research-paper-writing

1%%%%% NEW MATH DEFINITIONS %%%%%2 3\usepackage{amsmath,amsfonts,bm}4 5% Mark sections of captions for referring to divisions of figures6\newcommand{\figleft}{{\em (Left)}}7\newcommand{\figcenter}{{\em (Center)}}8\newcommand{\figright}{{\em (Right)}}9\newcommand{\figtop}{{\em (Top)}}10\newcommand{\figbottom}{{\em (Bottom)}}11\newcommand{\captiona}{{\em (a)}}12\newcommand{\captionb}{{\em (b)}}13\newcommand{\captionc}{{\em (c)}}14\newcommand{\captiond}{{\em (d)}}15 16% Highlight a newly defined term17\newcommand{\newterm}[1]{{\bf #1}}18 19 20% Figure reference, lower-case.21\def\figref#1{figure~\ref{#1}}22% Figure reference, capital. For start of sentence23\def\Figref#1{Figure~\ref{#1}}24\def\twofigref#1#2{figures \ref{#1} and \ref{#2}}25\def\quadfigref#1#2#3#4{figures \ref{#1}, \ref{#2}, \ref{#3} and \ref{#4}}26% Section reference, lower-case.27\def\secref#1{section~\ref{#1}}28% Section reference, capital.29\def\Secref#1{Section~\ref{#1}}30% Reference to two sections.31\def\twosecrefs#1#2{sections \ref{#1} and \ref{#2}}32% Reference to three sections.33\def\secrefs#1#2#3{sections \ref{#1}, \ref{#2} and \ref{#3}}34% Reference to an equation, lower-case.35\def\eqref#1{equation~\ref{#1}}36% Reference to an equation, upper case37\def\Eqref#1{Equation~\ref{#1}}38% A raw reference to an equation---avoid using if possible39\def\plaineqref#1{\ref{#1}}40% Reference to a chapter, lower-case.41\def\chapref#1{chapter~\ref{#1}}42% Reference to an equation, upper case.43\def\Chapref#1{Chapter~\ref{#1}}44% Reference to a range of chapters45\def\rangechapref#1#2{chapters\ref{#1}--\ref{#2}}46% Reference to an algorithm, lower-case.47\def\algref#1{algorithm~\ref{#1}}48% Reference to an algorithm, upper case.49\def\Algref#1{Algorithm~\ref{#1}}50\def\twoalgref#1#2{algorithms \ref{#1} and \ref{#2}}51\def\Twoalgref#1#2{Algorithms \ref{#1} and \ref{#2}}52% Reference to a part, lower case53\def\partref#1{part~\ref{#1}}54% Reference to a part, upper case55\def\Partref#1{Part~\ref{#1}}56\def\twopartref#1#2{parts \ref{#1} and \ref{#2}}57 58\def\ceil#1{\lceil #1 \rceil}59\def\floor#1{\lfloor #1 \rfloor}60\def\1{\bm{1}}61\newcommand{\train}{\mathcal{D}}62\newcommand{\valid}{\mathcal{D_{\mathrm{valid}}}}63\newcommand{\test}{\mathcal{D_{\mathrm{test}}}}64 65\def\eps{{\epsilon}}66 67 68% Random variables69\def\reta{{\textnormal{$\eta$}}}70\def\ra{{\textnormal{a}}}71\def\rb{{\textnormal{b}}}72\def\rc{{\textnormal{c}}}73\def\rd{{\textnormal{d}}}74\def\re{{\textnormal{e}}}75\def\rf{{\textnormal{f}}}76\def\rg{{\textnormal{g}}}77\def\rh{{\textnormal{h}}}78\def\ri{{\textnormal{i}}}79\def\rj{{\textnormal{j}}}80\def\rk{{\textnormal{k}}}81\def\rl{{\textnormal{l}}}82% rm