From 892759e5d6b361b3531e53592d17e48a73929a5c Mon Sep 17 00:00:00 2001 From: Frank Dellaert Date: Mon, 16 Jan 2023 20:00:07 -0800 Subject: [PATCH] Add math related to hybrid classes --- doc/Hybrid.lyx | 719 +++++++++++++++++++++++++++++++++++++++++++++++++ doc/Hybrid.pdf | Bin 0 -> 231745 bytes 2 files changed, 719 insertions(+) create mode 100644 doc/Hybrid.lyx create mode 100644 doc/Hybrid.pdf diff --git a/doc/Hybrid.lyx b/doc/Hybrid.lyx new file mode 100644 index 000000000..c854a4845 --- /dev/null +++ b/doc/Hybrid.lyx @@ -0,0 +1,719 @@ +#LyX 2.3 created this file. For more info see http://www.lyx.org/ +\lyxformat 544 +\begin_document +\begin_header +\save_transient_properties true +\origin unavailable +\textclass article +\use_default_options true +\maintain_unincluded_children false +\language english +\language_package default +\inputencoding auto +\fontencoding global +\font_roman "default" "default" +\font_sans "default" "default" +\font_typewriter "default" "default" +\font_math "auto" "auto" +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 100 +\font_tt_scale 100 100 +\use_microtype false +\use_dash_ligatures true +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize 11 +\spacing single +\use_hyperref false +\papersize default +\use_geometry true +\use_package amsmath 1 +\use_package amssymb 1 +\use_package cancel 1 +\use_package esint 1 +\use_package mathdots 1 +\use_package mathtools 1 +\use_package mhchem 1 +\use_package stackrel 1 +\use_package stmaryrd 1 +\use_package undertilde 1 +\cite_engine basic +\cite_engine_type default +\biblio_style plain +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\justification true +\use_refstyle 1 +\use_minted 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\leftmargin 1in +\topmargin 1in +\rightmargin 1in +\bottommargin 1in +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation indent +\paragraph_indentation default +\is_math_indent 0 +\math_numbering_side default +\quotes_style english +\dynamic_quotes 0 +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Title +Hybrid Inference +\end_layout + +\begin_layout Author +Frank Dellaert & Varun Agrawal +\end_layout + +\begin_layout Date +January 2023 +\end_layout + +\begin_layout Section +Hybrid Conditionals +\end_layout + +\begin_layout Standard +Here we develop a hybrid conditional density, on continuous variables (typically + a measurement +\begin_inset Formula $x$ +\end_inset + +), given a mix of continuous variables +\begin_inset Formula $y$ +\end_inset + + and discrete variables +\begin_inset Formula $m$ +\end_inset + +. + We start by reviewing a Gaussian conditional density and its invariants + (relationship between density, error, and normalization constant), and + then work out what needs to happen for a hybrid version. +\end_layout + +\begin_layout Subsubsection* +GaussianConditional +\end_layout + +\begin_layout Standard +A +\emph on +GaussianConditional +\emph default + is a properly normalized, multivariate Gaussian conditional density: +\begin_inset Formula +\[ +P(x|y)=\frac{1}{\sqrt{|2\pi\Sigma|}}\exp\left\{ -\frac{1}{2}\|Rx+Sy-d\|_{\Sigma}^{2}\right\} +\] + +\end_inset + +where +\begin_inset Formula $R$ +\end_inset + + is square and upper-triangular. + For every +\emph on +GaussianConditional +\emph default +, we have the following +\series bold +invariant +\series default +, +\begin_inset Formula +\begin{equation} +\log P(x|y)=K_{gc}-E_{gc}(x,y),\label{eq:gc_invariant} +\end{equation} + +\end_inset + +with the +\series bold +log-normalization constant +\series default + +\begin_inset Formula $K_{gc}$ +\end_inset + + equal to +\begin_inset Formula +\begin{equation} +K_{gc}=\log\frac{1}{\sqrt{|2\pi\Sigma|}}\label{eq:log_constant} +\end{equation} + +\end_inset + + and the +\series bold +error +\series default + +\begin_inset Formula $E_{gc}(x,y)$ +\end_inset + + equal to the negative log-density, up to a constant: +\begin_inset Formula +\begin{equation} +E_{gc}(x,y)=\frac{1}{2}\|Rx+Sy-d\|_{\Sigma}^{2}.\label{eq:gc_error} +\end{equation} + +\end_inset + +. +\end_layout + +\begin_layout Subsubsection* +GaussianMixture +\end_layout + +\begin_layout Standard +A +\emph on +GaussianMixture +\emph default + (maybe to be renamed to +\emph on +GaussianMixtureComponent +\emph default +) just indexes into a number of +\emph on +GaussianConditional +\emph default + instances, that are each properly normalized: +\end_layout + +\begin_layout Standard +\begin_inset Formula +\[ +P(x|y,m)=P_{m}(x|y). +\] + +\end_inset + +We store one +\emph on +GaussianConditional +\emph default + +\begin_inset Formula $P_{m}(x|y)$ +\end_inset + + for every possible assignment +\begin_inset Formula $m$ +\end_inset + + to a set of discrete variables. + As +\emph on +GaussianMixture +\emph default + is a +\emph on +Conditional +\emph default +, it needs to satisfy the a similar invariant to +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:gc_invariant" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +: +\begin_inset Formula +\begin{equation} +\log P(x|y,m)=K_{gm}-E_{gm}(x,y,m).