\@writefile{lof}{\contentsline{figure}{\numberline{1}{\ignorespaces Convergence of different algorithms for k-Means clustering with fashion MNIST dataset. The solution rank for the template\nobreakspace{}\textup{\hbox{\mathsurround\z@ \normalfont (\ignorespaces\ref{eq:sdp_svx}\unskip\@@italiccorr )}} is known and it is equal to number of clusters $k$ (Theorem\nobreakspace{}1. \cite{kulis2007fast}). As discussed in\nobreakspace{}\cite{tepper2018clustering}, setting rank $r>k$ leads more accurate reconstruction in expense of speed. Therefore, we set the rank to 20.}}{10}{figure.1}}
\newlabel{fig:clustering}{{1}{10}{Convergence of different algorithms for k-Means clustering with fashion MNIST dataset. The solution rank for the template~\eqref{eq:sdp_svx} is known and it is equal to number of clusters $k$ (Theorem~1. \cite{kulis2007fast}). As discussed in~\cite{tepper2018clustering}, setting rank $r>k$ leads more accurate reconstruction in expense of speed. Therefore, we set the rank to 20}{figure.1}{}}
\@writefile{lof}{\contentsline{figure}{\numberline{2}{\ignorespaces Convergence with different subsolvers for the aforementioned nonconvex relaxation. }}{11}{figure.2}}
\newlabel{fig:bp1}{{2}{11}{Convergence with different subsolvers for the aforementioned nonconvex relaxation}{figure.2}{}}
\@writefile{toc}{\contentsline{subsection}{\numberline{E.1}$\ell _\infty$ Denoising with a Generative Prior}{20}{subsection.E.1}}
\citation{ge2016efficient}
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\newlabel{fig:comparison_fab}{{E.1}{21}{$\ell _\infty$ Denoising with a Generative Prior}{equation.E.67}{}}
\@writefile{lof}{\contentsline{figure}{\numberline{3}{\ignorespaces Augmented Lagrangian vs Adam for $\ell _\infty$ denoising (left). $\ell _2$ vs $\ell _\infty$ denoising as defense against adversarial examples}}{21}{figure.3}}
\@writefile{lof}{\contentsline{figure}{\numberline{4}{\ignorespaces{\it{(Top)}} Objective convergence for calculating top generalized eigenvalue and eigenvector of $B$ and $C$. {\it{(Bottom)}} Eigenvalue structure of the matrices. For (i),(ii) and (iii), $C$ is positive semidefinite; for (iv), (v) and (vi), $C$ contains negative eigenvalues. {[(i): Generated by taking symmetric part of iid Gaussian matrix. (ii): Generated by randomly rotating diag($1^{-p}, 2^{-p}, \cdots , 1000^{-p}$)($p=1$). (iii): Generated by randomly rotating diag($10^{-p}, 10^{-2p}, \cdots , 10^{-1000p}$)($p=0.0025$).]}}}{22}{figure.4}}
\newlabel{fig:geig1}{{4}{22}{{\it{(Top)}} Objective convergence for calculating top generalized eigenvalue and eigenvector of $B$ and $C$. {\it{(Bottom)}} Eigenvalue structure of the matrices. For (i),(ii) and (iii), $C$ is positive semidefinite; for (iv), (v) and (vi), $C$ contains negative eigenvalues. {[(i): Generated by taking symmetric part of iid Gaussian matrix. (ii): Generated by randomly rotating diag($1^{-p}, 2^{-p}, \cdots , 1000^{-p}$)($p=1$). (iii): Generated by randomly rotating diag($10^{-p}, 10^{-2p}, \cdots , 10^{-1000p}$)($p=0.0025$).]}}{figure.4}{}}