is already a command, just don't name any random variables m83\def\rn{{\textnormal{n}}}84\def\ro{{\textnormal{o}}}85\def\rp{{\textnormal{p}}}86\def\rq{{\textnormal{q}}}87\def\rr{{\textnormal{r}}}88\def\rs{{\textnormal{s}}}89\def\rt{{\textnormal{t}}}90\def\ru{{\textnormal{u}}}91\def\rv{{\textnormal{v}}}92\def\rw{{\textnormal{w}}}93\def\rx{{\textnormal{x}}}94\def\ry{{\textnormal{y}}}95\def\rz{{\textnormal{z}}}96 97% Random vectors98\def\rvepsilon{{\mathbf{\epsilon}}}99\def\rvtheta{{\mathbf{\theta}}}100\def\rva{{\mathbf{a}}}101\def\rvb{{\mathbf{b}}}102\def\rvc{{\mathbf{c}}}103\def\rvd{{\mathbf{d}}}104\def\rve{{\mathbf{e}}}105\def\rvf{{\mathbf{f}}}106\def\rvg{{\mathbf{g}}}107\def\rvh{{\mathbf{h}}}108\def\rvu{{\mathbf{i}}}109\def\rvj{{\mathbf{j}}}110\def\rvk{{\mathbf{k}}}111\def\rvl{{\mathbf{l}}}112\def\rvm{{\mathbf{m}}}113\def\rvn{{\mathbf{n}}}114\def\rvo{{\mathbf{o}}}115\def\rvp{{\mathbf{p}}}116\def\rvq{{\mathbf{q}}}117\def\rvr{{\mathbf{r}}}118\def\rvs{{\mathbf{s}}}119\def\rvt{{\mathbf{t}}}120\def\rvu{{\mathbf{u}}}121\def\rvv{{\mathbf{v}}}122\def\rvw{{\mathbf{w}}}123\def\rvx{{\mathbf{x}}}124\def\rvy{{\mathbf{y}}}125\def\rvz{{\mathbf{z}}}126 127% Elements of random vectors128\def\erva{{\textnormal{a}}}129\def\ervb{{\textnormal{b}}}130\def\ervc{{\textnormal{c}}}131\def\ervd{{\textnormal{d}}}132\def\erve{{\textnormal{e}}}133\def\ervf{{\textnormal{f}}}134\def\ervg{{\textnormal{g}}}135\def\ervh{{\textnormal{h}}}136\def\ervi{{\textnormal{i}}}137\def\ervj{{\textnormal{j}}}138\def\ervk{{\textnormal{k}}}139\def\ervl{{\textnormal{l}}}140\def\ervm{{\textnormal{m}}}141\def\ervn{{\textnormal{n}}}142\def\ervo{{\textnormal{o}}}143\def\ervp{{\textnormal{p}}}144\def\ervq{{\textnormal{q}}}145\def\ervr{{\textnormal{r}}}146\def\ervs{{\textnormal{s}}}147\def\ervt{{\textnormal{t}}}148\def\ervu{{\textnormal{u}}}149\def\ervv{{\textnormal{v}}}150\def\ervw{{\textnormal{w}}}151\def\ervx{{\textnormal{x}}}152\def\ervy{{\textnormal{y}}}153\def\ervz{{\textnormal{z}}}154 155% Random matrices156\def\rmA{{\mathbf{A}}}157\def\rmB{{\mathbf{B}}}158\def\rmC{{\mathbf{C}}}159\def\rmD{{\mathbf{D}}}160\def\rmE{{\mathbf{E}}}161\def\rmF{{\mathbf{F}}}162\def\rmG{{\mathbf{G}}}163\def\rmH{{\mathbf{H}}}164\def\rmI{{\mathbf{I}}}165\def\rmJ{{\mathbf{J}}}166\def\rmK{{\mathbf{K}}}167\def\rmL{{\mathbf{L}}}168\def\rmM{{\mathbf{M}}}169\def\rmN{{\mathbf{N}}}170\def\rmO{{\mathbf{O}}}171\def\rmP{{\mathbf{P}}}172\def\rmQ{{\mathbf{Q}}}173\def\rmR{{\mathbf{R}}}174\def\rmS{{\mathbf{S}}}175\def\rmT{{\mathbf{T}}}176\def\rmU{{\mathbf{U}}}177\def\rmV{{\mathbf{V}}}178\def\rmW{{\mathbf{W}}}179\def\rmX{{\mathbf{X}}}180\def\rmY{{\mathbf{Y}}}181\def\rmZ{{\mathbf{Z}}}182 