riccardo/phd-thesis-beamer
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

Add animation of the convolution network

Browse Source 
Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
This commit is contained in:
Riccardo
2020-12-01 09:39:37 +01:00
parent b6f3da5b5b
commit 1714e1aa21
85 changed files with 311 additions and 267 deletions
Download Patch File Download Diff File Expand all files Collapse all files

518
thesis.tex
View File

@@ -12,6 +12,7 @@
\usepackage{upgreek}
\usepackage{physics}
\usepackage{tensor}
\usepackage{animate}
\usepackage{graphicx}
\usepackage{transparent}
\usepackage{tikz}
@@ -26,7 +27,7 @@
\usecolortheme{crane}
\usefonttheme{structurebold}
\setbeamertemplate{navigation symbols}{}
\addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{}
\addtobeamertemplate{background canvas}{\transfade[duration=0.15]}{}
\author[Finotello]{Riccardo Finotello}
\title[D-branes and Deep Learning]{D-branes and Deep Learning}
@@ -58,7 +59,7 @@
\newcommand{\highlight}[1]{\fcolorbox{yellow}{yellow!20}{#1}}
\renewcommand{\cite}[1]{{\tiny{\fcolorbox{red}{red!10}{[#1]}}}}
\newcommand{\firstlogo}{img/unito}
\newcommand{\firstlogo}{img/unito.pdf}
\newcommand{\thefirstlogo}{%
\begin{figure}
\centering
@@ -66,7 +67,7 @@
\end{figure}
}
\newcommand{\secondlogo}{img/infn}
\newcommand{\secondlogo}{img/infn.pdf}
\newcommand{\thesecondlogo}{%
\begin{figure}
\centering
@@ -160,7 +161,7 @@
}
{%
% \setbeamertemplate{footline}{}
\setbeamertemplate{footline}{}
\usebackgroundtemplate{%
\transparent{0.1}
\includegraphics[width=\paperwidth]{img/torino.png}
@@ -203,79 +204,52 @@
\pause
\begin{columns}
\begin{column}[t]{0.5\linewidth}
\highlight{Symmetries:}
\begin{itemize}
\item \textbf{Poincaré transf.}\ $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$
\item \textbf{2D diff.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$
\item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \upgamma_{\upalpha \upbeta}$
\end{itemize}
\begin{columns}[T, totalwidth=0.935\linewidth]
\begin{column}{0.45\linewidth}
\begin{tabular}{@{}ll@{}}
Symmetries: &
\\
\toprule
\textbf{Poincaré transf.}: &
$X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$
\\
\textbf{2D diff.}: &
$\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$
\\
\textbf{Weyl transf.}: &
$\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \upgamma_{\upalpha \upbeta}$
\\
\end{tabular}
\end{column}
\hfill
\begin{column}{0.45\linewidth}
\begin{tabular}{@{}ll@{}}
Conformal symmetry: &
\\
\toprule
\textbf{vanishing} stress-energy tensor: &
$\mathcal{T}_{\upalpha \upbeta} = 0$
\\
\textbf{traceless} stress-energy tensor: &
$\trace{\mathcal{T}} = 0$
\\
\textbf{conformal gauge}: &
$\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$
\\
\end{tabular}
\pause
\pause
\begin{column}[t]{0.5\linewidth}
\highlight{Conformal symmetry:}
\begin{itemize}
\item \textbf{vanishing} stress-energy tensor: $\mathcal{T}_{\upalpha \upbeta} = 0$
\item \textbf{traceless} stress-energy tensor: $\trace{\mathcal{T}} = 0$
\item \textbf{conformal gauge} $\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$
\end{itemize}
\begin{center}
\highlight{Conformal properties fixed by \textbf{OPE}s.}
\end{center}
\end{column}
\end{columns}
\end{frame}
% \begin{frame}{Action Principle and Conformal Symmetry}
% \begin{columns}
% \begin{column}{0.6\linewidth}
% \highlight{%
% Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$:
