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Commit 20c640a2 authored by Luc Giffon's avatar Luc Giffon
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remarques Luc + ajouter fichiers dummy tex au gitignore

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*/.kile/*
*.kilepr
# Byte-compiled / optimized / DLL files
__pycache__/
*.py[cod]
......@@ -87,3 +89,282 @@ target/
# Mypy cache
.mypy_cache/
# latex
## Core latex/pdflatex auxiliary files:
*.aux
*.lof
*.log
*.lot
*.fls
*.out
*.toc
*.fmt
*.fot
*.cb
*.cb2
.*.lb
## Intermediate documents:
*.dvi
*.xdv
*-converted-to.*
# these rules might exclude image files for figures etc.
# *.ps
# *.eps
# *.pdf
## Generated if empty string is given at "Please type another file name for output:"
.pdf
## Bibliography auxiliary files (bibtex/biblatex/biber):
*.bbl
*.bcf
*.blg
*-blx.aux
*-blx.bib
*.run.xml
## Build tool auxiliary files:
*.fdb_latexmk
*.synctex
*.synctex(busy)
*.synctex.gz
*.synctex.gz(busy)
*.pdfsync
## Build tool directories for auxiliary files
# latexrun
latex.out/
## Auxiliary and intermediate files from other packages:
# algorithms
*.alg
*.loa
# achemso
acs-*.bib
# amsthm
*.thm
# beamer
*.nav
*.pre
*.snm
*.vrb
# changes
*.soc
# comment
*.cut
# cprotect
*.cpt
# elsarticle (documentclass of Elsevier journals)
*.spl
# endnotes
*.ent
# fixme
*.lox
# feynmf/feynmp
*.mf
*.mp
*.t[1-9]
*.t[1-9][0-9]
*.tfm
#(r)(e)ledmac/(r)(e)ledpar
*.end
*.?end
*.[1-9]
*.[1-9][0-9]
*.[1-9][0-9][0-9]
*.[1-9]R
*.[1-9][0-9]R
*.[1-9][0-9][0-9]R
*.eledsec[1-9]
*.eledsec[1-9]R
*.eledsec[1-9][0-9]
*.eledsec[1-9][0-9]R
*.eledsec[1-9][0-9][0-9]
*.eledsec[1-9][0-9][0-9]R
# glossaries
*.acn
*.acr
*.glg
*.glo
*.gls
*.glsdefs
*.lzo
*.lzs
# uncomment this for glossaries-extra (will ignore makeindex's style files!)
# *.ist
# gnuplottex
*-gnuplottex-*
# gregoriotex
*.gaux
*.gtex
# htlatex
*.4ct
*.4tc
*.idv
*.lg
*.trc
*.xref
# hyperref
*.brf
# knitr
*-concordance.tex
# TODO Comment the next line if you want to keep your tikz graphics files
*.tikz
*-tikzDictionary
# listings
*.lol
# luatexja-ruby
*.ltjruby
# makeidx
*.idx
*.ilg
*.ind
# minitoc
*.maf
*.mlf
*.mlt
*.mtc[0-9]*
*.slf[0-9]*
*.slt[0-9]*
*.stc[0-9]*
# minted
_minted*
*.pyg
# morewrites
*.mw
# nomencl
*.nlg
*.nlo
*.nls
# pax
*.pax
# pdfpcnotes
*.pdfpc
# sagetex
*.sagetex.sage
*.sagetex.py
*.sagetex.scmd
# scrwfile
*.wrt
# sympy
*.sout
*.sympy
sympy-plots-for-*.tex/
# pdfcomment
*.upa
*.upb
# pythontex
*.pytxcode
pythontex-files-*/
# tcolorbox
*.listing
# thmtools
*.loe
# TikZ & PGF
*.dpth
*.md5
*.auxlock
# todonotes
*.tdo
# vhistory
*.hst
*.ver
# easy-todo
*.lod
# xcolor
*.xcp
# xmpincl
*.xmpi
# xindy
*.xdy
# xypic precompiled matrices and outlines
*.xyc
*.xyd
# endfloat
*.ttt
*.fff
# Latexian
TSWLatexianTemp*
## Editors:
# WinEdt
*.bak
*.sav
# Texpad
.texpadtmp
# LyX
*.lyx~
# Kile
*.backup
# gummi
.*.swp
# KBibTeX
*~[0-9]*
# auto folder when using emacs and auctex
./auto/*
*.el
# expex forward references with \gathertags
*-tags.tex
# standalone packages
*.sta
# Makeindex log files
*.lpz
reports/*.pdf
......@@ -7,7 +7,8 @@
\usepackage{float}
\usepackage{dsfont}
\usepackage{amsmath}
\usepackage{todonotes}
\input{math_commands.tex}
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
......@@ -20,6 +21,8 @@
\renewcommand{\ALG@beginalgorithmic}{\small}
\makeatother
\def\rvalpha{{\boldsymbol{\alpha}}}
\title{bolsonaro}
\date{September 2019}
......@@ -31,16 +34,16 @@
introduire le pb et les motivation ...