183% Elements of random matrices184\def\ermA{{\textnormal{A}}}185\def\ermB{{\textnormal{B}}}186\def\ermC{{\textnormal{C}}}187\def\ermD{{\textnormal{D}}}188\def\ermE{{\textnormal{E}}}189\def\ermF{{\textnormal{F}}}190\def\ermG{{\textnormal{G}}}191\def\ermH{{\textnormal{H}}}192\def\ermI{{\textnormal{I}}}193\def\ermJ{{\textnormal{J}}}194\def\ermK{{\textnormal{K}}}195\def\ermL{{\textnormal{L}}}196\def\ermM{{\textnormal{M}}}197\def\ermN{{\textnormal{N}}}198\def\ermO{{\textnormal{O}}}199\def\ermP{{\textnormal{P}}}200\def\ermQ{{\textnormal{Q}}}201\def\ermR{{\textnormal{R}}}202\def\ermS{{\textnormal{S}}}203\def\ermT{{\textnormal{T}}}204\def\ermU{{\textnormal{U}}}205\def\ermV{{\textnormal{V}}}206\def\ermW{{\textnormal{W}}}207\def\ermX{{\textnormal{X}}}208\def\ermY{{\textnormal{Y}}}209\def\ermZ{{\textnormal{Z}}}210 211% Vectors212\def\vzero{{\bm{0}}}213\def\vone{{\bm{1}}}214\def\vmu{{\bm{\mu}}}215\def\vtheta{{\bm{\theta}}}216\def\va{{\bm{a}}}217\def\vb{{\bm{b}}}218\def\vc{{\bm{c}}}219\def\vd{{\bm{d}}}220\def\ve{{\bm{e}}}221\def\vf{{\bm{f}}}222\def\vg{{\bm{g}}}223\def\vh{{\bm{h}}}224\def\vi{{\bm{i}}}225\def\vj{{\bm{j}}}226\def\vk{{\bm{k}}}227\def\vl{{\bm{l}}}228\def\vm{{\bm{m}}}229\def\vn{{\bm{n}}}230\def\vo{{\bm{o}}}231\def\vp{{\bm{p}}}232\def\vq{{\bm{q}}}233\def\vr{{\bm{r}}}234\def\vs{{\bm{s}}}235\def\vt{{\bm{t}}}236\def\vu{{\bm{u}}}237\def\vv{{\bm{v}}}238\def\vw{{\bm{w}}}239\def\vx{{\bm{x}}}240\def\vy{{\bm{y}}}241\def\vz{{\bm{z}}}242 243% Elements of vectors244\def\evalpha{{\alpha}}245\def\evbeta{{\beta}}246\def\evepsilon{{\epsilon}}247\def\evlambda{{\lambda}}248\def\evomega{{\omega}}249\def\evmu{{\mu}}250\def\evpsi{{\psi}}251\def\evsigma{{\sigma}}252\def\evtheta{{\theta}}253\def\eva{{a}}254\def\evb{{b}}255\def\evc{{c}}256\def\evd{{d}}257\def\eve{{e}}258\def\evf{{f}}259\def\evg{{g}}260\def\evh{{h}}261\def\evi{{i}}262\def\evj{{j}}263\def\evk{{k}}264\def\evl{{l}}265\def\evm{{m}}266\def\evn{{n}}267\def\evo{{o}}268\def\evp{{p}}269\def\evq{{q}}270\def\evr{{r}}271\def\evs{{s}}272\def\evt{{t}}273\def\evu{{u}}274\def\evv{{v}}275\def\evw{{w}}276\def\evx{{x}}277\def\evy{{y}}278\def\evz{{z}}279 280% Matrix281\def\mA{{\bm{A}}}282\def\mB{{\bm{B}}}283\def\mC{{\bm{C}}}284\def\mD{{\bm{D}}}285\def\mE{{\bm{E}}}286\def\mF{{\bm{F}}}287\def\mG{{\bm{G}}}288\def\mH{{\bm{H}}}289\def\mI{{\bm{I}}}290\def\mJ{{\bm{J}}}291\def\mK{{\bm{K}}}292\def\mL{{\bm{L}}}293\def\mM{{\bm{M}}}294\def\mN{{\bm{N}}}295\def\mO{{\bm{O}}}296\def\mP{{\bm{P}}}297\def\mQ{{\bm{Q}}}298\def\mR{{\bm{R}}}299\def\mS{{\bm{S}}}300\def\mT{{\bm{T}}}301\def\mU{{\bm{U}}}302\def\mV{{\bm{V}}}303\def\mW{{\bm{W}}}304\def\mX{{\bm{X}}}305\def\mY{{\bm{Y}}}306\def\mZ{{\bm{Z}}}307\def\mBeta{{\bm{\beta}}}308\def\mPhi{{\bm{\Phi}}}309\def\mLambda{{\bm{\Lambda}}}310\def\mSigma{{\bm{\Sigma}}}311 