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IPCookbook.tex
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% Freely inspired from https://www.charles-deledalle.fr/pages/files/cookbook_datascience.pdf
% Copyright (C) Ferréol Soulez, 2020
%
% This program is free software: you can redistribute it and/or modify
% it under the terms of the GNU General Public License as published by
% the Free Software Foundation, either version 3 of the License, or
% (at your option) any later version.
%
% This program is distributed in the hope that it will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
% GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program. If not, see <http://www.gnu.org/licenses/>.
%
%% Ferréol Soulez, 2020
\documentclass[a4paper]{cookbook}
\input{preamble.tex}
\usepackage{lipsum} % Dummy text
\usepackage[figwidth = 0.98\linewidth]{todonotes} % Dummy image (and more!)
\usepackage[absolute, overlay]{textpos} % Figure placement
\setlength{\TPHorizModule}{\paperwidth}
\setlength{\TPVertModule}{\paperheight}
\usepackage{tabls}
\title{Selected recipes for inverse problems}
\author
{%
Ferréol Soulez
}
%\institute{Centre de Recherche Astrophysique de Lyon}
%% Remove footline:
%\setbeamertemplate{footline}{}
\begin{document}
\begin{frame}
\vspace{-1.5em}
\begin{columns}[onlytextwidth]
\begin{column}{0.5\textwidth - .5cm}
\begin{exampleblock}{Notations}
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\alpha$ & (lower case) scalar \\
$\alpha^*$ & complex conjugate of $\alpha$\\
$\V{x} \in \Complexes^N$ & (boldface lower case) complex vector\\
$N$ & (upper case) cardinal (number of elements)\\
$\M{H} \in \Complexes^{N\times M}$ & (boldface upper case) linear operator (matrix) \\% $\Complexes^N \to \Complexes^M$\\
$f: \Complexes^N \to \Complexes^M $ & (lower case) function \\
$ L^{2}(\Reals^N)$ & space of squared-integrable function \\
$\mathcal{H}: L^{2}(\Reals^N) \to L^{2}(\Reals^M)$ & (calligraphic upper case) operator \\ & acting on functions
\end{tabular*}
\end{exampleblock}
\begin{block}{Basic properties}
{\tablinesep=1.2ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\left[\M{H}\,\V{x}\right]_m = \sum_n H_{m,n}\,x_n$ & vector matrix product\\
$\left[\M{H}\,\M{G}\right]_{m,k} = \sum_{n} H_{m,n}\,G_{n,k}$ & matrix product\\
$\M{H}\left(\alpha\,\V{x}+\beta\,\V{y}\right) = \alpha\,\M{H}\,\V{x}+\beta\,\M{H}\,\V{y}$ & linearity\\
$\left({\V{x}}\times {\V{y}}\right)_n = x_n\,y_n$ & element wise product \\
$\M{I} =
\begin{bmatrix}
1\ & 0\ & 0\ &\dots &0\\
0&1&0&\dots &0\\
0&0&1&\dots &0\\
\vdots &\vdots &\vdots &\ddots &\vdots \\
0&0&0&\dots &1\end{bmatrix}$ & identity matrix\\
\end{tabular*}}
{\tablinesep=1.2ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\M{A} = \diag(\V{a}) =
\begin{bmatrix}
a_0&0&0&\dots &0\\
0&a_1&0&\dots &0\\
0&0&a_2&\dots &0\\
\vdots &\vdots &\vdots &\ddots &\vdots \\
