835 lines
14 KiB
Plaintext
835 lines
14 KiB
Plaintext
#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\begin_document
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\begin_header
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\index Index
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\shortcut idx
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\end_index
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\leftmargin 1in
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\topmargin 1in
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\end_header
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\begin_body
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\begin_layout Title
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Hybrid Inference
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\end_layout
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\begin_layout Author
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Frank Dellaert
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\end_layout
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\begin_layout Date
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January 2023
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\end_layout
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\begin_layout Section
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Hybrid Conditionals
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\end_layout
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\begin_layout Standard
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Here we develop a hybrid conditional density, on continuous variables (typically
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a measurement
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\begin_inset Formula $x$
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\end_inset
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), given a mix of continuous variables
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\begin_inset Formula $y$
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\end_inset
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and discrete variables
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\begin_inset Formula $m$
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\end_inset
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.
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We start by reviewing a Gaussian conditional density and its invariants
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(relationship between density, error, and normalization constant), and
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then work out what needs to happen for a hybrid version.
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\end_layout
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\begin_layout Subsubsection*
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GaussianConditional
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\end_layout
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\begin_layout Standard
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A
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\emph on
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GaussianConditional
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\emph default
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is a properly normalized, multivariate Gaussian conditional density:
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\begin_inset Formula
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\[
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P(x|y)=\frac{1}{\sqrt{|2\pi\Sigma|}}\exp\left\{ -\frac{1}{2}\|Rx+Sy-d\|_{\Sigma}^{2}\right\}
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\]
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\end_inset
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where
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\begin_inset Formula $R$
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\end_inset
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is square and upper-triangular.
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For every
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\emph on
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GaussianConditional
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\emph default
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, we have the following
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\series bold
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invariant
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\series default
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,
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\begin_inset Formula
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\begin{equation}
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\log P(x|y)=K_{gc}-E_{gc}(x,y),\label{eq:gc_invariant}
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\end{equation}
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\end_inset
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with the
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\series bold
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log-normalization constant
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\series default
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\begin_inset Formula $K_{gc}$
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\end_inset
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equal to
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\begin_inset Formula
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\begin{equation}
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K_{gc}=\log\frac{1}{\sqrt{|2\pi\Sigma|}}\label{eq:log_constant}
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\end{equation}
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\end_inset
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and the
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\series bold
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error
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\series default
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\begin_inset Formula $E_{gc}(x,y)$
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\end_inset
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equal to the negative log-density, up to a constant:
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\begin_inset Formula
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\begin{equation}
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E_{gc}(x,y)=\frac{1}{2}\|Rx+Sy-d\|_{\Sigma}^{2}.\label{eq:gc_error}
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\end{equation}
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\end_inset
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.
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\end_layout
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\begin_layout Subsubsection*
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GaussianMixture
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\end_layout
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\begin_layout Standard
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A
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\emph on
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GaussianMixture
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\emph default
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(maybe to be renamed to
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\emph on
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GaussianMixtureComponent
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\emph default
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) just indexes into a number of
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\emph on
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GaussianConditional
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\emph default
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instances, that are each properly normalized:
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\end_layout
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\begin_layout Standard
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\begin_inset Formula
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\[
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P(x|y,m)=P_{m}(x|y).
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\]
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\end_inset
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We store one
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\emph on
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GaussianConditional
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\emph default
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\begin_inset Formula $P_{m}(x|y)$
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\end_inset
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for every possible assignment
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\begin_inset Formula $m$
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\end_inset
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to a set of discrete variables.
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As
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\emph on
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GaussianMixture
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\emph default
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is a
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\emph on
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Conditional
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\emph default
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, it needs to satisfy the a similar invariant to
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:gc_invariant"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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:
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\begin_inset Formula
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\begin{equation}
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\log P(x|y,m)=K_{gm}-E_{gm}(x,y,m).\label{eq:gm_invariant}
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\end{equation}
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\end_inset
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If we take the log of
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\begin_inset Formula $P(x|y,m)$
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\end_inset
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we get
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\begin_inset Formula
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\begin{equation}
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\log P(x|y,m)=\log P_{m}(x|y)=K_{gc}(m)-E_{gcm}(x,y).\label{eq:gm_log}
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\end{equation}
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\end_inset
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\end_layout
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\begin_layout Standard
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\noindent
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For conciseness, we will write
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\begin_inset Formula $K_{gc}(m)$
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\end_inset
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as
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\begin_inset Formula $K_{gcm}$
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\end_inset
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.
