Add animation of the convolution network
Signed-off-by: Riccardo Finotello <riccardo.finotello@gmail.com>
This commit is contained in:
518
thesis.tex
518
thesis.tex
@@ -12,6 +12,7 @@
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\usepackage{upgreek}
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\usepackage{physics}
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\usepackage{tensor}
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\usepackage{animate}
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\usepackage{graphicx}
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\usepackage{transparent}
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\usepackage{tikz}
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@@ -26,7 +27,7 @@
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\usecolortheme{crane}
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\usefonttheme{structurebold}
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\setbeamertemplate{navigation symbols}{}
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\addtobeamertemplate{background canvas}{\transfade[duration=0.25]}{}
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\addtobeamertemplate{background canvas}{\transfade[duration=0.15]}{}
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\author[Finotello]{Riccardo Finotello}
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\title[D-branes and Deep Learning]{D-branes and Deep Learning}
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@@ -58,7 +59,7 @@
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\newcommand{\highlight}[1]{\fcolorbox{yellow}{yellow!20}{#1}}
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\renewcommand{\cite}[1]{{\tiny{\fcolorbox{red}{red!10}{[#1]}}}}
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\newcommand{\firstlogo}{img/unito}
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\newcommand{\firstlogo}{img/unito.pdf}
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\newcommand{\thefirstlogo}{%
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\begin{figure}
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\centering
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@@ -66,7 +67,7 @@
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\end{figure}
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}
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\newcommand{\secondlogo}{img/infn}
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\newcommand{\secondlogo}{img/infn.pdf}
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\newcommand{\thesecondlogo}{%
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\begin{figure}
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\centering
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@@ -160,7 +161,7 @@
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}
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{%
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% \setbeamertemplate{footline}{}
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\setbeamertemplate{footline}{}
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\usebackgroundtemplate{%
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\transparent{0.1}
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\includegraphics[width=\paperwidth]{img/torino.png}
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@@ -203,79 +204,52 @@
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\pause
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\begin{columns}
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\begin{column}[t]{0.5\linewidth}
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\highlight{Symmetries:}
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\begin{itemize}
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\item \textbf{Poincaré transf.}\ $X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$
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\item \textbf{2D diff.}\ $\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$
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\item \textbf{Weyl transf.}\ $\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \upgamma_{\upalpha \upbeta}$
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\end{itemize}
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\begin{columns}[T, totalwidth=0.935\linewidth]
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\begin{column}{0.45\linewidth}
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\begin{tabular}{@{}ll@{}}
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Symmetries: &
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\\
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\toprule
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\textbf{Poincaré transf.}: &
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$X'^{\upmu} = \tensor{\Uplambda}{^{\upmu}_{\upnu}} X^{\upnu} + c^{\upmu}$
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\\
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\textbf{2D diff.}: &
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$\upgamma'_{\upalpha \upbeta} = \tensor{\qty( \mathrm{J}^{-1} )}{_{\upalpha \upbeta}^{\uplambda \uprho}}\, \upgamma_{\uplambda \uprho}$
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\\
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\textbf{Weyl transf.}: &
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$\upgamma'_{\upalpha \upbeta} = e^{2 \upomega}\, \upgamma_{\upalpha \upbeta}$
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\\
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\end{tabular}
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\end{column}
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\hfill
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\begin{column}{0.45\linewidth}
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\begin{tabular}{@{}ll@{}}
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Conformal symmetry: &
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\\
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\toprule
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\textbf{vanishing} stress-energy tensor: &
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$\mathcal{T}_{\upalpha \upbeta} = 0$
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\\
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\textbf{traceless} stress-energy tensor: &
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$\trace{\mathcal{T}} = 0$
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\\
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\textbf{conformal gauge}: &
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$\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$
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\\
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\end{tabular}
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\pause
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\pause
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\begin{column}[t]{0.5\linewidth}
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\highlight{Conformal symmetry:}
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\begin{itemize}
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\item \textbf{vanishing} stress-energy tensor: $\mathcal{T}_{\upalpha \upbeta} = 0$
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\item \textbf{traceless} stress-energy tensor: $\trace{\mathcal{T}} = 0$
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\item \textbf{conformal gauge} $\upgamma_{\upalpha \upbeta} = e^{\upphi}\, \upeta_{\upalpha \upbeta}$
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\end{itemize}
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\begin{center}
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\highlight{Conformal properties fixed by \textbf{OPE}s.}
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\end{center}
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\end{column}
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\end{columns}
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\end{frame}
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% \begin{frame}{Action Principle and Conformal Symmetry}
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% \begin{columns}
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% \begin{column}{0.6\linewidth}
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% \highlight{%
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% Let $z = e^{\uptau_E + i \upsigma} \Rightarrow \overline{\partial} \mathcal{T}( z ) = \partial \overline{\mathcal{T}}( \overline{z} ) = 0$:
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% }