% }
% \begin{equation*}
% \mathcal{T}( z )\, \Upphi_h( w )
% \stackrel{z \to w}{\sim}
% \frac{h}{(z - w)^2} \Upphi_h( w )
% +
% \frac{1}{z - w} \partial_w \Upphi_h( w )
% \end{equation*}
% \begin{equation*}
% \mathcal{T}( z )\, \mathcal{T}( w )
% \stackrel{z \to w}{\sim}
% \frac{\frac{c}{2}}{(z - w)^4}
% +
% \order{(z - w)^{-2}}
% \end{equation*}
% \begin{equationblock}{Virasoro algebra $\mathscr{V} \oplus \overline{\mathscr{V}}$}
% \begin{eqnarray*}
% \qty[ \overset{\scriptscriptstyle(-)}{L}_n,\, \overset{\scriptscriptstyle(-)}{L}_m ]
% & = &
% (n - m) \overset{\scriptscriptstyle(-)}{L}_{n + m} + \frac{c}{12} n \qty(n^2 - 1) \updelta_{n + m,\, 0}
% \\
% \qty[ L_n,\, \overline{L}_m ]
% & = &
% 0
% \end{eqnarray*}
% \end{equationblock}
% \end{column}
% \begin{column}{0.4\linewidth}
% \begin{figure}[h]
% \centering
% \resizebox{0.9\columnwidth}{!}{\import{img}{complex_plane.pgf}}
% \end{figure}
% \end{column}
% \end{columns}
% \end{frame}
\begin{frame}{Action Principle and Conformal Symmetry}
\highlight{Superstrings in $D$ dimensions:}
Superstrings in $D$ dimensions $\longrightarrow$ \emph{Virasoro algebra} (central extension of de Witt's algebra):
\begin{equation*}
\mathcal{T}( z )
=
@@ -311,7 +285,7 @@
\pause
\highlight{Consequence:}
Consequence:
\begin{equation*}
c_{\text{full}} = c + c_{\text{ghost}} = 0
\quad
@@ -324,152 +298,145 @@
\begin{frame}{Extra Dimensions and Compactification}
\begin{block}{Compactification}
\begin{columns}
\begin{column}{0.7\linewidth}
\begin{columns}[T, totalwidth=0.95\linewidth]
\begin{column}{0.8\linewidth}
\begin{equation*}
\mathscr{M}^{1,\, 9}
=
\mathscr{M}^{1,\, 3} \otimes \mathscr{X}_6
\end{equation*}
\vspace{-1em}
\begin{itemize}
\item $\mathscr{X}_6$ is a \textbf{compact} manifold
\item $N = 1$ \textbf{supersymmetry} is preserved in 4D
\item $N = 1$ \textbf{supersymmetry} preserved in 4D
\item algebra of $\mathrm{SU}(3) \otimes \mathrm{SU}(2) \otimes \mathrm{U}(1)$ in arising \textbf{gauge group}
\end{itemize}
\end{column}
\begin{tikzpicture}[remember picture, overlay]
\node[anchor=base] at (8em,-3.3em) {\cite{code in Hanson (1994)}};
\node[anchor=base] at (3em,-3.75em) {\includegraphics[width=0.3\linewidth]{img/cy}};
\node[anchor=base] at (-2em,-6em) {\cite{code in Hanson (1994)}};
\node[anchor=base] at (-7em,-6em) {\includegraphics[width=0.25\linewidth]{img/cy.png}};
\end{tikzpicture}
\begin{column}{0.3\linewidth}
% \centering
% \includegraphics[width=0.9\columnwidth]{img/cy}
\end{column}
\end{columns}
\end{block}
\pause
\begin{columns}
\begin{column}[t]{0.5\linewidth}
\highlight{Kähler manifolds} $\qty( M,\, g )$ such that
\vfill
\begin{columns}[T, totalwidth=0.95\linewidth]
\begin{column}{0.475\linewidth}
\textbf{Kähler manifolds} $\qty( M,\, g )$ such that:
\begin{itemize}
\item $\dim\limits_{\mathds{C}} M = m$
\item $\mathrm{Hol}( g ) \subseteq \mathrm{SU}( m )$
\item $g$ is Ricci-flat (equiv.\ $c_1\qty( M )$ vanishes)
\item $\mathrm{Ric}( g ) \equiv 0$ (equiv.\ $c_1\qty( M ) \equiv 0$)
\end{itemize}
\end{column}
\hfill
\pause
\begin{column}[t]{0.5\linewidth}
Characterised by \highlight{Hodge numbers}
\begin{column}{0.475\linewidth}
Characterised by \textbf{Hodge numbers}
\begin{equation*}
h^{r,\,s} = \dim\limits_{\mathds{C}} H_{\overline{\partial}}^{r,\,s}\qty( M,\, \mathds{C} )
\end{equation*}
counting the no.\ of harmonic $(r,\,s)$-forms.
counting the no.\ of harmonic $(r,s)$-forms.