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\end_layout
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\begin_layout Standard
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\SpecialChar allowbreak
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\end_layout
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\begin_layout Standard
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\noindent
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The key point here is that
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\family roman
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\series medium
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\shape up
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\size normal
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\emph off
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\bar no
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\strikeout off
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\xout off
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\uuline off
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\uwave off
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\noun off
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\color none
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\begin_inset Formula $K_{gm}$
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\end_inset
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\family default
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\series default
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\shape default
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\size default
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\emph default
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\bar default
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\strikeout default
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\xout default
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\uuline default
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\uwave default
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\noun default
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\color inherit
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is the log-normalization constant for the complete
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\emph on
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GaussianMixture
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\emph default
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across all values of
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\begin_inset Formula $m$
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\end_inset
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, and cannot be dependent on the value of
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\begin_inset Formula $m$
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\end_inset
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.
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In contrast,
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\begin_inset Formula $K_{gcm}$
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\end_inset
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is the log-normalization constant for a specific
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\emph on
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GaussianConditional
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\emph default
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mode (thus dependent on
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\begin_inset Formula $m$
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\end_inset
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) and can have differing values based on the covariance matrices for each
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mode.
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Thus to obtain a constant
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\begin_inset Formula $K_{gm}$
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\end_inset
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which satisfies the invariant, we need to specify
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\begin_inset Formula $E_{gm}(x,y,m)$
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\end_inset
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accordingly.
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\end_layout
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\begin_layout Standard
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\SpecialChar allowbreak
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\end_layout
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\begin_layout Standard
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\noindent
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By equating
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:gm_invariant"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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and
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:gm_log"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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, we see that this can be achieved by defining the error
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\begin_inset Formula $E_{gm}(x,y,m)$
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\end_inset
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as
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\begin_inset Formula
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\begin{equation}
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E_{gm}(x,y,m)=E_{gcm}(x,y)+K_{gm}-K_{gcm}\label{eq:gm_error}
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\end{equation}
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\end_inset
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where choose
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\begin_inset Formula $K_{gm}=\max K_{gcm}$
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\end_inset
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, as then the error will always be positive.
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\end_layout
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\begin_layout Section
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Hybrid Factors
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\end_layout
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\begin_layout Standard
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In GTSAM, we typically condition on known measurements, and factors encode
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the resulting negative log-likelihood of the unknown variables
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\begin_inset Formula $y$
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\end_inset
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given the measurements
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\begin_inset Formula $x$
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\end_inset
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.
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We review how a Gaussian conditional density is converted into a Gaussian
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factor, and then develop a hybrid version satisfying the correct invariants
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as well.
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\end_layout
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\begin_layout Subsubsection*
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JacobianFactor
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\end_layout
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\begin_layout Standard
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A
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\emph on
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JacobianFactor
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\emph default
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typically results from a
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\emph on
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GaussianConditional
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\emph default
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by having known values
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\begin_inset Formula $\bar{x}$
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\end_inset
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for the
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\begin_inset Quotes eld
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\end_inset
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measurement
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\begin_inset Quotes erd
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\end_inset
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\begin_inset Formula $x$
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\end_inset
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:
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\begin_inset Formula
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\begin{equation}
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L(y)\propto P(\bar{x}|y)\label{eq:likelihood}
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\end{equation}
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\end_inset
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In GTSAM factors represent the negative log-likelihood
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\begin_inset Formula $E_{jf}(y)$
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\end_inset
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and hence we have
|
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\begin_inset Formula
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\[
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E_{jf}(y)=-\log L(y)=C-\log P(\bar{x}|y),
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\]
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\end_inset
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with
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\begin_inset Formula $C$
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\end_inset
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the log of the proportionality constant in
|
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\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:likelihood"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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.
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Substituting in
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\begin_inset Formula $\log P(\bar{x}|y)$
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\end_inset
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from the invariant
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\begin_inset CommandInset ref
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|
LatexCommand eqref
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reference "eq:gc_invariant"
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plural "false"
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caps "false"
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noprefix "false"
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\end_inset
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we obtain
|
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\begin_inset Formula
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\[
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E_{jf}(y)=C-K_{gc}+E_{gc}(\bar{x},y).