312% Tensor313\DeclareMathAlphabet{\mathsfit}{\encodingdefault}{\sfdefault}{m}{sl}314\SetMathAlphabet{\mathsfit}{bold}{\encodingdefault}{\sfdefault}{bx}{n}315\newcommand{\tens}[1]{\bm{\mathsfit{#1}}}316\def\tA{{\tens{A}}}317\def\tB{{\tens{B}}}318\def\tC{{\tens{C}}}319\def\tD{{\tens{D}}}320\def\tE{{\tens{E}}}321\def\tF{{\tens{F}}}322\def\tG{{\tens{G}}}323\def\tH{{\tens{H}}}324\def\tI{{\tens{I}}}325\def\tJ{{\tens{J}}}326\def\tK{{\tens{K}}}327\def\tL{{\tens{L}}}328\def\tM{{\tens{M}}}329\def\tN{{\tens{N}}}330\def\tO{{\tens{O}}}331\def\tP{{\tens{P}}}332\def\tQ{{\tens{Q}}}333\def\tR{{\tens{R}}}334\def\tS{{\tens{S}}}335\def\tT{{\tens{T}}}336\def\tU{{\tens{U}}}337\def\tV{{\tens{V}}}338\def\tW{{\tens{W}}}339\def\tX{{\tens{X}}}340\def\tY{{\tens{Y}}}341\def\tZ{{\tens{Z}}}342 343 344% Graph345\def\gA{{\mathcal{A}}}346\def\gB{{\mathcal{B}}}347\def\gC{{\mathcal{C}}}348\def\gD{{\mathcal{D}}}349\def\gE{{\mathcal{E}}}350\def\gF{{\mathcal{F}}}351\def\gG{{\mathcal{G}}}352\def\gH{{\mathcal{H}}}353\def\gI{{\mathcal{I}}}354\def\gJ{{\mathcal{J}}}355\def\gK{{\mathcal{K}}}356\def\gL{{\mathcal{L}}}357\def\gM{{\mathcal{M}}}358\def\gN{{\mathcal{N}}}359\def\gO{{\mathcal{O}}}360\def\gP{{\mathcal{P}}}361\def\gQ{{\mathcal{Q}}}362\def\gR{{\mathcal{R}}}363\def\gS{{\mathcal{S}}}364\def\gT{{\mathcal{T}}}365\def\gU{{\mathcal{U}}}366\def\gV{{\mathcal{V}}}367\def\gW{{\mathcal{W}}}368\def\gX{{\mathcal{X}}}369\def\gY{{\mathcal{Y}}}370\def\gZ{{\mathcal{Z}}}371 372% Sets373\def\sA{{\mathbb{A}}}374\def\sB{{\mathbb{B}}}375\def\sC{{\mathbb{C}}}376\def\sD{{\mathbb{D}}}377% Don't use a set called E, because this would be the same as our symbol378% for expectation.379\def\sF{{\mathbb{F}}}380\def\sG{{\mathbb{G}}}381\def\sH{{\mathbb{H}}}382\def\sI{{\mathbb{I}}}383\def\sJ{{\mathbb{J}}}384\def\sK{{\mathbb{K}}}385\def\sL{{\mathbb{L}}}386\def\sM{{\mathbb{M}}}387\def\sN{{\mathbb{N}}}388\def\sO{{\mathbb{O}}}389\def\sP{{\mathbb{P}}}390\def\sQ{{\mathbb{Q}}}391\def\sR{{\mathbb{R}}}392\def\sS{{\mathbb{S}}}393\def\sT{{\mathbb{T}}}394\def\sU{{\mathbb{U}}}395\def\sV{{\mathbb{V}}}396\def\sW{{\mathbb{W}}}397\def\sX{{\mathbb{X}}}398\def\sY{{\mathbb{Y}}}399\def\sZ{{\mathbb{Z}}}400 401% Entries of a matrix402\def\emLambda{{\Lambda}}403\def\emA{{A}}404\def\emB{{B}}405\def\emC{{C}}406\def\emD{{D}}407\def\emE{{E}}408\def\emF{{F}}409\def\emG{{G}}410\def\emH{{H}}411\def\emI{{I}}412\def\emJ{{J}}413\def\emK{{K}}414\def\emL{{L}}415\def\emM{{M}}416\def\emN{{N}}417\def\emO{{O}}418\def\emP{{P}}419\def\emQ{{Q}}420\def\emR{{R}}421\def\emS{{S}}422\def\emT{{T}}423\def\emU{{U}}424\def\emV{{V}}425\def\emW{{W}}426\def\emX{{X}}427\def\emY{{Y}}428\def\emZ{{Z}}429\def\emSigma{{\Sigma}}430 