Template for submissions to the Symmetry and Geometry in Neural Representations Workshop at NeurIPS 2023.
\begin
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%\usepackage{rotating}% for sideways figures and tables
\usepackage{longtable}% for long tables
% The booktabs package is used by this sample document
% (it provides \toprule, \midrule and \bottomrule).
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\usepackage{booktabs}
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% to align numbers in a column by their decimal point.
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\usepackage[load-configurations=version-1]{siunitx} % newer version
%\usepackage{siunitx}
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\newcommand{\cs}[1]{\texttt{\char`\\#1}}
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\theoremheaderfont{\scshape}
\theorempostheader{:}
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\newtheorem*{note}{Note}
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\jmlrvolume{}
\firstpageno{1}
\editors{List of editors' names}
\jmlryear{2023}
\jmlrworkshop{Symmetry and Geometry in Neural Representations}
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\title[Short Title]{Full Title of Article\titlebreak This Title Has
A Line Break\titletag{\thanks{sample footnote}}}
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% Information about authors (Full names, emails, affiliations) have to be provided only for the submission of the camera-ready version. Only in that case, you can uncomment and use the next blocks.
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% Use \Name{Author Name} to specify the name.
% Spaces are used to separate forenames from the surname so that
% the surnames can be picked up for the page header and copyright footer.
% If the surname contains spaces, enclose the surname
% in braces, e.g. \Name{John {Smith Jones}} similarly
% if the name has a "von" part, e.g \Name{Jane {de Winter}}.
% If the first letter in the forenames is a diacritic
% enclose the diacritic in braces, e.g. \Name{{\'E}louise Smith}
% *** Make sure there's no spurious space before \nametag ***
% Two authors with the same address
% \author{\Name{Author Name1\nametag{\thanks{with a note}}} \Email{abc@sample.com}\and
% \Name{Author Name2} \Email{xyz@sample.com}\\
% \addr Address}
%Three or more authors with the same address:
% \author{\Name{Author Name1} \Email{an1@sample.com}\\
% \Name{Author Name2} \Email{an2@sample.com}\\
% \Name{Author Name3} \Email{an3@sample.com}\\
% \Name{Author Name4} \Email{an4@sample.com}\\
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% \addr Address}
% Authors with different addresses:
% \author{\Name{Author Name1} \Email{abc@sample.com}\\
% \addr Address 1
% \AND
% \Name{Author Name2} \Email{xyz@sample.com}\\
% \addr Address 2
%}
\begin{document}
\maketitle
\begin{abstract}
This is the abstract for this article.
\end{abstract}
\begin{keywords}
List of keywords
\end{keywords}
\section{Introduction}
\label{sec:intro}
This is a sample article that uses the \textsf{jmlr} class with
the \texttt{pmlr} class option. Please follow the guidelines in
this sample document as it can help to reduce complications when
combining the articles into a book. Please avoid using obsolete
commands, such as \verb|\rm|, and obsolete packages, such as
\textsf{epsfig}.\footnote{See
\url{http://www.ctan.org/pkg/l2tabu}}
Please also ensure that your document will compile with PDF\LaTeX.
If you have an error message that's puzzling you, first check for it
at the UK TUG FAQ
\url{https://texfaq.org/FAQ-man-latex}. If
that doesn't help, create a minimal working example (see
\url{https://www.dickimaw-books.com/latex/minexample}) and post
to somewhere like TeX on StackExchange
(\url{https://tex.stackexchange.com/}) or the LaTeX Community Forum
(\url{https://latex.org/forum/}).
\begin{note}
This is an numbered theorem-like environment that was defined in
this document's preamble.
\end{note}
\subsection{Sub-sections}
Sub-sections are produced using \verb|\subsection|.
\subsubsection{Sub-sub-sections}
Sub-sub-sections are produced using \verb|\subsubsection|.
\paragraph{Sub-sub-sub-sections}
Sub-sub-sub-sections are produced using \verb|\paragraph|.
These are unnumbered with a running head.
\subparagraph{Sub-sub-sub-sub-sections}
Sub-sub-sub-sub-sections are produced using \verb|\subparagraph|.
These are unnumbered with a running head.