\subsection{Notation}
Let $ X \in \mathbb{R}^{n \times d}$ be the matrix data and $Y \in \mathbb{R}^{n}$ be the labels vector associated to the matrix $X$, where for each $i$, $\textbf{x}_i \in \cal{X} \subseteq \mathbb{R}^d$ and $y_i \in {\cal Y} \subseteq \mathbb{R}$. \\
Let $ X \in \mathbb{R}^{n \times d}$ \todo{est-ce-que le non-gras majuscule est fréquent pour les matrices?} be the matrix data and $Y \in \mathbb{R}^{n}$ be the labels vector associated to the matrix $X$, where for each $i$, $\textbf{x}_i \in \mathcal{X} \subseteq \mathbb{R}^{d}$ and $y_i \in {\mathcal Y} \subseteq \mathbb{R}$. \\
A random forest $F_{t_1, \dots, t_l}$ is a classifier made of a collection of $l$ trees ${t_1, \dots, t_l}$. This forest can be seen as a function, and noted as:
A random forest $F_{t_1, \dots, t_l}$ \todo[inline]{confusion possible notation: majuscule non gras: fonction ou matrice? les deux, à éclaircir} is a classifier made of a collection of $l$ trees ${t_1, \dots, t_l}$. This forest can be seen as a function, and noted as:
%
$$\begin{array}{ccccc}
F_{t_1, \dots, t_l} & : & \cal{X} & \to & \cal{Y} \\
& & \textbf{x} & \mapsto & F_{t_1, \dots, t_l}(\textbf{x}) = f(\{t_1, \dots, t_l\} , \textbf{x}) \\
\end{array}$$
%
where $f$ is a function which depend on the task. In a regression setup, where ${\cal Y} = \mathbb{R}$, this function can be defined as:
where $f$ is a function which depend on the task\todo{f unclear: why to introduce it?}. In a regression setup, where ${\cal Y} = \mathbb{R}$\todo{I don't think it is usefull}, this function can be defined as:
%
$$f(\{t_1, \dots, t_l \} , \textbf{x}) = \sum_{i = 1}^{l} \alpha_i t_i(x) \ \text{ where } \alpha_i \in \mathbb{R},$$
%
......@@ -48,7 +51,7 @@ while in a classification setup, in which ${\cal Y} = \{ c_1, \dots, c_m \}$, $f
%
$$f(\{t_1, \dots, t_l \} , \textbf{x}) = \argmax_{c \in {\cal Y}} \sum_{i = 1}^{l} \mathds{1}(t_i(\textbf{x}) = c).$$
%
We will need to define the vector prediction of a forest for all the data matrix: $F_{t_1, \dots, t_l}(X) = \begin{pmatrix}
\todo{$\mathds{1}$ not defined}We \todo{no we}will need to define the vector prediction of a forest for all the data matrix: $F_{t_1, \dots, t_l}(X) = \begin{pmatrix}
F_{t_1, \dots, t_l}(x_1) \\
\dots \\
F_{t_1, \dots, t_l}(x_n)
......@@ -57,7 +60,10 @@ We will need to define the vector prediction of a forest for all the data matrix
%
%
All these notations can be summarized in the following table:\\
\begin{tabular}{l c}%\caption{Notation table}
\begin{table}
\begin{tabular}{l c}
%\hline
\textbf{x} & the vector x \\
$k$ & the desired (pruned) forest size \\
......@@ -71,13 +77,18 @@ All these notations can be summarized in the following table:\\
$F_{t_1, \dots, t_l}(\textbf{x}) \in {\cal Y}$ & the predicted label of \textbf{x} by the forest $F_{t_1, \dots, t_l}$ \\
$F_{t_1, \dots, t_l}(X) \in {\cal Y}^n$ & the predicted label of all the data of $X$ by the forest $F_{t_1, \dots, t_l}$\\
%\hline
\end{tabular}
\end{tabular}
\caption{Notations}
\end{table}\todo[inline]{ajouter les codifications des notations: bold minuscule: vecteur; non-bold majuscule: matrix, etc..}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Orthogonal Matching Pursuit (OMP)}
Given a matrix $D = [d_1, \dots , d_l] \in \mathbb{R}^n\times l$ (also called a dictionary) and a signal $\textbf{y}\in \mathbb{R}^n$, finding a $k$-sparse vector $\textbf{w} \in \mathbb{R}^l$ (i.e. $|| \textbf{w} ||_0 \leq k$) that minimize $|| X\textbf{w} - \textbf{y}||$ is an NP-hard problem (ref).