\end{column}
\end{columns}
\end{frame}
\begin{frame}{D-branes and Open Strings}
Polyakov's action naturally introduces \highlight{Neumann b.c.:}
Polyakov's action naturally introduces \textbf{Neumann b.c.} for \textbf{open strings}:
\begin{equation*}
\eval{\partial_{\upsigma} X\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
\end{equation*}
satisfied by \textbf{open and closed} strings living in $D$ dimensions s.t.\ $\square X = 0$.
satisfied by \highlight{\textbf{open and closed strings} in $D$ dim.} s.t.\ $\square X = 0 \Rightarrow X( z, \overline{z} ) = X( z ) + \overline{X}( \overline{z} )$.
\pause
\begin{equationblock}{T-duality}
% \begin{equationblock}{Equivalent Theories of Closed String Compactification}
% \begin{equation*}
% X( z, \overline{z} )
% =
% X( z ) + \overline{X}( \overline{z} )
% \quad
% \stackrel{T-dual}{\Rightarrow}
% \quad
% X( z ) - \overline{X}( \overline{z} )
% =
% Y( z, \overline{z} )
% =
% Y( z ) + \overline{Y}( \overline{z} )
% \end{equation*}
% \end{equationblock}
% \pause
\begin{block}{T-duality}
\textbf{Dirichlet b.c.} consequence of \textbf{T-duality} on $p$ directions:
\begin{equation*}
X( z, \overline{z} )
=
X( z ) + \overline{X}( \overline{z} )
\overline{X}( z ) \mapsto - \overline{X}( z )
\quad
\stackrel{T}{\Rightarrow}
\Rightarrow
\quad
X( z ) - \overline{X}( \overline{z} )
=
Y( z, \overline{z} )
=
Y( z ) + \overline{Y}( \overline{z} )
\eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
\quad
\stackrel{T-duality}{\longrightarrow}
\quad
\eval{\partial_{\uptau} \widetilde{X}^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
\end{equation*}
\end{equationblock}
\pause
Resulting effect (repeated $p \le D - 1$ times) leads to \highlight{Dirichlet b.c.:}
\begin{equation*}
\eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
\quad
\stackrel{T}{\Rightarrow}
\quad
\eval{\partial_{\uptau} Y^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
\quad
\forall i = 1, 2,\, \dots,\, p
\end{equation*}
thus \textbf{open strings} can be \textbf{constrained} to \highlight{$D(D - p - 1)$-branes.}
\hfill
\cite{Polchinski (1995, 1996)}
thus \textbf{open strings} can be \textbf{constrained} to $D(D - p - 1)$-branes.
\hfill
\cite{Polchinski (1995, 1996)}
\end{block}
\end{frame}
\begin{frame}{D-branes and Open Strings}
Introducing $Dp$-branes breaks \highlight{$\mathrm{ISO}(1,\, D-1) \rightarrow \mathrm{ISO}( 1, p ) \otimes \mathrm{SO}( D - 1 - p )$.}
Introducing $Dp$-branes breaks \highlight{$\mathrm{ISO}(1,\, D-1) \rightarrow \mathrm{ISO}( 1, p ) \otimes \mathrm{SO}( D - 1 - p )$}:
\begin{equation*}
\mathcal{A}^{\upmu} \rightarrow \qty( \mathcal{A}^A,\, \mathcal{A}^a )
\quad
\Rightarrow
\quad
\mathrm{U}( 1 )~\text{theory~in}~p+1~\text{dimensions~(and~scalars)}
\end{equation*}
\pause
\begin{equationblock}{Massless Spectrum (\emph{irrep} of \textbf{little group} $\mathrm{SO}( D - 2 )$)}
\begin{equation*}
\mathcal{A}^{\upmu}
\quad \leftrightarrow \quad
\upalpha_{-1}^{\upmu} \ket{0}
\qquad
\longrightarrow
\qquad
\begin{tabular}{@{}llll@{}}
$\mathcal{A}^A$
&
$\leftrightarrow$
&
$\upalpha_{-1}^A \ket{0},$
&
$A = 0,\, 1,\, \dots,\, p$
\\
$\mathcal{A}^a$
&
$\leftrightarrow$
&
$\upalpha_{-1}^a \ket{0},$
&
$a = 1,\, 2,\, \dots,\, D - p - 1$