\label{eq:gm_invariant} +\end{equation} + +\end_inset + +If we take the log of +\begin_inset Formula $P(x|y,m)$ +\end_inset + + we get +\begin_inset Formula +\begin{equation} +\log P(x|y,m)=\log P_{m}(x|y)=K_{gcm}-E_{gcm}(x,y).\label{eq:gm_log} +\end{equation} + +\end_inset + +Equating +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:gm_invariant" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + and +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:gm_log" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + we see that this can be achieved by defining the error +\begin_inset Formula $E_{gm}(x,y,m)$ +\end_inset + + as +\begin_inset Formula +\begin{equation} +E_{gm}(x,y,m)=E_{gcm}(x,y)+K_{gm}-K_{gcm}\label{eq:gm_error} +\end{equation} + +\end_inset + +where choose +\begin_inset Formula $K_{gm}=\max K_{gcm}$ +\end_inset + +, as then the error will always be positive. +\end_layout + +\begin_layout Section +Hybrid Factors +\end_layout + +\begin_layout Standard +In GTSAM, we typically condition on known measurements, and factors encode + the resulting negative log-likelihood of the unknown variables +\begin_inset Formula $y$ +\end_inset + + given the measurements +\begin_inset Formula $x$ +\end_inset + +. + We review how a Gaussian conditional density is converted into a Gaussian + factor, and then develop a hybrid version satisfying the correct invariants + as well. +\end_layout + +\begin_layout Subsubsection* +JacobianFactor +\end_layout + +\begin_layout Standard +A +\emph on +JacobianFactor +\emph default + typically results from a +\emph on +GaussianConditional +\emph default + by having known values +\begin_inset Formula $\bar{x}$ +\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 +\begin{equation} +L(y)\propto P(\bar{x}|y)\label{eq:likelihood} +\end{equation} + +\end_inset + +In GTSAM factors represent the negative log-likelihood +\begin_inset Formula $E_{jf}(y)$ +\end_inset + + and hence we have +\begin_inset Formula +\[ +E_{jf}(y)=-\log L(y)=C-\log P(\bar{x}|y), +\] + +\end_inset + +with +\begin_inset Formula $C$ +\end_inset + + the log of the proportionality constant in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:likelihood" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. + Substituting in +\begin_inset Formula $\log P(\bar{x}|y)$ +\end_inset + + from the invariant +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:gc_invariant" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + we obtain +\begin_inset Formula +\[ +E_{jf}(y)=C-K_{gc}+E_{gc}(\bar{x},y). +\] + +\end_inset + +The +\emph on +likelihood +\emph default + function in +\emph on +GaussianConditional +\emph default + chooses +\begin_inset Formula $C=K_{gc}$ +\end_inset + +, and the +\emph on +JacobianFactor +\emph default + does not store any constant; it just implements: +\begin_inset Formula +\[ +E_{jf}(y)=E_{gc}(\bar{x},y)=\frac{1}{2}\|R\bar{x}+Sy-d\|_{\Sigma}^{2}=\frac{1}{2}\|Ay-b\|_{\Sigma}^{2} +\] + +\end_inset + +with +\begin_inset Formula $A=S$ +\end_inset + + and +\begin_inset Formula $b=d-R\bar{x}$ +\end_inset + +. +\end_layout + +\begin_layout Subsubsection* +GaussianMixtureFactor +\end_layout + +\begin_layout Standard +Analogously, a +\emph on +GaussianMixtureFactor +\emph default + typically results from a GaussianMixture by having known values +\begin_inset Formula $\bar{x}$ +\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 +\[ +L(y,m)\propto P(\bar{x}|y,m). +\] + +\end_inset + +We will similarly implement the negative log-likelihood +\begin_inset Formula $E_{mf}(y,m)$ +\end_inset + +: +\begin_inset Formula +\[ +E_{mf}(y,m)=-\log L(y,m)=C-\log P(\bar{x}|y,m). +\] + +\end_inset + +Since we know the log-density from the invariant +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:gm_invariant" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +, we obtain +\begin_inset Formula +\[ +\log P(\bar{x}|y,m)=K_{gm}-E_{gm}(\bar{x},y,m), +\] + +\end_inset + + and hence +\begin_inset Formula +\[ +E_{mf}(y,m)=C+E_{gm}(\bar{x},y,m)-K_{gm}. +\] + +\end_inset + +Substituting in +\begin_inset CommandInset ref +LatexCommand eqref +reference "eq:gm_error" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + we finally have an expression where +\begin_inset Formula $K_{gm}$ +\end_inset + + canceled out, but we have a dependence on the individual component constants + +\begin_inset Formula $K_{gcm}$ +\end_inset + +: +\begin_inset Formula +\[ +E_{mf}(y,m)=C+E_{gcm}(\bar{x},y)-K_{gcm}. +\] + +\end_inset + +Unfortunately, we can no longer choose +\begin_inset Formula $C$ +\end_inset + + independently from +\begin_inset Formula $m$ +\end_inset + + to make the constant disappear. + 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 diff --git a/doc/Hybrid.pdf b/doc/Hybrid.pdf new file mode 100644 index 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