Given a matrix $D = [d_1, \dots , d_l] \in \mathbb{R}^{n\times l}$ \todo{l undefined in text} (also called a dictionary) and a signal $\textbf{y}\in \mathbb{R}^n$, finding a $k$-sparse vector $\textbf{w} \in \mathbb{R}^l$ (i.e. $|| \textbf{w} ||_0 \leq k$) that minimize $|| X\textbf{w} - \textbf{y}||$ is an NP-hard problem \todo{(ref np-hardness)}.
The Orthogonal Matching Pursuit (OMP) algorithm is a greedy algorithm that aim to give an approximate solution of this problem.
The approximation of $\textbf{y}$ is build one term at a time. Noting $\textbf{y}_k$ the current
approximation and $r_k = \textbf{y}_k - \textbf{y}_k$ the so-called residual, we select at each time step the atom (i.e. the column of $X$) which has the largest inner product with $r_k$, and update the approximation.
The approximation of $\textbf{y}$ is built one term at a time. Noting $\textbf{y}_k$ the current
approximation and $r_k = \textbf{y} - \textbf{y}_k$ the so-called residual, we select at each time step the atom (i.e. the column of $X$) which has the largest inner product with $r_k$, and update the approximation.
This step is repeated until a satisfactory approximation. This can be summarized in Algorithm \ref{algo: OMP}
......@@ -93,19 +104,20 @@ This step is repeated until a satisfactory approximation. This can be summarized
\ForEach {$k \in \{0, \dots, K\}$}
\State $d^* \gets \underset{d \in \{d_1, \dots, d_l\}}{\text{argmax}} \ |<d, r_k>|$
\State $\lambda \gets \lambda \cup \{d^*\}$
\State $w_{k+1} \gets \underset{\substack{\alpha \in \mathbb{R}^n \text{ s.c. } \\ D\alpha \ \in \ \text{span}(\lambda)}}{\text{argmin}} \ ||\textbf{y} - D\alpha||^2_2$
\State $r_{k + 1} \rightarrow \textbf{y} - D_{w_{k+1}}$
\State $\rvw_{k+1} \gets \underset{\substack{\rvalpha \in \mathbb{R}^n \text{ s.t. } \\ D\rvalpha \ \in \ \text{span}(\lambda)}}{\text{argmin}} \ ||\textbf{y} - D\rvalpha||^2_2$
\State $r_{k + 1} \rightarrow \textbf{y} - D \rvw_{k+1}$
\EndFor
\end{algorithmic}
\end{algorithm}
%
In general, the OMP algorithm can be seen as a algorithm that 'summarize' the most useful column of the dictionary for expressing the signal \textbf{y}.
In general, the OMP algorithm can be seen as an algorithm that 'summarize' \todo{summarize: unclear}the most useful column of the dictionary for expressing the signal \textbf{y}.
In this paper, we use this algorithm to reduce the forest's size by selecting the most informative trees in the forest (see Section \ref{sec: forest pruning} for more details).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Related Work}
\todo[inline]{réduire: sélectionner l'essentiel, regrouper en catégorie d'algorithmes et se positionner (forward vs backward selection, etc.)}
\begin{itemize}
\item \cite{Yang2012}: once the forest $(F = t_1, \dots, t_n)$ is built, he gives each tree a score (which measures the importance of the tree in the forest). The tree with the lowest score is removed from the forest. To eliminate the next tree, all the scores are recomputed, and the tree with the lowest score is removed...\\
They present in this paper 4 different tree's score. For each tree $t_i$, we compute:
\item \cite{Yang2012}: once the forest $(F = t_1, \dots, t_n)$\todo{inconsistent notations} is built, he \todo{who?} gives each tree a score (which measures the importance of the tree in the forest). The tree with the lowest score is removed from the forest. To eliminate the next tree, all the scores are recomputed, and the tree with the lowest score is removed... \todo{and so forth and so on}\\