0&0&0&\dots &a_{N-1}\end{bmatrix}$ & diagonal matrix % $\V{a}= \begin{bmatrix} a_0\\ a_1\\ a_2\\ \vdots \\ a_{N-1} \end{bmatrix}$
\end{tabular*}}
\end{block}
\begin{block}{Adjoint, scalar product}
{\tablinesep=1.2ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\left(H\T\right)_{i,j} = H^*_{j,i} $ & adjoint (transpose conjugate)\\
$\Inner{\V{x}}{\V{y}} = \sum_{n=0}^{N-1} x_n^*\,y_n = \V{x}\T \V{y}$ & dot (or scalar or inner) product\\
$\Inner{\M{H}\,\V{x}}{\V{y}} = \Inner{\V{x}}{\M{H}\T\,\V{y}}$ & adjoint formal definition\\
$\left(\alpha\,\M{H} + \beta\,\M{G}\right)\T = \alpha^*\,\M{H}\T + \beta^*\,\M{G}\T$&\\
$\left(\M{H}\,\M{G}\right)\T = \M{G}\T\,\M{H}\T$&
\end{tabular*}}
\end{block}
\begin{block}{Norms}
{\tablinesep=1.5ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$ \Norm{\V{x}}_2^2 = \Inner{\V{x}}{\V{x}} = \V{x}\T\V{x} = \sum_{n=0}^{N-1} \Abs{x_n}^2 = \sum_{n=0}^{N-1} x_n^*\,x_n$ & $\ell^2$ norm
\end{tabular*}}
{\tablinesep=1.2ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$ \Abs{ \Inner{\V{x}}{\V{y}}} \leq \Norm{\V{x}}_2 \, \Norm{\V{y}}_2 $ & (Cauchy-Schwartz) \\
$\Norm{\V{x} + \V{y} }^2_2 = \Norm{\V{x}}^2_2 + \Norm{\V{y}}^2_2 + 2\, \Inner{\V{x}}{\V{y}} $ & \\
$ \Norm{\V{x}}_p^p =\sum_{n=0}^{N-1} \Abs{x_n}^p\ , \,p\geq 1 $ & $\ell^p$ norm\\
$ \Norm{\V{x} + \V{y} }_p \leq \Norm{\V{x}}_p + \Norm{\V{y}}_p$ & triangular inequality\\
$\Norm{\M{H}}^2_F = \sum_{n,m} \Abs{H_{i,j}}^2 = \tr(\M{H}\T\M{H}) $& Frobenius norm
\end{tabular*}}
\end{block}
\begin{block}{Inverse}
$\M{H}\in \Complexes^{N\times N}$ is called invertible (also nonsingular or nondegenerate) if there exists a matrix $\M{G}$ such that:
\begin{align*}
\M{H} \M{G} = \M{G} \M{H} = \M{I}
\end{align*}
$\M{G} = \M{H}^{-1}$ is unique and is the inverse of $\M{H}$\\
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\left(\M{H}\,\M{M}\right)^{-1} = \M{M}^{-1}\,\M{H}^{-1}$& for any $\M{H}\,,\M{M}$ invertible\\
$\left(\M{H}\T\right)^{-1} = \left(\M{H}^{-1}\right)\T$&
\end{tabular*}}
\end{block}
\end{column}
\begin{column}{0.5\textwidth - .5cm}
\begin{block}{Eigenvalue \& eigenvectors of $\M{H} \in \Complexes^{N\times N}$}
$\lambda_i\in \Complexes$ and $\V{v}_i\in \Complexes^{N}$ are the $i^\text{th}$ eigenvalue and eigenvector respectively. They satisfy:
\begin{align*}
\M{H}\,\V{v}_i = \lambda_i\,\V{v}_i
\end{align*}
It leads to the eigendecomposition:
\begin{align*}
\M{H} = \M{Q}\,\diag(\V{\lambda})\,\M{Q}^{-1}\,,
\end{align*}
where the columns of $\M{V}$ are the $N$ eigenvectors of $\M{H}$.
\begin{itemize}
\item $\V{\lambda} = \eig(\M{H})$ is the spectrum of $\M{H}$,
\item $C = \frac{\max\Abs{\V{\lambda}}}{\min\Abs{\V{\lambda}}}$ is the condition number,
\item $\rank(\M{H})$ is the number of non-zero element of $\V{\lambda}$,
\item if $\lambda_i\neq 0,\,\forall i$, $\M{H} $ is invertible and $\M{H}^{-1} = \M{Q}\,\diag(\V{\lambda})^{-1}\,\M{Q}^{-1}\,$,
