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Notes for "Kullback-Leibler appr

Notes for "Kullback-Leibler appr

作者: jjx323 | 来源:发表于2019-03-01 11:55 被阅读0次

Article Information: F. J. Pinski, G. Simpson, A. M. Stuart, and H. Weber, SIAM Journal of Mathematical Analysis, 47(6), 4091-4122

Proof of Corollary 2.2:
Since D_{KL}(\nu || \mu) < \infty for some probability measure \nu, we know that there exists a sequence \nu_n such that
\begin{align} \lim_{n\rightarrow \infty}D_{KL}(\nu_n || \mu) = \mathop{\text{inf}}_{\nu} D_{KL}(\nu || \mu) < \infty. \end{align}
Noticing the second statement in Proposition 2.1, the sequence \{\nu_n\}_{n=1,2,\cdots} must contain a weak convergence subsequence. We still denote the subsequence as \{\nu_n\}_{n=1,2,\cdots} and assume the convergent measure as \nu_{*}. From the first statement in Proposition 2.1, we have
\begin{align} \lim_{n\rightarrow\infty}D_{KL}(\nu_n || \mu) = D_{KL}(\nu_{*} || \mu) = \mathop{\text{inf}}_{\nu} D_{KL}(\nu || \mu) . \end{align}
Hence, the proof is completed.

Notes for the Proof of Proposition 2.1:
The lower semicontinuity of (\nu, \mu) \rightarrow D_{KL}(\nu || \mu) can be seen from the following formula
\begin{align} & \sup_{\Theta}\sup_{n\in \mathbb{N}}\inf_{k \geq n} \left\{ \int \Theta d\nu_n - \log\Big[ \int \exp(\Theta) d\mu_n \Big] \right\} = D_{KL}(\nu_{*} || \mu_{*}) \\ & \quad \leq \sup_{n\in\mathbb{N}}\inf_{k\geq n}\sup_{\Theta} \left\{ \int \Theta d\nu_n - \log\Big[ \int \exp(\Theta) d\mu_n \Big] \right\} = \liminf_{n\rightarrow \infty} D_{KL}(\nu_n || \mu_n). \end{align}

Notes for the Proof of Lemma 3.3

As C_{0}^{-1}+\Gamma has form domain \mathcal{H}^{1}, the operator
(C_{0}^{-1}+\Gamma)^{-1/2}C_{0}^{-1/2} is bounded on \mathcal{H} by the closed graph theorem

Proof: Let \varphi_{n} \rightarrow \varphi in \mathcal{H}, and (C_{0}^{-1}+\Gamma)^{-1/2}C_{0}^{-1/2}\varphi_{n}\rightarrow \psi as n\rightarrow \infty, we need to prove (C_{0}^{-1}+\Gamma)^{-1/2}C_{0}^{-1/2}\varphi = \psi.
The key point is the following
\begin{align} \|(C_{0}^{-1}+\Gamma)^{-1/2}C_{0}^{-1/2}(\varphi_n - \varphi)\| & \leq C \|\varphi_n-\varphi\|_{\mathcal{H}^{-1}} \\ & \leq C \|\varphi_n - \varphi\| \rightarrow 0, \end{align}
where we used the condition C_{0}^{-1}+\Gamma has form domain \mathcal{H}^{1}.

Notes for Lemma 3.19:
In the proof of Lemma 3.19, the last few lines

\cdots \nu_{n}(K_{\delta}) \leq \delta for any n\geq 1 .......
\nu_{n}^{i}(K_{\delta}) \leq \frac{1}{\hat{p}}\nu(K_{\delta}) \leq \frac{\delta}{\hat{p}}

By my understanding, these statements should be modified as follow
\cdots \nu_{n}(\mathcal{H}\backslash K_{\delta}) \leq \delta for any n\geq 1 .......
\nu_{n}^{i}(\mathcal{H}\backslash K_{\delta}) \leq \frac{1}{\hat{p}}\nu(\mathcal{H}\backslash K_{\delta}) \leq \frac{\delta}{\hat{p}}

Notes for the last part of Appendix B:
How to obtain \sup_{n\geq 1} \| C_{*}^{-1/2}C_{n}^{1/2} \|_{\mathcal{L}(H)} < \infty
Proof: Since
\begin{align} \|C_{*}^{1/2}(C_{n}^{-1}-C_{*}^{-1})C_{*}^{1/2}\|_{\mathcal{HS}(\mathcal{H})}=\|(C_{*}^{1/2}C_{n}^{-1/2})(C_{*}^{1/2}C_{n}^{-1/2})^{*}-Id\|_{\mathcal{HS}(\mathcal{H})} \rightarrow 0, \end{align}
we find that
\begin{align} 1- \|C_{*}^{1/2}C_{n}^{-1/2}\|^2 \leq \delta, \quad \|C_{*}^{1/2}C_{n}^{-1/2}\|^2 - 1 \leq \delta. \end{align}
Then, we obtain 1-\delta\leq \|C_{*}^{1/2}C_{n}^{-1/2}\|^2 \leq 1+\delta. Hence, naturally, we finally arrive at the conclusion.

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