They present in this paper 4 \todo{numbers} different tree's score. For each tree $t_i$, we compute:
\begin{itemize}
\item $score_1 = mean_{(x,y) \in train}( margin((x,y), F) - margin((x,y),F\backslash t_i))$
\item $score_2 = min_{(x,y) \in train}( margin((x,y), F) - min_{(x,y) \in train} (margin(F\backslash t_i)))$
......@@ -135,7 +147,7 @@ For the experiments, they use breast cancer prognosis. They reduce the size of a
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Forest pruning}\label{sec: forest pruning}
In this section, we will describe our method for pruning the forest, and thus reduce its size. \\
Consider a forest $F_{t_1, \dots, t_l}$ of $l = 100$ trees, trained using the training data set, witch consist of the 60\% of the data. For every $i \in \{ 1, \dots , l\}$, we note the vector of prediction of the tree $t_i$ in all the $n$ data by:
Consider a forest $F_{t_1, \dots, t_l}$ of $l = 100$ \todo{trop concret}trees, trained using the training data set, witch consist of the 60\% of the data. For every $i \in \{ 1, \dots , l\}$, we note the vector of prediction of the tree $t_i$ in all the $n$ data by:
$$\textbf{y}_i = \begin{pmatrix}
t_1(\textbf{x}_1) \\
\dots \\
......@@ -143,7 +155,7 @@ $$\textbf{y}_i = \begin{pmatrix}
\end{pmatrix},$$
and the matrix of all the forest prediction in all the data by:
$$Y = [\textbf{y}_1 , \dots , \textbf{y}_l ] \in \mathbb{R}^{n \times l}.$$
We apply the OMP algorithm to the $Y$ matrix and to the reals labels vector $\textbf{y}$. Thus, we will look for the $k$ most informative trees to predict the true labels.
We apply the OMP algorithm to the $Y$ matrix and to the reals labels vector $\textbf{y}$. Thus, we will look for the $k$ most informative trees to predict the true labels. \todo{détailler plus comment est utilisé OMP}\todo{ajouter l'algorithme}
\section{Reference}
......
%%%%% NEW MATH DEFINITIONS %%%%%
\usepackage{amsmath,amsfonts,bm}
% Mark sections of captions for referring to divisions of figures
\newcommand{\figleft}{{\em (Left)}}
\newcommand{\figcenter}{{\em (Center)}}
\newcommand{\figright}{{\em (Right)}}
\newcommand{\figtop}{{\em (Top)}}
\newcommand{\figbottom}{{\em (Bottom)}}
\newcommand{\captiona}{{\em (a)}}
\newcommand{\captionb}{{\em (b)}}
\newcommand{\captionc}{{\em (c)}}
\newcommand{\captiond}{{\em (d)}}
% Highlight a newly defined term
\newcommand{\newterm}[1]{{\bf #1}}
% Figure reference, lower-case.
\def\figref#1{figure~\ref{#1}}
% Figure reference, capital. For start of sentence
\def\Figref#1{Figure~\ref{#1}}
\def\twofigref#1#2{figures \ref{#1} and \ref{#2}}
\def\quadfigref#1#2#3#4{figures \ref{#1}, \ref{#2}, \ref{#3} and \ref{#4}}
% Section reference, lower-case.
\def\secref#1{section~\ref{#1}}
% Section reference, capital.
\def\Secref#1{Section~\ref{#1}}
% Reference to two sections.
\def\twosecrefs#1#2{sections \ref{#1} and \ref{#2}}
% Reference to three sections.
\def\secrefs#1#2#3{sections \ref{#1}, \ref{#2} and \ref{#3}}
% Reference to an equation, lower-case.
%\def\eqref#1{equation~\ref{#1}}
% Reference to an equation, upper case
%\def\Eqref#1{Equation~\ref{#1}}
% A raw reference to an equation---avoid using if possible
\def\plaineqref#1{\ref{#1}}
% Reference to a chapter, lower-case.
\def\chapref#1{chapter~\ref{#1}}
% Reference to an equation, upper case.
\def\Chapref#1{Chapter~\ref{#1}}
% Reference to a range of chapters
\def\rangechapref#1#2{chapters\ref{#1}--\ref{#2}}
% Reference to an algorithm, lower-case.
\def\algref#1{algorithm~\ref{#1}}
% Reference to an algorithm, upper case.
