametric approach to learning HMM representations in RKHSs.”
Kernel Mean Estimation and Stein's Effect (2013, positive): “As a result, there have been successful applications that benefit from the kernel mean representation, namely, a maximum mean discrepancy (MMD) [8], kernel dependency measure [9], kernel two-sample-test [10], Hilbert space embedding of HMMs [11], and kernel Bayes rule [12].”
Hilbert Space Embeddings of POMDPs (2012, methodology): “Recently, inference methods which make use of the embeddings have been proposed, including for Hidden Markov Models [Song et al., 2009, Song et al., 2010], and belief propagation [Song et al., 2010, Song et al., 2011].”
Modelling transition dynamics in MDPs with RKHS embeddings (2012, methodology): “Suppose K is a bounded kernel and the conditions of (Song et al., 2010b) e ) , and thus by choosing λ → 0 , λ 3 m → ∞ we have that, for any ǫ > 0 , By the reproducing property of L , we have In this work, for theoretical analysis, we consider a normalised version of (3): where b .”
Past, Present, and Future of Simultaneous Localization and Mapping: Toward the Robust-Perception Age (2016, neutral): “Recent approaches to forecasting future robot states can be found in the machine learning literature, and involve the use of spectral techniques [230] and deep learning [252].”
Deep Transfer Learning with Joint Adaptation Networks (2016, background): “The joint embeddings can be viewed as an uncentered crosscovariance operator CX1:m by the standard equivalence between tensor and linear map (Song et al., 2010).”
Recurrent Recommender Networks (2017, methodology): “Likewise, we could resort to spectral methods [22, 1] or nonparametric estimators, such as Recurrent Neural Networks (RNNs).”
Multi-modal estimation with kernel embeddings for learning motion models (2013, unknown): “An estimate to the filtering problem posed in the form of a hidden Markov model was recently developed [29] that can learn complex non-linear models, but is limited to producing a maximum a-posteriori estimate.”
Conditional Adversarial Domain Adaptation (2017, methodology): “And the multilinear map of infinite-dimensional nonlinear feature maps has been successfully applied to embed joint distribution or conditional distribution into reproducing kernel Hilbert spaces [47, 44, 45, 30].”
Support Measure Data Description A One-Class Classifier for Group Anomaly Detection (2017, background): “Output: The radius and the norm of the center of a minimum support vectors in RKHS. Finally, the set { P i | i ∈ I α } induces mean regularization parameter λ .”
Predictive-State Decoders: Encoding the Future into Recurrent Networks (2017, background): “Extensions to handle nonlinear systems rely on RKHS embeddings [43], which themselves can be computationally infeasible to use with large datasets.”
Spectral Methods for Indian Buffet Process Inference (2014, background): “In particular, the work of [6, 5, 11, 21, 29] demonstrates that it is possible to infer latent variable structure accurately, despite the problem being nonconvex, thus exhibiting many local minima.”
Kernel Embeddings of Latent Tree Graphical Models (2011, background): “Then a singular value decomposition of Ĉ can be carried out to obtain an estimate for Û (See [24] for more details).”