\section{Cross-Referencing}
Always use \verb|\label| and \verb|\ref| (or one of the commands
described below) when cross-referencing. For example, the next
section is Section~\ref{sec:math}. The \textsf{jmlr} class
provides some convenient cross-referencing commands:
\verb|\sectionref|, \verb|\equationref|, \verb|\tableref|,
\verb|\figureref|, \verb|\algorithmref|, \verb|\theoremref|,
\verb|\lemmaref|, \verb|\remarkref|, \verb|\corollaryref|,
\verb|\definitionref|, \verb|\conjectureref|, \verb|\axiomref|,
\verb|\exampleref| and \verb|\appendixref|. The argument of these
commands may either be a single label or a comma-separated list
of labels. Examples:
Referencing sections: \sectionref{sec:math} or
\sectionref{sec:intro,sec:math} or
\sectionref{sec:intro,sec:math,sec:tables,sec:figures}.
Referencing equations: \equationref{eq:trigrule} or
\equationref{eq:trigrule,eq:df} or
\equationref{eq:trigrule,eq:f,eq:df,eq:y}.
Referencing tables: \tableref{tab:operatornames} or
\tableref{tab:operatornames,tab:example} or
\tableref{tab:operatornames,tab:example,tab:example-booktabs}.
Referencing figures: \figureref{fig:image} or
\figureref{fig:image,fig:teximage} or
\figureref{fig:image,fig:teximage,fig:subfigex} or
\figureref{fig:image-a,fig:image-b}.
Referencing algorithms: \algorithmref{alg:gauss} or
\algorithmref{alg:gauss,alg:moore} or
\algorithmref{alg:gauss,alg:moore,alg:net}.
Referencing theorem-like environments: \theoremref{thm:eigenpow},
\lemmaref{lem:sample}, \remarkref{rem:sample},
\corollaryref{cor:sample}, \definitionref{def:sample},
\conjectureref{con:sample}, \axiomref{ax:sample} and
\exampleref{ex:sample}.
Referencing appendices: \appendixref{apd:first} or
\appendixref{apd:first,apd:second}.
\section{Equations}
\label{sec:math}
The \textsf{jmlr} class loads the \textsf{amsmath} package, so
you can use any of the commands and environments defined there.
(See the \textsf{amsmath} documentation for further
details.\footnote{Either \texttt{texdoc amsmath} or
\url{http://www.ctan.org/pkg/amsmath}})
Unnumbered single-lined equations should be displayed using
\verb|\[| and \verb|\]|. For example:
\[E = m c^2\]
Numbered single-line equations should be displayed using the
\texttt{equation} environment. For example:
\begin{equation}\label{eq:trigrule}
\cos^2\theta + \sin^2\theta \equiv 1
\end{equation}
This can be referenced using \verb|\label| and \verb|\equationref|.
For example, \equationref{eq:trigrule}.
Multi-lined numbered equations should be displayed using the
\texttt{align} environment.\footnote{For reasons why you
shouldn't use the obsolete \texttt{eqnarray} environment, see
Lars Madsen, \emph{Avoid eqnarray!} TUGboat 33(1):21--25, 2012.} For example:
\begin{align}
f(x) &= x^2 + x\label{eq:f}\\
f'(x) &= 2x + 1\label{eq:df}
\end{align}
Unnumbered multi-lined equations should be displayed using the
\texttt{align*} environment. For example:
\begin{align*}
f(x) &= (x+1)(x-1)\\
&= x^2 - 1
\end{align*}
If you want to mix numbered with unnumbered lines use the
align environment and suppress unwanted line numbers with
\verb|\nonumber|. For example:
\begin{align}
y &= x^2 + 3x - 2x + 1\nonumber\\
&= x^2 + x + 1\label{eq:y}
\end{align}
An equation that is too long to fit on a single line can be
displayed using the \texttt{split} environment.
Text can be embedded in an equation using \verb|\text| or
\verb|\intertext| (as used in \theoremref{thm:eigenpow}).
See the \textsf{amsmath} documentation for further details.
\subsection{Operator Names}
\label{sec:op}
Predefined operator names are listed in \tableref{tab:operatornames}.
