From 7fbbb3593c5335b1e2727fcb7f8a01b20fdfdb5e Mon Sep 17 00:00:00 2001 From: Varun Agrawal Date: Thu, 31 Aug 2023 13:19:59 -0400 Subject: [PATCH] updated Hybrid.lyx to clarify the invariant structure of the GaussianMixture invariant --- doc/Hybrid.lyx | 108 +++++++++++++++++++++++++++++++++++++++++++++---- doc/Hybrid.pdf | Bin 231745 -> 170632 bytes 2 files changed, 100 insertions(+), 8 deletions(-) diff --git a/doc/Hybrid.lyx b/doc/Hybrid.lyx index c854a4845..3067d5c26 100644 --- a/doc/Hybrid.lyx +++ b/doc/Hybrid.lyx @@ -86,7 +86,7 @@ Hybrid Inference \end_layout \begin_layout Author -Frank Dellaert & Varun Agrawal +Frank Dellaert \end_layout \begin_layout Date @@ -269,7 +269,82 @@ If we take the log of \end_inset -Equating +The key point here is that +\family roman +\series medium +\shape up +\size normal +\emph off +\bar no +\strikeout off +\xout off +\uuline off +\uwave off +\noun off +\color none + +\begin_inset Formula $K_{gm}$ +\end_inset + + +\family default +\series default +\shape default +\size default +\emph default +\bar default +\strikeout default +\xout default +\uuline default +\uwave default +\noun default +\color inherit + is the log-normalization constant for the complete +\emph on +GaussianMixture +\emph default + across all values of +\begin_inset Formula $m$ +\end_inset + +, and is not dependent on the value of +\begin_inset Formula $m$ +\end_inset + +. + In contrast, +\begin_inset Formula $K_{gcm}$ +\end_inset + + is the log-normalization constant for a specific +\emph on +GaussianConditional +\emph default +mode (thus dependent on +\begin_inset Formula $m$ +\end_inset + +) and can have differing values based on the covariance matrices for each + mode. + Thus to obtain a constant +\begin_inset Formula $K_{gm}$ +\end_inset + + which satisfies the invariant, we need to specify +\begin_inset Formula $E_{gm}(x,y,m)$ +\end_inset + + accordingly. +\end_layout + +\begin_layout Standard +\SpecialChar allowbreak + +\end_layout + +\begin_layout Standard +\noindent +By equating \begin_inset CommandInset ref LatexCommand eqref reference "eq:gm_invariant" @@ -289,7 +364,7 @@ noprefix "false" \end_inset - we see that this can be achieved by defining the error +, we see that this can be achieved by defining the error \begin_inset Formula $E_{gm}(x,y,m)$ \end_inset @@ -541,9 +616,9 @@ noprefix "false" : \begin_inset Formula -\[ -E_{mf}(y,m)=C+E_{gcm}(\bar{x},y)-K_{gcm}. -\] +\begin{equation} +E_{mf}(y,m)=C+E_{gcm}(\bar{x},y)-K_{gcm}\label{eq:mixture_factor} +\end{equation} \end_inset @@ -555,8 +630,25 @@ Unfortunately, we can no longer choose \begin_inset Formula $m$ \end_inset - to make the constant disappear. - There are two possibilities: + to make the constant disappear, since +\begin_inset Formula $C$ +\end_inset + + has to be a constant applicable across all +\begin_inset Formula $m$ +\end_inset + +. +\end_layout + +\begin_layout Standard +\SpecialChar allowbreak + +\end_layout + +\begin_layout Standard +\noindent +There are 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