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% \begin{equation*}
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% \mathcal{T}( z )\, \Upphi_h( w )
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% \stackrel{z \to w}{\sim}
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% \frac{h}{(z - w)^2} \Upphi_h( w )
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% +
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% \frac{1}{z - w} \partial_w \Upphi_h( w )
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% \end{equation*}
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% \begin{equation*}
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% \mathcal{T}( z )\, \mathcal{T}( w )
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% \stackrel{z \to w}{\sim}
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% \frac{\frac{c}{2}}{(z - w)^4}
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% +
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% \order{(z - w)^{-2}}
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% \end{equation*}
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% \begin{equationblock}{Virasoro algebra $\mathscr{V} \oplus \overline{\mathscr{V}}$}
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% \begin{eqnarray*}
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% \qty[ \overset{\scriptscriptstyle(-)}{L}_n,\, \overset{\scriptscriptstyle(-)}{L}_m ]
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% & = &
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% (n - m) \overset{\scriptscriptstyle(-)}{L}_{n + m} + \frac{c}{12} n \qty(n^2 - 1) \updelta_{n + m,\, 0}
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% \\
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% \qty[ L_n,\, \overline{L}_m ]
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% & = &
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% 0
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% \end{eqnarray*}
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% \end{equationblock}
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% \end{column}
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% \begin{column}{0.4\linewidth}
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% \begin{figure}[h]
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% \centering
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% \resizebox{0.9\columnwidth}{!}{\import{img}{complex_plane.pgf}}
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% \end{figure}
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% \end{column}
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% \end{columns}
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% \end{frame}
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\begin{frame}{Action Principle and Conformal Symmetry}
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\highlight{Superstrings in $D$ dimensions:}
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Superstrings in $D$ dimensions $\longrightarrow$ \emph{Virasoro algebra} (central extension of de Witt's algebra):
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\begin{equation*}
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\mathcal{T}( z )
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=
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@@ -311,7 +285,7 @@
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\pause
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\highlight{Consequence:}
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Consequence:
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\begin{equation*}
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c_{\text{full}} = c + c_{\text{ghost}} = 0
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\quad
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@@ -324,152 +298,145 @@
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\begin{frame}{Extra Dimensions and Compactification}
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\begin{block}{Compactification}
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\begin{columns}
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\begin{column}{0.7\linewidth}
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\begin{columns}[T, totalwidth=0.95\linewidth]
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\begin{column}{0.8\linewidth}
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\begin{equation*}
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\mathscr{M}^{1,\, 9}
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=
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\mathscr{M}^{1,\, 3} \otimes \mathscr{X}_6
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\end{equation*}
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\vspace{-1em}
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\begin{itemize}
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\item $\mathscr{X}_6$ is a \textbf{compact} manifold
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\item $N = 1$ \textbf{supersymmetry} is preserved in 4D
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\item $N = 1$ \textbf{supersymmetry} preserved in 4D
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\item algebra of $\mathrm{SU}(3) \otimes \mathrm{SU}(2) \otimes \mathrm{U}(1)$ in arising \textbf{gauge group}
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\end{itemize}
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\end{column}
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\begin{tikzpicture}[remember picture, overlay]
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\node[anchor=base] at (8em,-3.3em) {\cite{code in Hanson (1994)}};
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\node[anchor=base] at (3em,-3.75em) {\includegraphics[width=0.3\linewidth]{img/cy}};
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\node[anchor=base] at (-2em,-6em) {\cite{code in Hanson (1994)}};
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\node[anchor=base] at (-7em,-6em) {\includegraphics[width=0.25\linewidth]{img/cy.png}};
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\end{tikzpicture}
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\begin{column}{0.3\linewidth}
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% \centering
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% \includegraphics[width=0.9\columnwidth]{img/cy}
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\end{column}
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\end{columns}
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\end{block}
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\pause
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\begin{columns}
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\begin{column}[t]{0.5\linewidth}
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\highlight{Kähler manifolds} $\qty( M,\, g )$ such that
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\vfill
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\begin{columns}[T, totalwidth=0.95\linewidth]
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\begin{column}{0.475\linewidth}
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\textbf{Kähler manifolds} $\qty( M,\, g )$ such that:
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\begin{itemize}
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\item $\dim\limits_{\mathds{C}} M = m$
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\item $\mathrm{Hol}( g ) \subseteq \mathrm{SU}( m )$
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\item $g$ is Ricci-flat (equiv.\ $c_1\qty( M )$ vanishes)
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\item $\mathrm{Ric}( g ) \equiv 0$ (equiv.\ $c_1\qty( M ) \equiv 0$)
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\end{itemize}
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\end{column}
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\hfill
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\pause
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\begin{column}[t]{0.5\linewidth}
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Characterised by \highlight{Hodge numbers}
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\begin{column}{0.475\linewidth}
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Characterised by \textbf{Hodge numbers}
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\begin{equation*}
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h^{r,\,s} = \dim\limits_{\mathds{C}} H_{\overline{\partial}}^{r,\,s}\qty( M,\, \mathds{C} )
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\end{equation*}
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counting the no.\ of harmonic $(r,\,s)$-forms.