For additional operators, either use \verb|\operatorname|,
for example $\operatorname{var}(X)$ or declare it with
\verb|\DeclareMathOperator|, for example
\begin{verbatim}
\DeclareMathOperator{\var}{var}
\end{verbatim}
and then use this new command. If you want limits that go above and
below the operator (like \verb|\sum|) use the starred versions
(\verb|\operatorname*| or \verb|\DeclareMathOperator*|).
\begin{table*}[htbp]
\floatconts
{tab:operatornames}%
{\caption{Predefined Operator Names (taken from
\textsf{amsmath} documentation)}}%
{%
\begin{tabular}{rlrlrlrl}
\cs{arccos} & $\arccos$ & \cs{deg} & $\deg$ & \cs{lg} & $\lg$ & \cs{projlim} & $\projlim$ \\
\cs{arcsin} & $\arcsin$ & \cs{det} & $\det$ & \cs{lim} & $\lim$ & \cs{sec} & $\sec$ \\
\cs{arctan} & $\arctan$ & \cs{dim} & $\dim$ & \cs{liminf} & $\liminf$ & \cs{sin} & $\sin$ \\
\cs{arg} & $\arg$ & \cs{exp} & $\exp$ & \cs{limsup} & $\limsup$ & \cs{sinh} & $\sinh$ \\
\cs{cos} & $\cos$ & \cs{gcd} & $\gcd$ & \cs{ln} & $\ln$ & \cs{sup} & $\sup$ \\
\cs{cosh} & $\cosh$ & \cs{hom} & $\hom$ & \cs{log} & $\log$ & \cs{tan} & $\tan$ \\
\cs{cot} & $\cot$ & \cs{inf} & $\inf$ & \cs{max} & $\max$ & \cs{tanh} & $\tanh$ \\
\cs{coth} & $\coth$ & \cs{injlim} & $\injlim$ & \cs{min} & $\min$ \\
\cs{csc} & $\csc$ & \cs{ker} & $\ker$ & \cs{Pr} & $\Pr$
\end{tabular}\par
\begin{tabular}{rlrl}
\cs{varlimsup} & $\varlimsup$
& \cs{varinjlim} & $\varinjlim$\\
\cs{varliminf} & $\varliminf$
& \cs{varprojlim} & $\varprojlim$
\end{tabular}
}
\end{table*}
\section{Vectors and Sets}
\label{sec:vec}
Vectors should be typeset using \cs{vec}. For example $\vec{x}$.
The \textsf{jmlr} class also provides \cs{set} to typeset a
set. For example $\set{S}$.
\section{Floats}
\label{sec:floats}
Floats, such as figures, tables and algorithms, are moving
objects and are supposed to float to the nearest convenient
location. Please don't force them to go in a particular place. In
general it's best to use the \texttt{htbp} specifier and don't
put the figure or table in the middle of a paragraph (that is
make sure there's a paragraph break above and below the float).
Floats are supposed to have a little extra space above and below
them to make them stand out from the rest of the text. This extra
spacing is put in automatically and shouldn't need modifying.
To ensure consistency, please \emph{don't} try changing the format of the caption by doing
something like:
\begin{verbatim}
\caption{\textit{A Sample Caption.}}
\end{verbatim}
or
\begin{verbatim}
\caption{\em A Sample Caption.}
\end{verbatim}
You can, of course, change the font for individual words or
phrases, for example:
\begin{verbatim}
\caption{A Sample Caption With
Some \emph{Emphasized Words}.}
\end{verbatim}
\subsection{Tables}
\label{sec:tables}
Tables should go in the \texttt{table} environment. Within this
environment use \verb|\floatconts| (defined by \textsf{jmlr})
to set the caption correctly and center the table contents.
\begin{table}[htbp]
% The first argument is the label.
% The caption goes in the second argument, and the table contents
% go in the third argument.