\end{tabular}
\end{equation*}
\end{equationblock}
\pause
\begin{columns}
\begin{column}{0.5\linewidth}
\vspace{2em}
\begin{columns}[T, totalwidth=0.95\linewidth]
\begin{column}{0.475\linewidth}
\centering
\resizebox{0.55\columnwidth}{!}{\import{img}{chanpaton.pgf}}
\end{column}
\resizebox{0.5\columnwidth}{!}{\import{img}{chanpaton.pgf}}
\begin{column}{0.5\linewidth}
Introducing \textbf{Chan--Paton factors} $\tensor{\uplambda}{^r_{ij}}$, when branes are \textbf{coincident}:
\hfill\cite{Chan, Paton (1969)}
\cite{Chan, Paton (1969)}
\begin{equation*}
\bigoplus\limits_{r = 1}^N \mathrm{U}_r(1)
\quad
\longrightarrow
\quad
\mathrm{U}( N )
\ket{n;\, r}
=
\sum\limits_{i,\, j = 1}^N
\ket{n;\, i,\, j}\,
\tensor{\uplambda}{^r_{ij}}
\end{equation*}
\end{column}
\hfill
\pause
\begin{column}{0.475\linewidth}
\begin{block}{Chan--Paton Factors}
When branes are \textbf{coincident}:
\begin{equation*}
\bigoplus\limits_{r = 1}^N \mathrm{U}_r(1)
\quad
\longrightarrow
\quad
\mathrm{U}( N )
\end{equation*}
\end{block}
\pause
\highlight{Build gauge bosons, fermions and scalars.}
\end{column}
@@ -542,6 +509,7 @@
\qty( X_{(t)} )^I
=
\tensor{\qty( R_{(t)} )}{^I_J}\, X^J - g_{(t)}^I
\in \mathds{R}^4
\end{equation*}
\pause
where
@@ -567,9 +535,9 @@
\begin{columns}
\begin{column}{0.6\linewidth}
\begin{itemize}
\item consider $u = x + i y = e^{\uptau_e + i \upsigma}$ and $\overline{u} = u^*$
\item $u = x + i y = e^{\uptau_e + i \upsigma}$ and $\overline{u} = u^*$
\item let $x_{(t)} < x_{(t-1)}$ be the \textbf{worldsheet intersection points} on \textbf{real axis}
\item $x_{(t)} < x_{(t-1)}$ \textbf{worldsheet intersection points}
\item $X_{(t)}^{1,\, 2}$ are \textbf{Neumann}, $X_{(t)}^{3,\, 4}$ are \textbf{Dirichlet}
\end{itemize}
@@ -685,7 +653,7 @@
\begin{split}
\partial_z \mathcal{X}( z )
& =
\sum\limits_{l,\, r} c_{lr}\,
\sum\limits_{l,\, r = -\infty}^{+\infty} c_{lr}\,
\qty( - \upomega_z )^{A_{lr}}\,
\qty( 1 - \upomega_z )^{B_{lr}}\,
B_{0,\, l}^{(L)}( \omega_z )\,
@@ -718,7 +686,7 @@
\end{frame}
\begin{frame}{The Solution}
\highlight{Operations sequence:}
Sequence of the operations:
\begin{enumerate}
\item rotation matrix $=$ monodromy matrix
@@ -742,20 +710,19 @@
\begin{columns}
\begin{column}{0.4\linewidth}
\centering
\resizebox{0.607\columnwidth}{!}{\import{img}{branesangles.pgf}}
\resizebox{0.6\columnwidth}{!}{\import{img}{branesangles.pgf}}
\end{column}
\hfill
\begin{column}{0.6\linewidth}
\begin{equation*}
\begin{split}
\eval{S_{\mathds{R}^4}}_{\text{on-shell}}
2 \uppi \upalpha' \eval{S_{\mathds{R}^4}}_{\text{on-shell}}
& =
\frac{1}{2\uppi \upalpha'}
\sum\limits_{t = 1}^3
\qty( \frac{1}{2} \abs{g_{(t)}^{\perp}} \abs{f_{(t-1)} - f_{(t)}} )
\\
& =
\text{Area}\qty( \qty{ f_{(t)} } )
\text{Area}\qty( \qty{ f_{(t)} }_{1 \le t \le N_B} )
\end{split}
\end{equation*}
\end{column}
@@ -763,8 +730,23 @@
\vfill
}
\only<6->{%
\centering
\resizebox{0.25\columnwidth}{!}{\import{img}{brane3d.pgf}}
\begin{columns}
\begin{column}{0.35\linewidth}
\centering
\resizebox{0.8\columnwidth}{!}{\import{img}{brane3d.pgf}}
\end{column}
\hfill
\begin{column}{0.6\linewidth}
\begin{itemize}
\item strings no longer confined to plane
\item strings form a \emph{small bump} from the D-brane