\def\Algref#1{Algorithm~\ref{#1}}
\def\twoalgref#1#2{algorithms \ref{#1} and \ref{#2}}
\def\Twoalgref#1#2{Algorithms \ref{#1} and \ref{#2}}
% Reference to a part, lower case
\def\partref#1{part~\ref{#1}}
% Reference to a part, upper case
\def\Partref#1{Part~\ref{#1}}
\def\twopartref#1#2{parts \ref{#1} and \ref{#2}}
\def\ceil#1{\lceil #1 \rceil}
\def\floor#1{\lfloor #1 \rfloor}
\def\1{\bm{1}}
\newcommand{\train}{\mathcal{D}}
\newcommand{\valid}{\mathcal{D_{\mathrm{valid}}}}
\newcommand{\test}{\mathcal{D_{\mathrm{test}}}}
\def\eps{{\epsilon}}
% Random variables
\def\reta{{\textnormal{$\eta$}}}
\def\ra{{\textnormal{a}}}
\def\rb{{\textnormal{b}}}
\def\rc{{\textnormal{c}}}
\def\rd{{\textnormal{d}}}
\def\re{{\textnormal{e}}}
\def\rf{{\textnormal{f}}}
\def\rg{{\textnormal{g}}}
\def\rh{{\textnormal{h}}}
\def\ri{{\textnormal{i}}}
\def\rj{{\textnormal{j}}}
\def\rk{{\textnormal{k}}}
\def\rl{{\textnormal{l}}}
% rm is already a command, just don't name any random variables m
\def\rn{{\textnormal{n}}}
\def\ro{{\textnormal{o}}}
\def\rp{{\textnormal{p}}}
\def\rq{{\textnormal{q}}}
\def\rr{{\textnormal{r}}}
\def\rs{{\textnormal{s}}}
\def\rt{{\textnormal{t}}}
\def\ru{{\textnormal{u}}}
\def\rv{{\textnormal{v}}}
\def\rw{{\textnormal{w}}}
\def\rx{{\textnormal{x}}}
\def\ry{{\textnormal{y}}}
\def\rz{{\textnormal{z}}}
% Random vectors
\def\rvepsilon{{\mathbf{\epsilon}}}
\def\rvtheta{{\mathbf{\theta}}}
\def\rva{{\mathbf{a}}}
\def\rvb{{\mathbf{b}}}
\def\rvc{{\mathbf{c}}}
\def\rvd{{\mathbf{d}}}
\def\rve{{\mathbf{e}}}
\def\rvf{{\mathbf{f}}}
\def\rvg{{\mathbf{g}}}
\def\rvh{{\mathbf{h}}}
\def\rvu{{\mathbf{i}}}
\def\rvj{{\mathbf{j}}}
\def\rvk{{\mathbf{k}}}
\def\rvl{{\mathbf{l}}}
\def\rvm{{\mathbf{m}}}
\def\rvn{{\mathbf{n}}}
\def\rvo{{\mathbf{o}}}
\def\rvp{{\mathbf{p}}}
\def\rvq{{\mathbf{q}}}
\def\rvr{{\mathbf{r}}}
\def\rvs{{\mathbf{s}}}
\def\rvt{{\mathbf{t}}}
\def\rvu{{\mathbf{u}}}
\def\rvv{{\mathbf{v}}}
\def\rvw{{\mathbf{w}}}
\def\rvx{{\mathbf{x}}}
\def\rvy{{\mathbf{y}}}
\def\rvz{{\mathbf{z}}}
% Elements of random vectors
\def\erva{{\textnormal{a}}}
\def\ervb{{\textnormal{b}}}
\def\ervc{{\textnormal{c}}}
\def\ervd{{\textnormal{d}}}
\def\erve{{\textnormal{e}}}
\def\ervf{{\textnormal{f}}}
\def\ervg{{\textnormal{g}}}
\def\ervh{{\textnormal{h}}}
\def\ervi{{\textnormal{i}}}
\def\ervj{{\textnormal{j}}}
\def\ervk{{\textnormal{k}}}
\def\ervl{{\textnormal{l}}}
\def\ervm{{\textnormal{m}}}
\def\ervn{{\textnormal{n}}}
\def\ervo{{\textnormal{o}}}