\end{itemize}
\end{block}
\begin{block}{Peculiar matrices}
\begin{itemize}
\item {\bf Unitary:} $\M{Q}\T\, \M{Q} = \M{Q}\, \M{Q}\T = \M{I}$
\item {\bf Hermitian:} $\M{H}\T= \M{H}$ and $\M{H} = \M{Q}\,\diag(\V{\lambda})\,\M{Q}\T$ with $\V{\lambda}\in\Reals^N$ and $\M{Q}$ unitary
\item {\bf Positive semi-definite:} $ \lambda_i\geq 0\,,\ \forall i \iff \V{x}\T\M{H}\V{x} \geq 0, \, \forall \V{x}$
\item {\bf Toeplitz:} $H_{i,j} = h_{i-j}$,
\item {\bf Circulant:} Toeplitz with $H_{i,j}= h_{(i-j) \mod N}$
\end{itemize}
\end{block}
\begin{block}{Trace and determinant}
\begin{itemize}
\item {\bf trace:} $\ \tr(\M{H}) = \sum_n H_{n,n} = \sum_n \lambda_n $
\end{itemize}
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\tr(\M{H}\T) = \tr(\M{H})^*$& $\tr(\M{H}\,\M{G}) =\tr(\M{G}\,\M{H})$\\
$\tr(\alpha\,\M{H}+\beta\,\M{G}) =\alpha\,\tr(\M{H})+\alpha\,\tr(\M{H})$& \\
\end{tabular*}}
\begin{itemize}
\item {\bf determinant:} $\ \det(\M{H}) = \prod_n \lambda_n $
\end{itemize}
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\det(\M{H}) \neq 0 \iff \M{H} $ is invertible & $\det(\M{H}\T) = \det(\M{H})^*$\\
$\det(\M{H}\,\M{G}) =\det(\M{H})\,\det(\M{G})$& $\det(\M{H}^{-1})= 1/ \det(\M{H})$
\end{tabular*}}
\end{block}
\begin{block}{Singular value decomposition}
For all matrices $\M{H}\in \Complexes^{N\times M}$ there exists two unitary matrices $\M{U} \in \Complexes^{N\times N}$ and $\M{V} \in \Complexes^{M\times M}$ and a real non-negative diagonal matrix $\M{\Sigma} \in \Complexes^{N\times M}$ (singular values: $\sigma_i = \Sigma_{i,i}\,,\ \forall i\leq \min(M,N)$) subject to:
\begin{align*}
\M{H} = \M{U}\,\M{\Sigma}\,\M{V}\T
\end{align*}
\vspace{-2em}\begin{itemize}
\item columns of $\M{U}$ are eigenvectors of $\M{H}\,\M{H}\T$,
\item columns of $\M{V}$ are eigenvectors of $\M{H}\T\,\M{H}$,
\item $\M{\Sigma} = \sqrt{\diag\left(\eig\left(\M{H}\,\M{H}\T\right)\right)} $,
\item $\Norm{\M{H}}^2_F = \sum_{i} \sigma_i^2 $
\item $\rank(\M{H})$: number of non zero singular values,
% \item if $\M{H}$ is Hermitian $\M{U} = \M{V} = \M{Q}$
\end{itemize}
\end{block}
\begin{block}{Inversion lemmas \& Woodbury identity}
$ \M{B}^{-1}\M{V} \left(\M{A} - \M{U}\M{B}^{-1}\M{V}\right)^{-1} = \left( \M{B} - \M{V}\M{A}^{-1}\M{U}\right)^{-1}\M{V}\M{A}^{-1}$\\
$ \left(\M{A} - \M{U}\M{B}^{-1}\M{V}\right)^{-1}\M{U}\M{B}^{-1} = \M{A}^{-1}\M{U}\left( \M{B} - \M{V}\M{A}^{-1}\M{U}\right)^{-1}$\\
$\left(\M{A} + \M{U}\,\M{B}\,\M{V} \right)^{-1} = \M{A}^{-1} - \M{A}^{-1}\,\M{U}\left( \M{B}^{-1} + \M{V}\,\M{A}^{-1}\,\M{U}\right)^{-1}\,\M{V}\,\M{A}^{-1}$
\end{block}
\begin{block}{Moore-Penrose pseudo inverse}
The pseudo inverse of $\M{H} = \M{U}\,\M{\Sigma}\,\M{V}\T$ writes $\M{H}^+ = \M{V}\,\M{\Sigma}^+\,\M{U}\T$ with $\Sigma_{i,i}^+ = \left\{
\begin{array}{ll}
\Sigma^{-1}_{i,i} & \text{if }\sigma_i\neq0 \,, \\
0 & \text{otherwise} \,.