\floatconts
{tab:example}%
{\caption{An Example Table}}%
{\begin{tabular}{ll}
\bfseries Dataset & \bfseries Result\\
Data1 & 0.12345\\
Data2 & 0.67890\\
Data3 & 0.54321\\
Data4 & 0.09876
\end{tabular}}
\end{table}
If you want horizontal rules you can use the \textsf{booktabs}
package which provides the commands \verb|\toprule|,
\verb|\midrule| and \verb|\bottomrule|. For example, see
\tableref{tab:example-booktabs}.
\begin{table}[hbtp]
\floatconts
{tab:example-booktabs}
{\caption{A Table With Horizontal Lines}}
{\begin{tabular}{ll}
\toprule
\bfseries Dataset & \bfseries Result\\
\midrule
Data1 & 0.12345\\
Data2 & 0.67890\\
Data3 & 0.54321\\
Data4 & 0.09876\\
\bottomrule
\end{tabular}}
\end{table}
If you want vertical lines as well, you can't use the
\textsf{booktabs} commands as there'll be some unwanted gaps.
Instead you can use \LaTeX's \verb|\hline|, but the rows may
appear a bit cramped. You can add extra space above or below a
row using \verb|\abovestrut| and \verb|\belowstrut|. For example,
see \tableref{tab:example-hline}.
\begin{table}[htbp]
\floatconts
{tab:example-hline}
{\caption{A Table With Horizontal and Vertical Lines}}%
{%
\begin{tabular}{|l|l|}
\hline
\abovestrut{2.2ex}\bfseries Dataset & \bfseries Result\\\hline
\abovestrut{2.2ex}Data1 & 0.12345\\
Data2 & 0.67890\\
Data3 & 0.54321\\
\belowstrut{0.2ex}Data4 & 0.09876\\\hline
\end{tabular}
}
\end{table}
If you want to align numbers on their decimal point, you can
use the \textsf{siunitx} package. For example, see
\tableref{tab:example-siunitx}. For further details see the
\textsf{siunitx} documentation\footnote{Either \texttt{texdoc
siunitx} or \url{http://www.ctan.org/pkg/siunitx}}.
\begin{table}[htbp]
\floatconts
{tab:example-siunitx}
{\caption{A Table With Numbers Aligned on the Decimal Point}}
{\begin{tabular}{lS}
\bfseries Dataset & {\bfseries Result}\\
Data1 & 0.12345\\
Data2 & 10.6789\\
Data3 & 50.543\\
Data4 & 200.09876
\end{tabular}}
\end{table}
If the table is too wide, you can adjust the inter-column
spacing by changing the value of \verb|\tabcolsep|. For
example:
\begin{verbatim}
\setlength{\tabcolsep}{3pt}
\end{verbatim}
If the table is very wide but not very long, you can use the
\texttt{sidewaystable} environment defined in the
\textsf{rotating} package (so use \verb|\usepackage{rotating}|).
If the table is too long to fit on a page, you should use the
\texttt{longtable} environment defined in the \textsf{longtable}
package (so use \verb|\usepackage{longtable}|).
\subsection{Figures}
\label{sec:figures}
Figures should go in the \texttt{figure} environment. Within this
environment, use \verb|\floatconts| to correctly position the
caption and center the image. Use \verb|\includegraphics|
for external graphics files but omit the file extension. Do not
use \verb|\epsfig| or \verb|\psfig|. If you want to scale the
image, it's better to use a fraction of the line width rather
than an explicit length. For example, see \figureref{fig:image}.
\begin{figure}[htbp]
% Caption and label go in the first argument and the figure contents
% go in the second argument
\floatconts
{fig:image}
{\caption{Example Image}}
{\includegraphics[width=0.5\linewidth]{example-image}}
\end{figure}
If your image is made up of \LaTeX\ code (for example, commands
provided by the \textsf{pgf} package) you can include it using
\cs{includeteximage} (defined by the \textsf{jmlr} class). This
can be scaled and rotated in the same way as \cs{includegraphics}.
For example, see \figureref{fig:teximage}.
\begin{figure}[htbp]
\floatconts
{fig:teximage}
{\caption{Image Created Using \LaTeX\ Code}}
{\includeteximage[angle=45]{example-teximage}}
\end{figure}
If the figure is too wide to fit on the page, you can use the
\texttt{sidewaysfigure} environment defined in the
\textsf{rotating} package.
