From 447f060e26379a91a7e457f9ee90d657999ba6a0 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?R=C3=A9my=20Degenne?= Date: Mon, 18 Mar 2024 16:40:50 +0100 Subject: [PATCH] rework Renyi div def --- blueprint/src/sections/hellinger.tex | 4 +-- blueprint/src/sections/renyi_divergence.tex | 29 +++++---------------- 2 files changed, 8 insertions(+), 25 deletions(-) diff --git a/blueprint/src/sections/hellinger.tex b/blueprint/src/sections/hellinger.tex index b5b949ff..15af1f21 100644 --- a/blueprint/src/sections/hellinger.tex +++ b/blueprint/src/sections/hellinger.tex @@ -20,8 +20,8 @@ \chapter{Hellinger distance} \end{lemma} \begin{proof} -\uses{lem:renyi_eq_log_fDiv, lem:fDiv_mul, lem:fDiv_add_linear} -By Lemma~\ref{lem:renyi_eq_log_fDiv}, $R_{1/2}(\mu, \nu) = -2 \log (1 - \frac{1}{2} D_f(\nu, \mu))$ for $f : x \mapsto -2 (\sqrt{x} - 1)$. Using Lemma~\ref{lem:fDiv_mul}, $R_{1/2}(\mu, \nu) = -2 \log (1 - D_g(\nu, \mu))$ for $g(x) = 1 - \sqrt{x}$. +\uses{lem:fDiv_mul, lem:fDiv_add_linear} +$R_{1/2}(\mu, \nu) = -2 \log (1 - \frac{1}{2} D_f(\nu, \mu))$ for $f : x \mapsto -2 (\sqrt{x} - 1)$. Using Lemma~\ref{lem:fDiv_mul}, $R_{1/2}(\mu, \nu) = -2 \log (1 - D_g(\nu, \mu))$ for $g(x) = 1 - \sqrt{x}$. It suffices then to show that $H^2(\mu, \nu) = D_g(\mu, \nu)$, which is true by an application of Lemma~\ref{lem:fDiv_add_linear}. \end{proof} diff --git a/blueprint/src/sections/renyi_divergence.tex b/blueprint/src/sections/renyi_divergence.tex index c5d3acfc..8620c1c8 100644 --- a/blueprint/src/sections/renyi_divergence.tex +++ b/blueprint/src/sections/renyi_divergence.tex @@ -9,31 +9,16 @@ \chapter{Rényi divergences} \begin{align*} R_\alpha(\mu, \nu) = \left\{ \begin{array}{ll} - - \log(\nu\{x \mid \frac{d \mu}{d \nu}(x) > 0\}) & \text{ for } \alpha = 0 + - \log(\nu\{x \mid \frac{d \mu}{d \nu}(x) > 0\}) & \text{for } \alpha = 0 \\ - \KL(\mu, \nu) & \text{ for } \alpha = 1 + \KL(\mu, \nu) & \text{for } \alpha = 1 \\ - \frac{1}{\alpha - 1}\log \nu\left[\left(\frac{d \mu}{d \nu}\right)^\alpha\right] & \text {for } \alpha \in (0,1) \text{ , or } \alpha > 1 \text{ and }\mu \ll \nu - \\ - +\infty & \text {otherwise} + \frac{1}{\alpha - 1} \log (1 + (\alpha - 1) D_f(\nu, \mu)) & \text{for } \alpha \in (0,+\infty) \backslash \{1\} \end{array}\right. \end{align*} + with $f : x \mapsto \frac{1}{\alpha - 1}(x^{\alpha} - 1)$. \end{definition} -\begin{lemma} - \label{lem:renyi_eq_log_fDiv} - %\lean{} - %\leanok - \uses{def:fDiv, def:Renyi} - For $\nu$ a probability measure and either $\alpha \in (0,1)$ or $\mu \ll \nu$, $R_\alpha(\mu, \nu) = \frac{1}{\alpha - 1} \log (1 + (\alpha - 1) D_f(\nu, \mu))$ for $f : x \mapsto \frac{1}{\alpha - 1}(x^{\alpha} - 1)$, which is convex with $f(1)=0$. It is thus a monotone transformation of a f-divergence. - - TODO: use this for the definition? -\end{lemma} - -\begin{proof} -Unfold the definitions. -\end{proof} - \begin{lemma} \label{lem:renyi_symm} %\lean{} @@ -96,8 +81,7 @@ \section{Properties inherited from f-divergences} \end{theorem} \begin{proof} -\uses{thm:fDiv_data_proc, lem:renyi_eq_log_fDiv} -By Lemma~\ref{lem:renyi_eq_log_fDiv}, $R_\alpha(\mu, \nu) = \frac{1}{\alpha - 1} \log (1 + (\alpha - 1) D_f(\nu, \mu))$ for $f : x \mapsto \frac{1}{\alpha - 1}(x^{\alpha} - 1)$. +\uses{thm:fDiv_data_proc} The function $x \mapsto \frac{1}{\alpha - 1}\log (1 + (\alpha - 1)x)$ is non-decreasing and $D_f$ satisfies the DPI (Theorem~\ref{thm:fDiv_data_proc}), hence we get the DPI for $R_\alpha$. \end{proof} @@ -111,8 +95,7 @@ \section{Properties inherited from f-divergences} \end{lemma} \begin{proof} -\uses{cor:data_proc_event, lem:renyi_eq_log_fDiv} -By Lemma~\ref{lem:renyi_eq_log_fDiv}, $R_\alpha(\mu, \nu) = \frac{1}{\alpha - 1} \log (1 + (\alpha - 1) D_f(\nu, \mu))$ for $f : x \mapsto \frac{1}{\alpha - 1}(x^{\alpha} - 1)$. +\uses{cor:data_proc_event} By Corollary~\ref{cor:data_proc_event}, $D_f(\mu, \nu) \ge D_f(\mu_E, \nu_E)$, hence $R_\alpha(\mu, \nu) \ge R_\alpha(\mu_E, \nu_E)$. \end{proof}