\end{array}
\right.$
\begin{itemize}
\item $\M{H}\,\M{H}^+\,\M{H} = \M{H} $ and $ \M{H}^+\,\M{H}\,\M{H}^+ = \M{H}^+ $
\item $\M{H}$ is square and $\ \rank(\M{H}) = N \Rightarrow \M{H}^+ = \M{H}^{-1}$
\item $\M{H}$ is broad: $\ \rank(\M{H}) \leq N$ and $ \M{H}^+ = \M{H}\T \left(\M{H}\,\M{H}\T\right)^{-1}$
\item $\M{H}$ is tall: $\ \rank(\M{H}) \leq M $ and $\M{H}^+ = \left(\M{H}\T\,\M{H}\right)^{-1}\,\M{H}\T $
\end{itemize}
\end{block}
\end{column}
\end{columns}
\end{frame}
\begin{frame}
\vspace{-1.5em}
\begin{columns}[onlytextwidth]
\begin{column}{0.5\textwidth - .5cm}
\begin{block}{Derivatives}
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$J_{i,j}=\left[\PDer{\V{x}}{\V{y}}\right]_{i,j} = \PDer{x_i}{y_j} $ & Jacobian matrix\\
$\left[\nabla f \right]_i= \PDer{f}{x_i}$ & Gradient vector\\
$\left[\nabla^2 f\right]_{i,j} =
\frac{\partial^2 f}{\partial x_i\,\partial x_j }$ & Hessian matrix
\end{tabular*}}
\end{block}
\begin{block}{Derivation rules}
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
% $\partial \M{A} = 0$ \\
$\partial\left(\alpha\, \M{A}+ \beta \,\M{B}\right) = \alpha\,\partial\M{A} +\beta\,\partial\M{B}$ & linearity \\
$\partial\left( \M{A}\,\M{B}\right) =\left( \partial\M{A}\right) \, \M{B} + \M{A}\, \left(\partial\M{B}\right)$ & \\
$\partial\left( \M{A}^{-1}\right) = -\M{A}^{-1}\left( \partial\M{A}\right)\,\M{A}^{-1}$ & \\
$\partial\left( \M{A}\T\right) = \left( \partial\M{A}\right)\T$ & \\
$\PDer{\V{x}\T\V{y}}{\V{x}} = \PDer{\V{y}\T\V{x}}{\V{x}} = \V{y} $&\\
$\PDer{\V{x}\T\M{A}\V{x}}{\V{x}} = \left(\M{A} + \M{A}\T\right) \V{x} $& \\
% $\PDer{f\comp g}{x} =\left( \PDer{f}{x}\comp g \right)\, \PDer{g}{\V{x}}$ & chain rule\\
% $\PDer{f\comp g(\V{x})}{\V{x}} = \sum_m^M \PDer{f}{\left[g(\V{x})\right]_m}\, \PDer{\left[g(\V{x})\right]_m}{\V{x}}$ & chain rule\\
$\PDer{f\comp g(\V{x})}{\V{x}} = \At{\sum_{m=0}^{M-1} \PDer{f}{u_m}\, \PDer{u_m}{\V{x}}}{\V{u} = g(\V{x})}$ & chain rule\\
$\PDer{\left(\M{A}\,\V{x}-\V{y}\right)\T\M{B}\left(\M{A}\,\V{x}-\V{y}\right)}{\M{A}} = \left(\M{B} + \M{B}\T \right)\,\left(\M{A}\,\V{x}- \V{y}\right)\,\V{x}\T$&
\end{tabular*}}
\end{block}
\begin{block}{Continuous Fourier transform}
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\FT{f}(\nu) = \F{f}(\nu) = \int_{-\infty}^{+\infty} f(t) \E^{-\TwoIPi\,\nu\,t}\,\D{t}$& forward\\
$f(t) = \IF{\FT{f}}(t) = \int_{-\infty}^{+\infty} f(\nu) \E^{\TwoIPi\,\nu\,t}\,\D{\nu}$& inverse
\end{tabular*}}
% {\bf Properties}
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\F{\alpha f + \beta g} = \alpha \FT{f} + \beta \FT{g} $ & linearity\\
$\F{f(t-t_0)} =\E^{-\TwoIPi\,\nu\,t_0} \FT{f}(\nu) $ & shift\\
$\F{\E^{\TwoIPi\,\nu_0\,t} f(t) } = \FT{f}(\nu-\nu_0) $ & modulation\\
$\F{f(a\,t)} =\frac{1}{\Abs{a}} \FT{f}\left(\frac{\nu}{a}\right) $ & scaling\\
$\F{f^*}(\nu) =\FT{f}^*(-\nu) $ & conjugation\\
$ \int_{-\infty}^{+\infty} \Abs{\FT{f}(\nu)}^2\,\D{\nu} = \int_{-\infty}^{+\infty} \Abs{f(t)}^2\,\D{t} $ & Plancherel-Parseval\\
$\FT{f}(0) = \int_{-\infty}^{+\infty} f(t)\,\D{t} $ & integration\\
$\F{f^{(n)}} = \left(\TwoIPi\,\nu\right)^n\,\FT{f}(\nu)$ & differentiation\\
$\F{f\ast g} =\FT{f}\,\FT{g}$ & convolution\\
$\F{f\star f} = \Abs{\FT{f}}^2$ & autocorrelation
\end{tabular*}}
\end{block}
\begin{block}{Discrete Fourier Transform}
{\tablinesep=1.2ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$ \M{F}=\begin{bmatrix}
1&1&1&\cdots &1\\
1&\omega &\omega^{2}&\cdots &\omega^{N-1}\\
1&\omega^{2}&\omega^{4}&\cdots &\omega^{2(N-1)}\\
% 1&\omega^{3}&\omega^{6}&\omega^{9}&\cdots &\omega^{3(N-1)}\\
\vdots &\vdots &\vdots &\ddots &\vdots \\
1&\omega^{N-1}&\omega^{2(N-1)}&\cdots &\omega^{(N-1)(N-1)}
\end{bmatrix}$ &
\begin{tabular}{r}
with $\omega = \E^{-\frac{-2\imath\,\pi }{N}}$ \\
the $N^\text{th}$ root of unity.