It's best not to use \verb|\graphicspath| with the \textsf{jmlr} class
as it can cause problems with the production editing process. If the
images are contained in a subdirectory, specify this when you
include the image, for example \verb|\includegraphics{figures/mypic}|.
\subsection{Sub-Figures}
\label{sec:subfigures}
Sub-figures can be created using \verb|\subfigure|, which is
defined by the \textsf{jmlr} class. The optional argument allows
you to provide a subcaption. The label should be placed in the
mandatory argument of \verb|\subfigure|. You can reference the
entire figure, for example \figureref{fig:subfigex}, or you can
reference part of the figure using \verb|\figureref|, for example
\figureref{fig:image-a}. Alternatively you can reference the
subfigure using \verb|\subfigref|, for example
\subfigref{fig:image-a,fig:image-b} in \figureref{fig:subfigex}.
\begin{figure}[htbp]
\floatconts
{fig:subfigex}
{\caption{An Example With Sub-Figures.}}
{%
\subfigure[Image A]{\label{fig:image-a}%
\includegraphics[width=0.2\linewidth]{example-image-a}}%
\qquad
\subfigure[Image B]{\label{fig:image-b}%
\includegraphics[width=0.25\linewidth]{example-image-b}}
}
\end{figure}
By default, the sub-figures are aligned on the baseline.
This can be changed using the second optional argument
of \verb|\subfigure|. This may be \texttt{t} (top), \texttt{c}
(centered) or \texttt{b} (bottom). For example, the subfigures
\subfigref{fig:image-a2,fig:image-b2} in \figureref{fig:subfigex2}
both have \verb|[c]| as the second optional argument.
\begin{figure}[htbp]
\floatconts
{fig:subfigex2}
{\caption{Another Example With Sub-Figures (with a Cramped
Sub-Caption).}}
{%
\subfigure[Image A][c]{\label{fig:image-a2}%
\includegraphics[width=0.1\linewidth]{example-image-a}}%
\qquad
\subfigure[Image B][c]{\label{fig:image-b2}%
\includegraphics[width=0.3\linewidth]{example-image-b}}
}%
\end{figure}
Note that a very narrow sub-float will leave little space for the
sub-caption, which will likely cause Underfull/Overfull hbox warnings.
You can set the length \verb|\jmlrminsubcaptionwidth| to indicate
the minimum width to be made available for the sub-captions, as in
\figureref{fig:subfigex3}.
\begin{figure}[htbp]
\floatconts
{fig:subfigex3}
{\caption{Another Example With Sub-Figures (with a Less Cramped
Sub-Caption).}}
{\setlength{\jmlrminsubcaptionwidth}{0.2\linewidth}%
\subfigure[Image A][c]{\label{fig:image-a3}%
\includegraphics[width=0.1\linewidth]{example-image-a}}%
\qquad
\subfigure[Image B][c]{\label{fig:image-b3}%
\includegraphics[width=0.2\linewidth]{example-image-b}}
}%
\end{figure}
\subsection{Sub-Tables}
\label{sec:subtables}
There is an analogous command \verb|\subtable| for sub-tables.
It has the same syntax as \verb|\subfigure| described above.
You can reference the table using \verb|\tableref|, for example
\tableref{tab:subtabex} or you can reference part of the table,
for example \tableref{tab:ab}. Alternatively you can reference the
subtable using \verb|\subtabref|, for example
\subtabref{tab:ab,tab:cd} in \tableref{tab:subtabex}.
\begin{table}[htbp]
\floatconts
{tab:subtabex}
{\caption{An Example With Sub-Tables}}
{%
\subtable{%
\label{tab:ab}%
\begin{tabular}{cc}
\bfseries A & \bfseries B\\
1 & 2
\end{tabular}
}\qquad
\subtable{%
\label{tab:cd}%
\begin{tabular}{cc}
\bfseries C & \bfseries D\\
3 & 4\\
5 & 6
\end{tabular}
}
}
\end{table}
By default, the sub-tables are aligned on the top.