\item classical action \textbf{larger} than factorised case
\end{itemize}
\end{column}
\end{columns}
\vfill
}
\end{block}
\end{frame}
@@ -773,7 +755,7 @@
\subsection[Fermions]{Fermions and Point-like Defect CFT}
\begin{frame}{Fermions on the Strip}
\begin{columns}
\begin{columns}[totalwidth=0.95\linewidth]
\begin{column}{0.4\linewidth}
\centering
\resizebox{0.9\columnwidth}{!}{\import{img}{defects.pgf}}
@@ -803,7 +785,7 @@
\pause
\begin{block}{Stress-energy Tensor}
\begin{equationblock}{Stress-energy Tensor}
\begin{equation*}
\mathcal{T}_{\pm\pm}( \upxi_{\pm} )
=
@@ -839,11 +821,11 @@
0
\end{cases}
\end{equation*}
\end{block}
\end{equationblock}
\end{frame}
\begin{frame}{Conserved Product and Operators}
Expand on a \highlight{basis of solutions}
Expand on a \textbf{basis of solutions}
\begin{equation*}
\uppsi_{\pm}( \upxi_{\pm} )
=
@@ -890,7 +872,7 @@
\pause
Derive the \highlight{algebra of operators:}
Derive the \textbf{algebra of operators:}
\begin{equation*}
\qty[ b_n,\, b_m^{\dagger} ]_+
=
@@ -956,7 +938,7 @@
\pause
Theories are subject to \highlight{consistency conditions:}
Theories are subject to \textbf{consistency conditions:}
\begin{columns}
\begin{column}{0.6\linewidth}
\begin{equation*}
@@ -977,7 +959,7 @@
\end{frame}
\begin{frame}{Stress-energy Tensor and CFT Approach}
Compute the OPEs leading to the \highlight{stress-energy tensor:}
Compute the OPEs leading to the \textbf{stress-energy tensor:}
\begin{equation*}
\mathcal{T}( z )
=
@@ -1070,7 +1052,7 @@
\begin{columns}
\begin{column}{0.5\linewidth}
\centering
\includegraphics[width=0.9\columnwidth]{img/cone}
\includegraphics[width=0.9\columnwidth]{img/cone.pdf}
\end{column}
\hfill
\begin{column}{0.5\linewidth}
@@ -1100,50 +1082,45 @@
\end{frame}
\begin{frame}{Orbifolds}
\begin{columns}[c]
\begin{columns}[T]
\begin{column}{0.475\linewidth}
\begin{center}
\begin{tabular}{@{}l@{}}
\textbf{Mathematics}
\begin{itemize}
\item manifold $M$
\item (Lie) group $G$
\item \emph{stabilizer}: $G_p = \qty{g \in G \mid gp = p \in M}$
\item \emph{orbit}: $Gp = \qty{gp \in M \mid g \in G}$
\item charts $\upphi = \uppi \circ \mathscr{P}$ where:
\begin{itemize}
\item $\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$
\item $\uppi\colon U / G \to M$
\end{itemize}
\end{itemize}
\end{center}
\end{column}
\begin{column}{0.05\linewidth}
\centering
$\Rightarrow$
\\
\toprule
manifold $M$
\\
(Lie) group $G$
\\
\emph{stabilizer}: $G_p = \qty{g \in G \mid gp = p \in M}$
\\
\emph{orbit}: $Gp = \qty{gp \in M \mid g \in G}$
\\
charts $\upphi = \uppi \circ \mathscr{P}$ where:
\\
\vspace{1em}$\mathscr{P}\colon U \subset \mathds{R}^n \to U / G$
\\
\vspace{1em}$\uppi\colon U / G \to M$
\\
\end{tabular}
\end{column}
\hfill
\begin{column}{0.475\linewidth}
\begin{center}
\begin{tabular}{@{}l@{}}
\textbf{Physics}
\begin{itemize}
\item global orbit space $M / G$
\item $G$ group of isometries
\item fixed points
\item additional d.o.f.\ (\emph{twisted states})
\item singular limits of CY manifolds
\end{itemize}
\end{center}
\\
\toprule
global orbit space $M / G$
\\
$G$ group of isometries
\\
fixed points
\\
additional d.o.f.\ (\emph{twisted states})
\\
singular limits of CY manifolds
\\
\end{tabular}
\end{column}
\end{columns}
@@ -1217,7 +1194,7 @@
\pause