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\]
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\end_inset
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The
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\emph on
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likelihood
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\emph default
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function in
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\emph on
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GaussianConditional
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\emph default
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chooses
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\begin_inset Formula $C=K_{gc}$
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\end_inset
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, and the
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\emph on
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JacobianFactor
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\emph default
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does not store any constant; it just implements:
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\begin_inset Formula
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\[
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E_{jf}(y)=E_{gc}(\bar{x},y)=\frac{1}{2}\|R\bar{x}+Sy-d\|_{\Sigma}^{2}=\frac{1}{2}\|Ay-b\|_{\Sigma}^{2}
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\]
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\end_inset
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|
with
|
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\begin_inset Formula $A=S$
|
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\end_inset
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|
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|
and
|
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\begin_inset Formula $b=d-R\bar{x}$
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\end_inset
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|
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|
.
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\end_layout
|
|
|
|
\begin_layout Subsubsection*
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GaussianMixtureFactor
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\end_layout
|
|
|
|
\begin_layout Standard
|
|
Analogously, a
|
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\emph on
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|
GaussianMixtureFactor
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|
\emph default
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|
typically results from a GaussianMixture by having known values
|
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\begin_inset Formula $\bar{x}$
|
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\end_inset
|
|
|
|
for the
|
|
\begin_inset Quotes eld
|
|
\end_inset
|
|
|
|
measurement
|
|
\begin_inset Quotes erd
|
|
\end_inset
|
|
|
|
|
|
\begin_inset Formula $x$
|
|
\end_inset
|
|
|
|
:
|
|
\begin_inset Formula
|
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\[
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L(y,m)\propto P(\bar{x}|y,m).
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\]
|
|
|
|
\end_inset
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We will similarly implement the negative log-likelihood
|
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\begin_inset Formula $E_{mf}(y,m)$
|
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\end_inset
|
|
|
|
:
|
|
\begin_inset Formula
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\[
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E_{mf}(y,m)=-\log L(y,m)=C-\log P(\bar{x}|y,m).
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\]
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|
|
\end_inset
|
|
|
|
Since we know the log-density from the invariant
|
|
\begin_inset CommandInset ref
|
|
LatexCommand eqref
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|
reference "eq:gm_invariant"
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|
plural "false"
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|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
, we obtain
|
|
\begin_inset Formula
|
|
\[
|
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\log P(\bar{x}|y,m)=K_{gm}-E_{gm}(\bar{x},y,m),
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\]
|
|
|
|
\end_inset
|
|
|
|
and hence
|
|
\begin_inset Formula
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|
\[
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E_{mf}(y,m)=C+E_{gm}(\bar{x},y,m)-K_{gm}.
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\]
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|
|
\end_inset
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|
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|
Substituting in
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|
\begin_inset CommandInset ref
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|
LatexCommand eqref
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|
reference "eq:gm_error"
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|
plural "false"
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|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
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|
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|
we finally have an expression where
|
|
\begin_inset Formula $K_{gm}$
|
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\end_inset
|
|
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|
canceled out, but we have a dependence on the individual component constants
|
|
|
|
\begin_inset Formula $K_{gcm}$
|
|
\end_inset
|
|
|
|
:
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
E_{mf}(y,m)=C+E_{gcm}(\bar{x},y)-K_{gcm}\label{eq:mixture_factor}
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|
\end{equation}
|
|
|
|
\end_inset
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|
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|
Unfortunately, we can no longer choose
|
|
\begin_inset Formula $C$
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\end_inset
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|
|
|
independently from
|
|
\begin_inset Formula $m$
|
|
\end_inset
|
|
|
|
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 two possibilities:
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Implement likelihood to yield both a hybrid factor
|
|
\emph on
|
|
and
|
|
\emph default
|
|
a discrete factor.
|
|
\end_layout
|
|
|
|
\begin_layout Enumerate
|
|
Hide the constant inside the collection of JacobianFactor instances, which
|
|
is the possibility we implement.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In either case, we implement the mixture factor
|
|
\begin_inset Formula $E_{mf}(y,m)$
|
|
\end_inset
|
|
|
|
as a set of
|
|
\emph on
|
|
JacobianFactor
|
|
\emph default
|
|
instances
|
|
\begin_inset Formula $E_{mf}(y,m)$
|
|
\end_inset
|
|
|
|
, indexed by the discrete assignment
|
|
\begin_inset Formula $m$
|
|
\end_inset
|
|
|
|
:
|
|
\begin_inset Formula
|
|
\[
|
|
E_{mf}(y,m)=E_{jfm}(y)=\frac{1}{2}\|A_{m}y-b_{m}\|_{\Sigma_{mfm}}^{2}.
