Characteristic Kernels and Infinitely Divisible Distributions

We connect shift-invariant characteristic kernels to infinitely divisible distributions on $\mathbb{R}^{d}$. Characteristic kernels play an important role in machine learning applications with their kernel means to distinguish any two probability measures. The contribution of this paper is two-fold. First, we show, using the L\'evy-Khintchine formula, that any shift-invariant kernel given by a bounded, continuous and symmetric probability density function (pdf) of an infinitely divisible distribution on $\mathbb{R}^d$ is characteristic. We also present some closure property of such characteristic kernels under addition, pointwise product, and convolution. Second, in developing various kernel mean algorithms, it is fundamental to compute the following values: (i) kernel mean values $m_P(x)$, $x \in \mathcal{X}$, and (ii) kernel mean RKHS inner products ${\left\langle m_P, m_Q \right\rangle_{\mathcal{H}}}$, for probability measures $P, Q$. If $P, Q$, and kernel $k$ are Gaussians, then computation (i) and (ii) results in Gaussian pdfs that is tractable. We generalize this Gaussian combination to more general cases in the class of infinitely divisible distributions. We then introduce a {\it conjugate} kernel and {\it convolution trick}, so that the above (i) and (ii) have the same pdf form, expecting tractable computation at least in some cases. As specific instances, we explore $\alpha$-stable distributions and a rich class of generalized hyperbolic distributions, where the Laplace, Cauchy and Student-t distributions are included

Hilbert Space Embeddings of POMDPs

A nonparametric approach for policy learning for POMDPs is proposed. The approach represents distributions over the states, observations, and actions as embeddings in feature spaces, which are reproducing kernel Hilbert spaces. Distributions over states given the observations are obtained by applying the kernel Bayes' rule to these distribution embeddings. Policies and value functions are defined on the feature space over states, which leads to a feature space expression for the Bellman equation. Value iteration may then be used to estimate the optimal value function and associated policy. Experimental results confirm that the correct policy is learned using the feature space representation.Comment: Appears in Proceedings of the Twenty-Eighth Conference on Uncertainty in Artificial Intelligence (UAI2012

カーネル法と確率分布の無限分解可能性

Open House, ISM in Tachikawa, 2014.6.13統計数理研究所オープンハウス（立川）、H26.6.13ポスター発

厳密なカーネル平均を利用した状態空間フィルタリングアルゴリズム

Open House, ISM in Tachikawa, 2013.6.14統計数理研究所オープンハウス（立川）、H25.6.14ポスター発

収束保証つき確率推論アルゴリズム

Open House, ISM in Tachikawa, 2011.7.14統計数理研究所オープンハウス（立川）、H23.7.14ポスター発

POMDP価値反復アルゴリズムのカーネル化

Open House, ISM in Tachikawa, 2012.6.15統計数理研究所オープンハウス（立川）、H24.6.15ポスター発

New Bifunctional Antioxidants : In tramolecular Synergistic Effects between Benzofuranol and Thiopropionate Group, Part II

The antioxidant activities of benzofuranols and chromanols with methyl, methyl thiomethyl and thiopropionate groups were evaluated for the oxidation of tetralin at 61 and 140℃. The antioxidants tested showed almost the same behaviour for the oxidation of tetralin initiated by an azo initiator at 61℃. However, benzofuranol and chromanol with a thiopropionate group at the meta position of the OH group were shown to improve antioxidant activity at high temperature to a greater extent than the methyl and methyl thiomethyl groups