\end{tabular}
\end{tabular*}}
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\M{F}^{-1} = \frac{1}{N}\,\M{F}\T\,$ & orthogonality\\
$\M{U} = \frac{1}{\sqrt{N}}\,\M{F}$ & $\M{U}$ is unitary\\
$\Norm{\V{x}}^2_2 =\frac{1}{N}\Norm{\M{F}\,\V{x}}_2^2$& Plancherel-Parseval\\
$\Norm{\V{x}}_1 \leq \Norm{\M{F}\,\V{x}}_1 \leq N\,\Norm{\V{x}}_1 $&
\end{tabular*}}
\end{block}
\begin{block}{Circular convolution matrix $\M{H}$}
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$ H_{0,i} = h_i $ & impulse response (PSF)\\
$\M{H}=\M{F}^{-1} \,\diag(\FT{\V{h}})\,\M{F}$& diagonalization by Fourier \\
$\FT{\V{h}}=\M{F}\,\V{h}\,\ $ & eigenvalues spectrum
\end{tabular*}}
\end{block}
\end{column}
\begin{column}{0.5\textwidth - .5cm}
\begin{block}{Continuous probability distribution}
$\V{x}\in \Complexes^N$ is a continuous random vector, it has a probability density function (pdf) $f_X(\V{x})$ such that, for all $\Set{A} \subseteq \Complexes^N$:
\begin{align*}
\Pr(\V{x}\in \Set{A})=\int_{\Set{A}} f_{X}(\V{x})\, \D{\V{x}}
\end{align*}
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\bar{\V{x}} = \Expect{\V{x}} = \Avg{\V{x}} = \int_{-\infty}^{+\infty} \V{x}\,f_X(\V{x})\,\D{\V{x}} $& Expectation (mean)\\
$\Expect{\alpha\,\V{x} +\beta\,\V{y} + \gamma} =\alpha\,\Expect{\V{x}} +\beta\,\Expect{\V{y}} + \gamma $ & linearity\\
$\M{C}_{\V{x},\V{y}} = \Cov\left(\V{x},\V{y}\right) = \Expect{\V{x}\V{y}\T} - \Expect{\V{x}}\,\Expect{\V{y}}\T$ & Cross-covariance \\
$\M{C}_{\V{x}}=\Cov\left(\V{x},\V{x}\right) = \Expect{\V{x}\V{x}\T} - \Expect{\V{x}}\,\Expect{\V{x}}\T$ & Covariance\\
$\M{C}_{\V{x}}$ is an Hermitian matrix of size $N\times N$.
\end{tabular*}}
\end{block}
\begin{block}{Independence \& Uncorrelation}
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$ \Pr(X\Given Y) = \Pr(X)$ & Independence \\
$ f_{X,Y}(\V{x},\V{y}) = f_X(\V{x})\, f_Y(\V{y})$ & Independence\\
$\Cov\left(\alpha\,\V{x} +\beta\,\V{y} + \gamma\right) = \alpha^2\,\Cov\left(\V{x}\right) +\beta^2\,\Cov\left(\V{y} \right)$ & Uncorrelation\\
$ \Cov\left(\V{x},\V{y}\right) = 0 $ & Uncorrelation \\
\end{tabular*}}
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
independence $\Rightarrow$ uncorrelation &
uncorrelation $\nRightarrow$ independence \\
$ \Pr(X\Given Y) = \frac{\Pr(Y\Given X)\,\Pr(X) }{\Pr(Y) }$& Bayes rule
\end{tabular*}}\vspace{-1ex}
\end{block}
% \begin{block}{Bayes rule}
% \begin{align*}
% \Pr(X\Given Y) = \frac{\Pr(Y\Given X)\,\Pr(Y) }{\Pr(Y) }
% \end{align*}
% \end{block}
\begin{block}{Convexity}
$ f: \Complexes^N \to \Reals $ is strictly convex if and only if:
\begin{align*}
f\left(\lambda\,\V{x} + (1-\lambda)\,\V{y}) < \lambda\,f(\V{x}) + (1-\lambda)\,f(\V{y}\right)\,\ \forall\,\V{x},\V{y}, \, \lambda\in ]0,1[\,.