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counting the no.\ of harmonic $(r,s)$-forms.
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\end{column}
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\end{columns}
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\end{frame}
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\begin{frame}{D-branes and Open Strings}
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Polyakov's action naturally introduces \highlight{Neumann b.c.:}
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Polyakov's action naturally introduces \textbf{Neumann b.c.} for \textbf{open strings}:
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\begin{equation*}
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\eval{\partial_{\upsigma} X\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
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\end{equation*}
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satisfied by \textbf{open and closed} strings living in $D$ dimensions s.t.\ $\square X = 0$.
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satisfied by \highlight{\textbf{open and closed strings} in $D$ dim.} s.t.\ $\square X = 0 \Rightarrow X( z, \overline{z} ) = X( z ) + \overline{X}( \overline{z} )$.
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\pause
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\begin{equationblock}{T-duality}
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% \begin{equationblock}{Equivalent Theories of Closed String Compactification}
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% \begin{equation*}
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% X( z, \overline{z} )
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% =
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% X( z ) + \overline{X}( \overline{z} )
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% \quad
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% \stackrel{T-dual}{\Rightarrow}
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% \quad
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% X( z ) - \overline{X}( \overline{z} )
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% =
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% Y( z, \overline{z} )
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% =
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% Y( z ) + \overline{Y}( \overline{z} )
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% \end{equation*}
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% \end{equationblock}
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% \pause
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\begin{block}{T-duality}
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\textbf{Dirichlet b.c.} consequence of \textbf{T-duality} on $p$ directions:
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\begin{equation*}
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X( z, \overline{z} )
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=
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X( z ) + \overline{X}( \overline{z} )
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\overline{X}( z ) \mapsto - \overline{X}( z )
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\quad
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\stackrel{T}{\Rightarrow}
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\Rightarrow
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\quad
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X( z ) - \overline{X}( \overline{z} )
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=
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Y( z, \overline{z} )
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=
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Y( z ) + \overline{Y}( \overline{z} )
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\eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
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\quad
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\stackrel{T-duality}{\longrightarrow}
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\quad
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\eval{\partial_{\uptau} \widetilde{X}^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
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\end{equation*}
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\end{equationblock}
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\pause
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Resulting effect (repeated $p \le D - 1$ times) leads to \highlight{Dirichlet b.c.:}
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\begin{equation*}
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\eval{\partial_{\upsigma} X^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
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\quad
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\stackrel{T}{\Rightarrow}
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\quad
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\eval{\partial_{\uptau} Y^i\qty( \uptau, \upsigma )}_{\upsigma = 0}^{\upsigma = \ell} = 0
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\quad
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\forall i = 1, 2,\, \dots,\, p
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\end{equation*}
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thus \textbf{open strings} can be \textbf{constrained} to \highlight{$D(D - p - 1)$-branes.}
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\hfill
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\cite{Polchinski (1995, 1996)}
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thus \textbf{open strings} can be \textbf{constrained} to $D(D - p - 1)$-branes.