This can be changed using the second optional argument
of \verb|\subtable|. This may be \texttt{t} (top), \texttt{c}
(centered) or \texttt{b} (bottom). For example, the sub-tables
\subtabref{tab:ab2,tab:cd2} in \tableref{tab:subtabex2}
both have \verb|[c]| as the second optional argument.
\begin{table}[htbp]
\floatconts
{tab:subtabex2}
{\caption{Another Example With Sub-Tables}}
{%
\subtable[][c]{%
\label{tab:ab2}%
\begin{tabular}{cc}
\bfseries A & \bfseries B\\
1 & 2
\end{tabular}
}\qquad
\subtable[][c]{%
\label{tab:cd2}%
\begin{tabular}{cc}
\bfseries C & \bfseries D\\
3 & 4\\
5 & 6
\end{tabular}
}
}
\end{table}
\subsection{Algorithms}
\label{sec:algorithms}
Enumerated textual algorithms can be displayed using the
\texttt{algorithm} environment. Within this environment, use
use an \texttt{enumerate} or nested \texttt{enumerate} environments.
For example, see \algorithmref{alg:gauss}. Note that algorithms
float like figures and tables.
\begin{algorithm}[htbp]
\floatconts
{alg:gauss}% label
{\caption{The Gauss-Seidel Algorithm}}
{% contents
\begin{enumerate}
\item For $k=1$ to maximum number of iterations
\begin{enumerate}
\item For $i=1$ to $n$
\begin{enumerate}
\item $x_i^{(k)} =
\frac{b_i - \sum_{j=1}^{i-1}a_{ij}x_j^{(k)}
- \sum_{j=i+1}^{n}a_{ij}x_j^{(k-1)}}{a_{ii}}$
\item If $\|\vec{x}^{(k)}-\vec{x}^{(k-1)} < \epsilon\|$,
where $\epsilon$ is a specified stopping criteria, stop.
\end{enumerate}
\end{enumerate}
\end{enumerate}
}
\end{algorithm}
If you'd rather have the same numbering throughout the algorithm
but still want the convenient indentation of nested
\texttt{enumerate} environments, you can use the
\texttt{enumerate*} environment provided by the \textsf{jmlr}
class. For example, see \algorithmref{alg:moore}.
\begin{algorithm}
\floatconts
{alg:moore}% label
{\caption{Moore's Shortest Path}}%caption
{% main float content
Given a connected graph $G$, where the length of each edge is 1:
\begin{enumerate*}
\item Set the label of vertex $s$ to 0
\item Set $i=0$
\begin{enumerate*}
\item \label{step:locate}Locate all unlabelled vertices
adjacent to a vertex labelled $i$ and label them $i+1$
\item If vertex $t$ has been labelled,
\begin{enumerate*}
\item[] the shortest path can be found by backtracking, and
the length is given by the label of $t$.
\end{enumerate*}
otherwise
\begin{enumerate*}
\item[] increment $i$ and return to step~\ref{step:locate}
\end{enumerate*}
\end{enumerate*}
\end{enumerate*}
}
\end{algorithm}
Pseudo code can be displayed using the \texttt{algorithm2e}
environment. This is defined by the \textsf{algorithm2e} package
(which is automatically loaded) so check the \textsf{algorithm2e}
documentation for further details.\footnote{Either \texttt{texdoc
algorithm2e} or \url{http://www.ctan.org/pkg/algorithm2e}}
For an example, see \algorithmref{alg:net}.
\begin{algorithm2e}
\caption{Computing Net Activation}
\label{alg:net}
% older versions of algorithm2e have \dontprintsemicolon instead
% of the following:
%\DontPrintSemicolon
% older versions of algorithm2e have \linesnumbered instead of the
% following:
%\LinesNumbered
\KwIn{$x_1, \ldots, x_n, w_1, \ldots, w_n$}
\KwOut{$y$, the net activation}
$y\leftarrow 0$\;
\For{$i\leftarrow 1$ \KwTo $n$}{
$y \leftarrow y + w_i*x_i$\;
}
\end{algorithm2e}
\section{Description Lists}
The \textsf{jmlr} class also provides a description-like
environment called \texttt{altdescription}. This has an
argument that should be the widest label in the list. Compare:
\begin{description}
\item[add] A method that adds two variables.
\item[differentiate] A method that differentiates a function.
\end{description}
with
\begin{altdescription}{differentiate}
\item[add] A method that adds two variables.