\def\ervp{{\textnormal{p}}}
\def\ervq{{\textnormal{q}}}
\def\ervr{{\textnormal{r}}}
\def\ervs{{\textnormal{s}}}
\def\ervt{{\textnormal{t}}}
\def\ervu{{\textnormal{u}}}
\def\ervv{{\textnormal{v}}}
\def\ervw{{\textnormal{w}}}
\def\ervx{{\textnormal{x}}}
\def\ervy{{\textnormal{y}}}
\def\ervz{{\textnormal{z}}}
% Random matrices
\def\rmA{{\mathbf{A}}}
\def\rmB{{\mathbf{B}}}
\def\rmC{{\mathbf{C}}}
\def\rmD{{\mathbf{D}}}
\def\rmE{{\mathbf{E}}}
\def\rmF{{\mathbf{F}}}
\def\rmG{{\mathbf{G}}}
\def\rmH{{\mathbf{H}}}
\def\rmI{{\mathbf{I}}}
\def\rmJ{{\mathbf{J}}}
\def\rmK{{\mathbf{K}}}
\def\rmL{{\mathbf{L}}}
\def\rmM{{\mathbf{M}}}
\def\rmN{{\mathbf{N}}}
\def\rmO{{\mathbf{O}}}
\def\rmP{{\mathbf{P}}}
\def\rmQ{{\mathbf{Q}}}
\def\rmR{{\mathbf{R}}}
\def\rmS{{\mathbf{S}}}
\def\rmT{{\mathbf{T}}}
\def\rmU{{\mathbf{U}}}
\def\rmV{{\mathbf{V}}}
\def\rmW{{\mathbf{W}}}
\def\rmX{{\mathbf{X}}}
\def\rmY{{\mathbf{Y}}}
\def\rmZ{{\mathbf{Z}}}
% Elements of random matrices
\def\ermA{{\textnormal{A}}}
\def\ermB{{\textnormal{B}}}
\def\ermC{{\textnormal{C}}}
\def\ermD{{\textnormal{D}}}
\def\ermE{{\textnormal{E}}}
\def\ermF{{\textnormal{F}}}
\def\ermG{{\textnormal{G}}}
\def\ermH{{\textnormal{H}}}
\def\ermI{{\textnormal{I}}}
\def\ermJ{{\textnormal{J}}}
\def\ermK{{\textnormal{K}}}
\def\ermL{{\textnormal{L}}}
\def\ermM{{\textnormal{M}}}
\def\ermN{{\textnormal{N}}}
\def\ermO{{\textnormal{O}}}
\def\ermP{{\textnormal{P}}}
\def\ermQ{{\textnormal{Q}}}
\def\ermR{{\textnormal{R}}}
\def\ermS{{\textnormal{S}}}
\def\ermT{{\textnormal{T}}}
\def\ermU{{\textnormal{U}}}
\def\ermV{{\textnormal{V}}}
\def\ermW{{\textnormal{W}}}
\def\ermX{{\textnormal{X}}}
\def\ermY{{\textnormal{Y}}}
\def\ermZ{{\textnormal{Z}}}
% Vectors
\def\vzero{{\bm{0}}}
\def\vone{{\bm{1}}}
\def\vmu{{\bm{\mu}}}
\def\vtheta{{\bm{\theta}}}
\def\va{{\bm{a}}}
\def\vb{{\bm{b}}}
\def\vc{{\bm{c}}}
\def\vd{{\bm{d}}}
\def\ve{{\bm{e}}}
\def\vf{{\bm{f}}}
\def\vg{{\bm{g}}}
\def\vh{{\bm{h}}}
\def\vi{{\bm{i}}}
\def\vj{{\bm{j}}}
\def\vk{{\bm{k}}}
\def\vl{{\bm{l}}}
\def\vm{{\bm{m}}}
\def\vn{{\bm{n}}}
\def\vo{{\bm{o}}}
\def\vp{{\bm{p}}}
\def\vq{{\bm{q}}}
\def\vr{{\bm{r}}}
\def\vs{{\bm{s}}}
\def\vt{{\bm{t}}}
\def\vu{{\bm{u}}}
\def\vv{{\bm{v}}}
\def\vw{{\bm{w}}}
\def\vx{{\bm{x}}}
\def\vy{{\bm{y}}}
\def\vz{{\bm{z}}}
% Elements of vectors
\def\evalpha{{\alpha}}
\def\evbeta{{\beta}}
\def\evepsilon{{\epsilon}}
\def\evlambda{{\lambda}}
\def\evomega{{\omega}}
\def\evmu{{\mu}}
\def\evpsi{{\psi}}
\def\evsigma{{\sigma}}
\def\evtheta{{\theta}}