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\hfill
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\cite{Polchinski (1995, 1996)}
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\end{block}
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\end{frame}
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\begin{frame}{D-branes and Open Strings}
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Introducing $Dp$-branes breaks \highlight{$\mathrm{ISO}(1,\, D-1) \rightarrow \mathrm{ISO}( 1, p ) \otimes \mathrm{SO}( D - 1 - p )$.}
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Introducing $Dp$-branes breaks \highlight{$\mathrm{ISO}(1,\, D-1) \rightarrow \mathrm{ISO}( 1, p ) \otimes \mathrm{SO}( D - 1 - p )$}:
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\begin{equation*}
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\mathcal{A}^{\upmu} \rightarrow \qty( \mathcal{A}^A,\, \mathcal{A}^a )
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\quad
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\Rightarrow
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\quad
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\mathrm{U}( 1 )~\text{theory~in}~p+1~\text{dimensions~(and~scalars)}
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\end{equation*}
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\pause
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\begin{equationblock}{Massless Spectrum (\emph{irrep} of \textbf{little group} $\mathrm{SO}( D - 2 )$)}
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\begin{equation*}
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\mathcal{A}^{\upmu}
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\quad \leftrightarrow \quad
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\upalpha_{-1}^{\upmu} \ket{0}
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\qquad
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\longrightarrow
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\qquad
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\begin{tabular}{@{}llll@{}}
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$\mathcal{A}^A$
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&
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$\leftrightarrow$
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&
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$\upalpha_{-1}^A \ket{0},$
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&
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$A = 0,\, 1,\, \dots,\, p$
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\\
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$\mathcal{A}^a$
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&
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$\leftrightarrow$
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&
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$\upalpha_{-1}^a \ket{0},$
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&
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$a = 1,\, 2,\, \dots,\, D - p - 1$
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\end{tabular}
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\end{equation*}
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\end{equationblock}
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\pause
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\begin{columns}
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\begin{column}{0.5\linewidth}
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\vspace{2em}
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\begin{columns}[T, totalwidth=0.95\linewidth]
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\begin{column}{0.475\linewidth}
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\centering
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\resizebox{0.55\columnwidth}{!}{\import{img}{chanpaton.pgf}}
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\end{column}
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\resizebox{0.5\columnwidth}{!}{\import{img}{chanpaton.pgf}}
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\begin{column}{0.5\linewidth}
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Introducing \textbf{Chan--Paton factors} $\tensor{\uplambda}{^r_{ij}}$, when branes are \textbf{coincident}:
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\hfill\cite{Chan, Paton (1969)}
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\cite{Chan, Paton (1969)}
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\begin{equation*}
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\bigoplus\limits_{r = 1}^N \mathrm{U}_r(1)
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\quad
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\longrightarrow
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\quad
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\mathrm{U}( N )
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\ket{n;\, r}
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=
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\sum\limits_{i,\, j = 1}^N
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\ket{n;\, i,\, j}\,
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\tensor{\uplambda}{^r_{ij}}
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\end{equation*}
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\end{column}
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\hfill
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\pause
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\begin{column}{0.475\linewidth}
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\begin{block}{Chan--Paton Factors}
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When branes are \textbf{coincident}:
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\begin{equation*}
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\bigoplus\limits_{r = 1}^N \mathrm{U}_r(1)
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\quad
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\longrightarrow
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\quad
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\mathrm{U}( N )
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\end{equation*}
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\end{block}
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\pause
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\highlight{Build gauge bosons, fermions and scalars.}
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\end{column}
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@@ -542,6 +509,7 @@
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\qty( X_{(t)} )^I
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=
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\tensor{\qty( R_{(t)} )}{^I_J}\, X^J - g_{(t)}^I
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\in \mathds{R}^4
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\end{equation*}
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\pause
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where
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@@ -567,9 +535,9 @@
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\begin{columns}
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\begin{column}{0.6\linewidth}
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\begin{itemize}
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\item consider $u = x + i y = e^{\uptau_e + i \upsigma}$ and $\overline{u} = u^*$
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\item $u = x + i y = e^{\uptau_e + i \upsigma}$ and $\overline{u} = u^*$
|
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|
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\item let $x_{(t)} < x_{(t-1)}$ be the \textbf{worldsheet intersection points} on \textbf{real axis}
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\item $x_{(t)} < x_{(t-1)}$ \textbf{worldsheet intersection points}
|
||||
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\item $X_{(t)}^{1,\, 2}$ are \textbf{Neumann}, $X_{(t)}^{3,\, 4}$ are \textbf{Dirichlet}
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||||
\end{itemize}
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||||
@@ -685,7 +653,7 @@
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\begin{split}
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\partial_z \mathcal{X}( z )
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& =
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||||
\sum\limits_{l,\, r} c_{lr}\,
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\sum\limits_{l,\, r = -\infty}^{+\infty} c_{lr}\,
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||||
\qty( - \upomega_z )^{A_{lr}}\,
|
||||
\qty( 1 - \upomega_z )^{B_{lr}}\,
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||||
B_{0,\, l}^{(L)}( \omega_z )\,
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@@ -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}
|
||||
|
||||
Reference in New Issue
Block a user