\end{align*}
% $f$ has only a global minimum $\V{x}^+=\argmin_{\V{x}} f(\V{x})$
%
{\tablinesep=1.ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\V{x}^+=\argmin_{\V{x}} f(\V{x})$& global minimum\\
$ \V{p}\in\partial f(\V{x}) \Leftrightarrow f(\V{x})-f(\V{x}') \geq\Inner{\V{p}}{\V{x}-\V{x}'}\,, \forall\V{x}'$ & subgradient
\end{tabular*}}\vspace{-1ex}
\end{block}
\begin{block}{Gradient descent}
$f: \Complexes^N \to \Reals $ is convex and differentiable with Lipschitz gradient $L$:
\begin{align*}
\Norm{\nabla \,\V{x} - \nabla \,\V{y}}_2 \leq L\,\Norm{\V{x} - \V{y}}_2
\end{align*}
The following sequence converges toward a minimizer of $f$ in $O(1/k)$:
\begin{align*}
\V{x}^{(k+1)} = \V{x}^{(k)} - \gamma \nabla f(\V{x}^{(k)}) \text{ with } \gamma\in]0,1/L]
\end{align*}
\end{block}
\begin{block}{Newton's method}
$f: \Complexes^N \to \Reals $ is convex and twice differentiable, the sequence:
\begin{align*}
\V{x}^{(k+1)} = \V{x}^{(k)} - \left(\nabla^2 f(\V{x}^{(k)})\right)^{-1}\,\nabla f(\V{x}^{(k)})
\end{align*}
converges toward a minimizer of $f$ in $O(1/k^2)$:
\end{block}
\begin{block}{Projection}
$\M{P} \in \Complexes^{N\times N}$ is a projection on a subset $\Set{S} \subseteq \Complexes^{N}$ and its indicator $\imath_{\Set{S}}$:
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\M{P} \,\V{x} = \argmin_{\V{y}} \left(\imath_{\Set{S}}(\V{y}) + \frac{1}{2}\Norm{\V{x} - \V{y}}^2\right)$ &
with $\imath_{\Set{S}}(\V{x})\left\{
\begin{array}{ll}
0 & \text{if } \V{x}\in\Set{S} \,, \\
+\infty & \text{otherwise} \,.
\end{array}
\right.$\\
$\M{P}\,\M{P} = \M{P}\,.$ & idempotent
\end{tabular*}}
{\tablinesep=1.1ex
\begin{tabular*}{\columnwidth}{@{\extracolsep{\fill}}l r }
$\Norm{\M{P} \,\V{x} - \M{P} \,\V{y}}_2 \leq \Norm{\V{x} - \V{y}}_2$ & $\Set{S}$ is convex $\Rightarrow\M{P}$ is non-expansive\\
$ \V{x}^{(k+1)} = \M{P}\left(\V{x}^{(k)} - \gamma \nabla f(\V{x}^{(k)}) \right)$ & projected gradient descent
\end{tabular*}}
\end{block}
\begin{block}{Proximity operator}
$f$: a lower semi-continuous convex function, its proximity operator is:
\begin{align*}
\prox_f(\V{y}) = \argmin_{\V{y}} \left( f(\V{y}) + \frac{1}{2}\Norm{\V{x} - \V{y}}^2\right)
\end{align*}
$ \V{p} = \prox_f(\V{x}) \Leftrightarrow \V{x} - \V{p} \in \partial f (\V{p})$ with $ \partial f (\V{p})$ the subgradient of $f$\\
$\V{x}^+ = \prox_f(\V{x}^+) \Leftrightarrow \V{x}^+=\argmin_{\V{x}} f(\V{x})$% are the minimizers of $f$.
% $\V{x}^{(k+1)} = \prox_f(\V{x}^{(k)})$ Algo du point proximal
\end{block}\vspace{-1ex}
\end{column}
\end{columns}
\end{frame}
\end{document}