Consider \highlight{scalar QED:}
Scalars on NBO:
\begin{equation*}
\upphi_{\qty{ k_+,\, l,\, \vec{k},\, r}}\qty( u,\, v,\, z,\, \vec{x} )
=
@@ -1266,8 +1243,9 @@
\pause
\begin{center}
\it
most terms \textbf{do not converge} due to \textbf{isolated zeros} ($l_{(*)} \equiv 0$) and cannot be recovered even with a \textbf{distributional interpretions} due to the term $\propto u^{-1}$ in the exponentatial
\emph{
most terms \textbf{do not converge} due to \textbf{isolated zeros} \emph{($l_{(*)} \equiv 0$)} and cannot be recovered even with a \textbf{distributional interpretions} due to the term $\propto u^{-1}$ in the exponentatial
}
\end{center}
\end{frame}
@@ -1464,6 +1442,12 @@
\pause
\begin{tikzpicture}[remember picture, overlay]
\draw[line width=4pt, red] (13em, 5.5em) rectangle (22em, 0em);
\end{tikzpicture}
\pause
\begin{block}{Machine Learning Approach}
What is $\mathscr{R}$?
\begin{equation*}
@@ -1490,13 +1474,15 @@
\pause
\item effectively use knowledge from \textbf{computer science, mathematics and physics} to solve problems
\item knowledge from \textbf{computer science, mathematics and physics} to solve problems
\pause
\item provide in-depth \textbf{data analysis} of the datasets
\end{itemize}
\pause
\begin{center}
\includegraphics[width=0.7\linewidth]{img/label-distribution_orig}
\includegraphics[width=0.7\linewidth]{img/label-distribution_orig.pdf}
\end{center}
\end{frame}
@@ -1504,31 +1490,6 @@
\subsection[Machine Learning]{Machine Learning for String Theory}
\begin{frame}{Exploratory Data Analysis}
Machine Learning \highlight{pipeline:}
\begin{center}
\textbf{exploratory} data analysis
$\rightarrow$
feature \textbf{selection}
$\rightarrow$
Hodge numbers
\end{center}
\pause
\begin{columns}
\begin{column}{0.5\linewidth}
\centering
\includegraphics[width=0.9\columnwidth]{img/corr-matrix_orig}
\end{column}
\hfill
\begin{column}{0.5\linewidth}
\centering
\includegraphics[width=0.9\columnwidth, trim={0 0 6in 0}, clip]{img/scalar-features_orig}
\end{column}
\end{columns}
\end{frame}
\begin{frame}{Dataset}
\begin{itemize}
\item $7890$ CICY manifolds (full dataset)
@@ -1553,13 +1514,38 @@
\pause
\begin{center}
\includegraphics[width=0.7\linewidth]{img/label-distribution-compare_orig}
\includegraphics[width=0.7\linewidth]{img/label-distribution-compare_orig.pdf}
\end{center}
\end{frame}
\begin{frame}{Exploratory Data Analysis}
Machine Learning \highlight{pipeline:}
\begin{center}
\textbf{exploratory} data analysis
$\rightarrow$
feature \textbf{selection}
$\rightarrow$
Hodge numbers
\end{center}
\pause
\begin{columns}
\begin{column}{0.5\linewidth}
\centering
\includegraphics[width=0.9\columnwidth]{img/corr-matrix_orig.pdf}
\end{column}
\hfill
\begin{column}{0.5\linewidth}
\centering
\includegraphics[width=0.9\columnwidth, trim={0 0 6in 0}, clip]{img/scalar-features_orig.pdf}
\end{column}
\end{columns}
\end{frame}
\begin{frame}{Machine Learning}
\centering
\includegraphics[width=0.85\linewidth]{img/ml_map}
\includegraphics[width=0.85\linewidth]{img/ml_map.png}
\begin{tikzpicture}[remember picture, overlay]
\node[anchor=base] at (16em,18em) {\cite{from scikit-learn.org}};
@@ -1584,9 +1570,9 @@
\begin{column}{0.4\linewidth}
What is PCA for a $X \in \mathds{R}^{n \times p}$?
