outline_content.tex

\begin{enumerate}[label=\textbf{\arabic*.}]
\item \textsc{Vector spaces}
\vspace{-0.2cm}
\begin{enumerate}[label=\arabic*.,noitemsep]
\item General definitions
\item Linear dependency
\item Basis, dimension
\end{enumerate}
\item \textsc{Linear transformations}
\vspace{-0.2cm}
\begin{enumerate}[label=\arabic*.,noitemsep]
\item Linear transformations
\item Matrix representation
\item Kernel and image
\end{enumerate}
\item \textsc{Rank}
\vspace{-0.2cm}
\begin{enumerate}[label=\arabic*.,noitemsep]
\item Definition of the rank
\item Properties of the rank
\item Invertible matrices
\item Transpose of a matrix, symmetric matrices
\end{enumerate}
\item \textsc{Norm and inner product}
\vspace{-0.2cm}
\begin{enumerate}[label=\arabic*.,noitemsep]
\item Norm
\item Inner product
\item Orthogonality
\item Orthogonal projection and distance to a subspace
\end{enumerate}
\item \textsc{Matrices and orthogonality}
\vspace{-0.2cm}
\begin{enumerate}[label=\arabic*.,noitemsep]
\item Gram-Schmidt orthogonalization method
\item Orthogonal matrices
\end{enumerate}
\item \textsc{Eigenvalues, eigenvectors and Markov chains}
\vspace{-0.2cm}
\begin{enumerate}[label=\arabic*.,noitemsep]
\item Eigenvalues and eigenvectors
\item Diagonalizable matrices
\item Application to Markov chains
\item Example: Google's PageRank algorithm
\end{enumerate}
\item \textsc{The spectral theorem and PCA}
\vspace{-0.2cm}
\begin{enumerate}[label=\arabic*.,noitemsep]
\item The Spectral Theorem
\item Application: Principal Component Analysis (PCA)
\item Singular value decomposition
\item Interpretations of the SVD
\end{enumerate}
\item \textsc{Graphs and Linear Algebra}
\vspace{-0.2cm}
\begin{enumerate}[label=\arabic*.,noitemsep]
\item Graphs
\item Graph Laplacian
\item Spectral clustering with the graph Laplacian
\item Spectral clustering as a relaxation
\item Spectral clustering beyond graphs
\end{enumerate}
\item \textsc{Convex functions}
\vspace{-0.2cm}
\begin{enumerate}[label=\arabic*.,noitemsep]
\item Convex sets
\item Convex functions
\end{enumerate}
\item \textsc{Linear regression}
\vspace{-0.2cm}
\begin{enumerate}[label=\arabic*.,noitemsep]
\item Least squares
\item Penalized least squares: Ridge regression and Lasso
\item Norms for matrices, application to matrix completion
\end{enumerate}
\item \textsc{Optimality conditions}
\vspace{-0.2cm}
\begin{enumerate}[label=\arabic*.,noitemsep]
\item Local and global minimizers
\item Constrained optimization
\item The Lagrangian and the dual problem
\item Kuhn Tucker Theorem
\end{enumerate}
\item \textsc{Gradient descent}
\vspace{-0.2cm}
\begin{enumerate}[label=\arabic*.,noitemsep]
\item Gradient descent
\item Newton's method
\item Stochastic gradient descent
\end{enumerate}
\end{enumerate}