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
In GTSAM, we define
|
|
\begin_inset Formula $A_{m}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $b_{m}$
|
|
\end_inset
|
|
|
|
strategically to make the
|
|
\emph on
|
|
JacobianFactor
|
|
\emph default
|
|
compute the constant, as well:
|
|
\begin_inset Formula
|
|
\[
|
|
\frac{1}{2}\|A_{m}y-b_{m}\|_{\Sigma_{mfm}}^{2}=C+E_{gcm}(\bar{x},y)-K_{gcm}.
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
Substituting in the definition
|
|
\begin_inset CommandInset ref
|
|
LatexCommand eqref
|
|
reference "eq:gc_error"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
for
|
|
\begin_inset Formula $E_{gcm}(\bar{x},y)$
|
|
\end_inset
|
|
|
|
we need
|
|
\begin_inset Formula
|
|
\[
|
|
\frac{1}{2}\|A_{m}y-b_{m}\|_{\Sigma_{mfm}}^{2}=C+\frac{1}{2}\|R_{m}\bar{x}+S_{m}y-d_{m}\|_{\Sigma_{m}}^{2}-K_{gcm}
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
which can achieved by setting
|
|
\begin_inset Formula
|
|
\[
|
|
A_{m}=\left[\begin{array}{c}
|
|
S_{m}\\
|
|
0
|
|
\end{array}\right],~b_{m}=\left[\begin{array}{c}
|
|
d_{m}-R_{m}\bar{x}\\
|
|
c_{m}
|
|
\end{array}\right],~\Sigma_{mfm}=\left[\begin{array}{cc}
|
|
\Sigma_{m}\\
|
|
& 1
|
|
\end{array}\right]
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
and setting the mode-dependent scalar
|
|
\begin_inset Formula $c_{m}$
|
|
\end_inset
|
|
|
|
such that
|
|
\begin_inset Formula $c_{m}^{2}=C-K_{gcm}$
|
|
\end_inset
|
|
|
|
.
|
|
This can be achieved by
|
|
\begin_inset Formula $C=\max K_{gcm}=K_{gm}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula $c_{m}=\sqrt{2(C-K_{gcm})}$
|
|
\end_inset
|
|
|
|
.
|
|
Note that in case that all constants
|
|
\begin_inset Formula $K_{gcm}$
|
|
\end_inset
|
|
|
|
are equal, we can just use
|
|
\begin_inset Formula $C=K_{gm}$
|
|
\end_inset
|
|
|
|
and
|
|
\begin_inset Formula
|
|
\[
|
|
A_{m}=S_{m},~b_{m}=d_{m}-R_{m}\bar{x},~\Sigma_{mfm}=\Sigma_{m}
|
|
\]
|
|
|
|
\end_inset
|
|
|
|
as before.
|
|
\end_layout
|
|
|
|
\begin_layout Standard
|
|
In summary, we have
|
|
\begin_inset Formula
|
|
\begin{equation}
|
|
E_{mf}(y,m)=\frac{1}{2}\|A_{m}y-b_{m}\|_{\Sigma_{mfm}}^{2}=E_{gcm}(\bar{x},y)+K_{gm}-K_{gcm}.\label{eq:mf_invariant}
|
|
\end{equation}
|
|
|
|
\end_inset
|
|
|
|
which is identical to the GaussianMixture error
|
|
\begin_inset CommandInset ref
|
|
LatexCommand eqref
|
|
reference "eq:gm_error"
|
|
plural "false"
|
|
caps "false"
|
|
noprefix "false"
|
|
|
|
\end_inset
|
|
|
|
.
|
|
\end_layout
|
|
|
|
\end_body
|
|
\end_document
|