\def\eva{{a}}
\def\evb{{b}}
\def\evc{{c}}
\def\evd{{d}}
\def\eve{{e}}
\def\evf{{f}}
\def\evg{{g}}
\def\evh{{h}}
\def\evi{{i}}
\def\evj{{j}}
\def\evk{{k}}
\def\evl{{l}}
\def\evm{{m}}
\def\evn{{n}}
\def\evo{{o}}
\def\evp{{p}}
\def\evq{{q}}
\def\evr{{r}}
\def\evs{{s}}
\def\evt{{t}}
\def\evu{{u}}
\def\evv{{v}}
\def\evw{{w}}
\def\evx{{x}}
\def\evy{{y}}
\def\evz{{z}}
% Matrix
\def\mA{{\bm{A}}}
\def\mB{{\bm{B}}}
\def\mC{{\bm{C}}}
\def\mD{{\bm{D}}}
\def\mE{{\bm{E}}}
\def\mF{{\bm{F}}}
\def\mG{{\bm{G}}}
\def\mH{{\bm{H}}}
\def\mI{{\bm{I}}}
\def\mJ{{\bm{J}}}
\def\mK{{\bm{K}}}
\def\mL{{\bm{L}}}
\def\mM{{\bm{M}}}
\def\mN{{\bm{N}}}
\def\mO{{\bm{O}}}
\def\mP{{\bm{P}}}
\def\mQ{{\bm{Q}}}
\def\mR{{\bm{R}}}
\def\mS{{\bm{S}}}
\def\mT{{\bm{T}}}
\def\mU{{\bm{U}}}
\def\mV{{\bm{V}}}
\def\mW{{\bm{W}}}
\def\mX{{\bm{X}}}
\def\mY{{\bm{Y}}}
\def\mZ{{\bm{Z}}}
\def\mBeta{{\bm{\beta}}}
\def\mPhi{{\bm{\Phi}}}
\def\mLambda{{\bm{\Lambda}}}
\def\mSigma{{\bm{\Sigma}}}
% Tensor
\DeclareMathAlphabet{\mathsfit}{\encodingdefault}{\sfdefault}{m}{sl}
\SetMathAlphabet{\mathsfit}{bold}{\encodingdefault}{\sfdefault}{bx}{n}
\newcommand{\tens}[1]{\bm{\mathsfit{#1}}}
\def\tA{{\tens{A}}}
\def\tB{{\tens{B}}}
\def\tC{{\tens{C}}}
\def\tD{{\tens{D}}}
\def\tE{{\tens{E}}}
\def\tF{{\tens{F}}}
\def\tG{{\tens{G}}}
\def\tH{{\tens{H}}}
\def\tI{{\tens{I}}}
\def\tJ{{\tens{J}}}
\def\tK{{\tens{K}}}
\def\tL{{\tens{L}}}
\def\tM{{\tens{M}}}
\def\tN{{\tens{N}}}
\def\tO{{\tens{O}}}
\def\tP{{\tens{P}}}
\def\tQ{{\tens{Q}}}
\def\tR{{\tens{R}}}
\def\tS{{\tens{S}}}
\def\tT{{\tens{T}}}
\def\tU{{\tens{U}}}
\def\tV{{\tens{V}}}
\def\tW{{\tens{W}}}
\def\tX{{\tens{X}}}
\def\tY{{\tens{Y}}}
\def\tZ{{\tens{Z}}}
% Graph
\def\gA{{\mathcal{A}}}
\def\gB{{\mathcal{B}}}
\def\gC{{\mathcal{C}}}
\def\gD{{\mathcal{D}}}
\def\gE{{\mathcal{E}}}
\def\gF{{\mathcal{F}}}
\def\gG{{\mathcal{G}}}
\def\gH{{\mathcal{H}}}
\def\gI{{\mathcal{I}}}
\def\gJ{{\mathcal{J}}}
\def\gK{{\mathcal{K}}}
\def\gL{{\mathcal{L}}}
\def\gM{{\mathcal{M}}}
\def\gN{{\mathcal{N}}}
\def\gO{{\mathcal{O}}}
\def\gP{{\mathcal{P}}}
\def\gQ{{\mathcal{Q}}}
\def\gR{{\mathcal{R}}}
\def\gS{{\mathcal{S}}}
\def\gT{{\mathcal{T}}}
\def\gU{{\mathcal{U}}}
\def\gV{{\mathcal{V}}}
\def\gW{{\mathcal{W}}}
\def\gX{{\mathcal{X}}}
\def\gY{{\mathcal{Y}}}
\def\gZ{{\mathcal{Z}}}
% Sets
\def\sA{{\mathbb{A}}}
\def\sB{{\mathbb{B}}}
\def\sC{{\mathbb{C}}}
\def\sD{{\mathbb{D}}}
% Don't use a set called E, because this would be the same as our symbol
% for expectation.