\item[differentiate] A method that differentiates a function.
\end{altdescription}
\section{Theorems, Lemmas etc}
\label{sec:theorems}
The following theorem-like environments are predefined by
the \textsf{jmlr} class: \texttt{theorem}, \texttt{example},
\texttt{lemma}, \texttt{proposition}, \texttt{remark},
\texttt{corollary}, \texttt{definition}, \texttt{conjecture}
and \texttt{axiom}. You can use the \texttt{proof} environment
to display the proof if need be, as in \theoremref{thm:eigenpow}.
\begin{theorem}[Eigenvalue Powers]\label{thm:eigenpow}
If $\lambda$ is an eigenvalue of $\vec{B}$ with eigenvector
$\vec{\xi}$, then $\lambda^n$ is an eigenvalue of $\vec{B}^n$
with eigenvector $\vec{\xi}$.
\begin{proof}
Let $\lambda$ be an eigenvalue of $\vec{B}$ with eigenvector
$\xi$, then
\begin{align*}
\vec{B}\vec{\xi} &= \lambda\vec{\xi}
\intertext{premultiply by $\vec{B}$:}
\vec{B}\vec{B}\vec{\xi} &= \vec{B}\lambda\vec{\xi}\\
\Rightarrow \vec{B}^2\vec{\xi} &= \lambda\vec{B}\vec{\xi}\\
&= \lambda\lambda\vec{\xi}\qquad
\text{since }\vec{B}\vec{\xi}=\lambda\vec{\xi}\\
&= \lambda^2\vec{\xi}
\end{align*}
Therefore true for $n=2$. Now assume true for $n=k$:
\begin{align*}
\vec{B}^k\vec{\xi} &= \lambda^k\vec{\xi}
\intertext{premultiply by $\vec{B}$:}
\vec{B}\vec{B}^k\vec{\xi} &= \vec{B}\lambda^k\vec{\xi}\\
\Rightarrow \vec{B}^{k+1}\vec{\xi} &= \lambda^k\vec{B}\vec{\xi}\\
&= \lambda^k\lambda\vec{\xi}\qquad
\text{since }\vec{B}\vec{\xi}=\lambda\vec{\xi}\\
&= \lambda^{k+1}\vec{\xi}
\end{align*}
Therefore true for $n=k+1$. Therefore, by induction, true for all
$n$.
\end{proof}
\end{theorem}
\begin{lemma}[A Sample Lemma]\label{lem:sample}
This is a lemma.
\end{lemma}
\begin{remark}[A Sample Remark]\label{rem:sample}
This is a remark.
\end{remark}
\begin{corollary}[A Sample Corollary]\label{cor:sample}
This is a corollary.
\end{corollary}
\begin{definition}[A Sample Definition]\label{def:sample}
This is a definition.
\end{definition}
\begin{conjecture}[A Sample Conjecture]\label{con:sample}
This is a conjecture.
\end{conjecture}
\begin{axiom}[A Sample Axiom]\label{ax:sample}
This is an axiom.
\end{axiom}
\begin{example}[An Example]\label{ex:sample}
This is an example.
\end{example}
\section{Citations and Bibliography}
\label{sec:cite}
The \textsf{jmlr} class automatically loads \textsf{natbib}.
This sample file has the citations defined in the accompanying
BibTeX file \texttt{pmlr-sample.bib}. For a parenthetical
citation use \verb|\citep|. For example
\citep{guyon-elisseeff-03}. For a textual citation use
\verb|\citet|. For example \citet{guyon2007causalreport}.
Both commands may take a comma-separated list, for example
\citet{guyon-elisseeff-03,guyon2007causalreport}.
These commands have optional arguments and have a starred
version. See the \textsf{natbib} documentation for further
details.\footnote{Either \texttt{texdoc natbib} or
\url{http://www.ctan.org/pkg/natbib}}
The bibliography is displayed using \verb|\bibliography|.
\acks{Acknowledgements go here.}
\bibliography{pmlr-sample}
\appendix
\section{First Appendix}\label{apd:first}
This is the first appendix.
\section{Second Appendix}\label{apd:second}
This is the second appendix.
\end{document}