431% entries of a tensor432% Same font as tensor, without \bm wrapper433\newcommand{\etens}[1]{\mathsfit{#1}}434\def\etLambda{{\etens{\Lambda}}}435\def\etA{{\etens{A}}}436\def\etB{{\etens{B}}}437\def\etC{{\etens{C}}}438\def\etD{{\etens{D}}}439\def\etE{{\etens{E}}}440\def\etF{{\etens{F}}}441\def\etG{{\etens{G}}}442\def\etH{{\etens{H}}}443\def\etI{{\etens{I}}}444\def\etJ{{\etens{J}}}445\def\etK{{\etens{K}}}446\def\etL{{\etens{L}}}447\def\etM{{\etens{M}}}448\def\etN{{\etens{N}}}449\def\etO{{\etens{O}}}450\def\etP{{\etens{P}}}451\def\etQ{{\etens{Q}}}452\def\etR{{\etens{R}}}453\def\etS{{\etens{S}}}454\def\etT{{\etens{T}}}455\def\etU{{\etens{U}}}456\def\etV{{\etens{V}}}457\def\etW{{\etens{W}}}458\def\etX{{\etens{X}}}459\def\etY{{\etens{Y}}}460\def\etZ{{\etens{Z}}}461 462% The true underlying data generating distribution463\newcommand{\pdata}{p_{\rm{data}}}464% The empirical distribution defined by the training set465\newcommand{\ptrain}{\hat{p}_{\rm{data}}}466\newcommand{\Ptrain}{\hat{P}_{\rm{data}}}467% The model distribution468\newcommand{\pmodel}{p_{\rm{model}}}469\newcommand{\Pmodel}{P_{\rm{model}}}470\newcommand{\ptildemodel}{\tilde{p}_{\rm{model}}}471% Stochastic autoencoder distributions472\newcommand{\pencode}{p_{\rm{encoder}}}473\newcommand{\pdecode}{p_{\rm{decoder}}}474\newcommand{\precons}{p_{\rm{reconstruct}}}475 476\newcommand{\laplace}{\mathrm{Laplace}} % Laplace distribution477 478\newcommand{\E}{\mathbb{E}}479\newcommand{\Ls}{\mathcal{L}}480\newcommand{\R}{\mathbb{R}}481\newcommand{\emp}{\tilde{p}}482\newcommand{\lr}{\alpha}483\newcommand{\reg}{\lambda}484\newcommand{\rect}{\mathrm{rectifier}}485\newcommand{\softmax}{\mathrm{softmax}}486\newcommand{\sigmoid}{\sigma}487\newcommand{\softplus}{\zeta}488\newcommand{\KL}{D_{\mathrm{KL}}}489\newcommand{\Var}{\mathrm{Var}}490\newcommand{\standarderror}{\mathrm{SE}}491\newcommand{\Cov}{\mathrm{Cov}}492% Wolfram Mathworld says $L^2$ is for function spaces and $\ell^2$ is for vectors493% But then they seem to use $L^2$ for vectors throughout the site, and so does494% wikipedia.495\newcommand{\normlzero}{L^0}496\newcommand{\normlone}{L^1}497\newcommand{\normltwo}{L^2}498\newcommand{\normlp}{L^p}499\newcommand{\normmax}{L^\infty}500 501\newcommand{\parents}{Pa} % See usage in notation.tex. Chosen to match Daphne's book.502 503\DeclareMathOperator*{\argmax}{arg\,max}504\DeclareMathOperator*{\argmin}{arg\,min}505 506\DeclareMathOperator{\sign}{sign}507\DeclareMathOperator{\Tr}{Tr}508\let\ab\allowbreak509