\def\sF{{\mathbb{F}}}
\def\sG{{\mathbb{G}}}
\def\sH{{\mathbb{H}}}
\def\sI{{\mathbb{I}}}
\def\sJ{{\mathbb{J}}}
\def\sK{{\mathbb{K}}}
\def\sL{{\mathbb{L}}}
\def\sM{{\mathbb{M}}}
\def\sN{{\mathbb{N}}}
\def\sO{{\mathbb{O}}}
\def\sP{{\mathbb{P}}}
\def\sQ{{\mathbb{Q}}}
\def\sR{{\mathbb{R}}}
\def\sS{{\mathbb{S}}}
\def\sT{{\mathbb{T}}}
\def\sU{{\mathbb{U}}}
\def\sV{{\mathbb{V}}}
\def\sW{{\mathbb{W}}}
\def\sX{{\mathbb{X}}}
\def\sY{{\mathbb{Y}}}
\def\sZ{{\mathbb{Z}}}
% Entries of a matrix
\def\emLambda{{\Lambda}}
\def\emA{{A}}
\def\emB{{B}}
\def\emC{{C}}
\def\emD{{D}}
\def\emE{{E}}
\def\emF{{F}}
\def\emG{{G}}
\def\emH{{H}}
\def\emI{{I}}
\def\emJ{{J}}
\def\emK{{K}}
\def\emL{{L}}
\def\emM{{M}}
\def\emN{{N}}
\def\emO{{O}}
\def\emP{{P}}
\def\emQ{{Q}}
\def\emR{{R}}
\def\emS{{S}}
\def\emT{{T}}
\def\emU{{U}}
\def\emV{{V}}
\def\emW{{W}}
\def\emX{{X}}
\def\emY{{Y}}
\def\emZ{{Z}}
\def\emSigma{{\Sigma}}
% entries of a tensor
% Same font as tensor, without \bm wrapper
\newcommand{\etens}[1]{\mathsfit{#1}}
\def\etLambda{{\etens{\Lambda}}}
\def\etA{{\etens{A}}}
\def\etB{{\etens{B}}}
\def\etC{{\etens{C}}}
\def\etD{{\etens{D}}}
\def\etE{{\etens{E}}}
\def\etF{{\etens{F}}}
\def\etG{{\etens{G}}}
\def\etH{{\etens{H}}}
\def\etI{{\etens{I}}}
\def\etJ{{\etens{J}}}
\def\etK{{\etens{K}}}
\def\etL{{\etens{L}}}
\def\etM{{\etens{M}}}
\def\etN{{\etens{N}}}
\def\etO{{\etens{O}}}
\def\etP{{\etens{P}}}
\def\etQ{{\etens{Q}}}
\def\etR{{\etens{R}}}
\def\etS{{\etens{S}}}
\def\etT{{\etens{T}}}
\def\etU{{\etens{U}}}
\def\etV{{\etens{V}}}
\def\etW{{\etens{W}}}
\def\etX{{\etens{X}}}
\def\etY{{\etens{Y}}}
\def\etZ{{\etens{Z}}}
% The true underlying data generating distribution
\newcommand{\pdata}{p_{\rm{data}}}
% The empirical distribution defined by the training set
\newcommand{\ptrain}{\hat{p}_{\rm{data}}}
\newcommand{\Ptrain}{\hat{P}_{\rm{data}}}
% The model distribution
\newcommand{\pmodel}{p_{\rm{model}}}
\newcommand{\Pmodel}{P_{\rm{model}}}
\newcommand{\ptildemodel}{\tilde{p}_{\rm{model}}}
% Stochastic autoencoder distributions
\newcommand{\pencode}{p_{\rm{encoder}}}
\newcommand{\pdecode}{p_{\rm{decoder}}}
\newcommand{\precons}{p_{\rm{reconstruct}}}
\newcommand{\laplace}{\mathrm{Laplace}} % Laplace distribution
\newcommand{\E}{\mathbb{E}}
\newcommand{\Ls}{\mathcal{L}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\emp}{\tilde{p}}
\newcommand{\lr}{\alpha}
\newcommand{\reg}{\lambda}
\newcommand{\rect}{\mathrm{rectifier}}
\newcommand{\softmax}{\mathrm{softmax}}
\newcommand{\sigmoid}{\sigma}
\newcommand{\softplus}{\zeta}
\newcommand{\KL}{D_{\mathrm{KL}}}
\newcommand{\Var}{\mathrm{Var}}
\newcommand{\standarderror}{\mathrm{SE}}
\newcommand{\Cov}{\mathrm{Cov}}
% Wolfram Mathworld says $L^2$ is for function spaces and $\ell^2$ is for vectors
% But then they seem to use $L^2$ for vectors throughout the site, and so does
% wikipedia.
\newcommand{\normlzero}{L^0}
\newcommand{\normlone}{L^1}
\newcommand{\normltwo}{L^2}
\newcommand{\normlp}{L^p}
\newcommand{\normmax}{L^\infty}
\newcommand{\parents}{Pa} % See usage in notation.tex. Chosen to match Daphne's book.
\DeclareMathOperator{\sign}{sign}
\DeclareMathOperator{\Tr}{Tr}
\let\ab\allowbreak
\newcommand{\intint}[1]{\left \llbracket#1\right \rrbracket}
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