\begin{itemize}
\item find new coordinates to \textbf{``put the variance in order''}
\item project data onto a \textbf{lower dimensional} space where variance is maximised
\item \highlight{equivalently} compute the \textbf{eigenvectors} of $X X^T$ or the \textbf{singular values} of $X$
\item equivalently compute the \textbf{eigenvectors} of $X X^T$ or the \textbf{singular values} of $X$
\item isolate \textbf{the signal} from the \textbf{background}
@@ -1594,10 +1580,11 @@
\end{itemize}
\end{column}
\hfill
\pause
\begin{column}{0.6\linewidth}
\centering
\includegraphics[width=0.5\columnwidth]{img/marchenko-pastur}
\includegraphics[width=\columnwidth]{img/svd_orig}
\includegraphics[width=0.5\columnwidth]{img/marchenko-pastur.pdf}
\includegraphics[width=\columnwidth]{img/svd_orig.pdf}
\end{column}
\end{columns}
\end{frame}
@@ -1607,13 +1594,13 @@
\begin{column}{0.5\linewidth}
\centering
\textbf{Configuration Matrix Only}
\includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_matrix_plots}
\includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_matrix_plots.pdf}
\end{column}
\hfill\pause
\begin{column}{0.5\linewidth}
\centering
\textbf{Best Training Set}
\includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_best_plots}
\includegraphics[width=0.8\columnwidth, trim={0 0 3.3in 0}, clip]{img/cicy_best_plots.pdf}
\end{column}
\end{columns}
\end{frame}
@@ -1635,14 +1622,14 @@
\begin{block}{Neural Networks}
\vspace{0.5em}
\begin{tabular}{@{}lc@{}}
fully connected:
\begin{tabular}{@{}ll@{}}
\textbf{fully connected}:
&
$a^{\qty(i)\, \qty{l+1}} = \upphi\qty( a^{\qty(i)\, \qty{l}} \cdot W^{\qty{l}} + b^{\qty{l}} \mathds{1} )$
$a^{\qty(i)\, \qty{l+1}} = \upphi\qty( a^{\qty(i)\, \qty{l}} \cdot W^{\qty{l}} + b^{\qty{l}} \mathds{1}_l )$
\\
convolutional:
\textbf{convolutional}:
&
$a^{\qty(i)\, \qty{l+1}} = \upphi\qty( a^{\qty(i)\, \qty{l}}\, *\, W^{\qty{l}} + b^{\qty{l}} \mathds{1} )$
$a^{\qty(i)\, \qty{l+1}} = \upphi\qty( a^{\qty(i)\, \qty{l}}\, *\, W^{\qty{l}} + b^{\qty{l}} \mathds{1}_l )$
\end{tabular}
\end{block}
@@ -1679,10 +1666,9 @@
\hfill
\begin{column}{0.6\linewidth}
\centering
\only<1>{\includegraphics[width=0.75\columnwidth, trim={12in 10in 0 0}, clip]{img/input_mat}}
\only<2>{\includegraphics[width=0.75\columnwidth, trim={12in 5in 0 5in}, clip]{img/input_mat}}
\only<3>{\includegraphics[width=0.75\columnwidth, trim={12in 0 0 10in}, clip]{img/input_mat}}
\only<4->{\resizebox{\columnwidth}{!}{\import{img}{ccnn.pgf}}}
\only<1-3>{\includegraphics[width=0.75\columnwidth, trim={12in 5in 0 5in}, clip]{img/input_mat.png}}
\only<4>{\animategraphics[autoplay,loop,controls,width=\linewidth]{8}{img/animation/sequence/conv-}{0}{79}}
\only<5>{\resizebox{\columnwidth}{!}{\import{img}{ccnn.pgf}}}
\end{column}
\end{columns}
\end{frame}
@@ -1709,13 +1695,13 @@
\begin{column}{0.5\linewidth}
\centering
\textbf{Best Training Set}
\includegraphics[width=\columnwidth]{img/cicy_best_plots}
\includegraphics[width=\columnwidth]{img/cicy_best_plots.pdf}
\end{column}
\hfill
\begin{column}{0.5\linewidth}
\centering
\includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve_h11}
\includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve}
\includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve_h11.pdf}
\includegraphics[width=0.75\columnwidth]{img/inc_nn_learning_curve.pdf}
\end{column}
\end{columns}
\end{frame}