Homogeneous Projection
If $\mathbf{ d } $ is a continuous dilation in $ \mathbb{ R } ^n$ then for any $x\in \mathbb{ R } ^n\backslash\{\mathbf{ 0} \}$ there exist $s\in \mathbb{ R } $ and $z\in S$ such that
$z=\mathbf{ d } (s)x.$
The corresponding point $z\in S$ is called a $\textit{homogeneous projection}$ of $x$ on the unit sphere $S$. If $\mathbf{ d } $ is a monotone dilation then homogeneous projection is unique.
Computation of homogeneous projection
The function ${\color{red} { \texttt{hproj }} } $ computes $\mathbf{d}$-homogeneous projection of a vector $x$ on the unit sphere $x^{\top}Px=1$ for a linear monotone dilation $\mathbf{d}$.
- $\textbf{Input parameters}: {\color{blue}x}\text{ (vector to be projected)}, {\color{blue}{G_{\mathbf{ d } } } }\text{ (generator of linear dilation)}, {\color{blue}P} \text{ (a shape matrix of the weighted Euclidean norm)}$
- $ \textbf{Output parameters}: {\color{magenta}{z}}$ (homogeneous projection of $x$)
${\color{red} { \texttt{demo_hproj.m}} } $
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Example of compuation of homogeneous projection %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tol=1e-6; % compuational tolerance %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Shape matrix of the wighted Euclidean norm %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P=[1 -1; -1 2]; if norm(P-P')>tol disp('Error: shape matrix P must be symmetric'); return; end; if min(real(eig(P)))< tol disp('Error: shape matrix P must be positive definite'); return; end; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Generator of the linear dilation in R^2 %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Gd=[5 0; 1 3]; %monotonicy check if min(real(eig(P*Gd+Gd'*P)))< tol disp('Error: dilation must be monotone'); return; end; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Computation of homogeneous projection %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% x=[0.5;-1.5]; % vector x in R^2 disp(['Vector = [', num2str(x(1)),';',num2str(x(2)),']']); z=hproj(x,Gd,@(x)hnorm(x,Gd,P)); % homogeneous projection of x % on the unit sphere x'*P*x=1 % computed by bisection method disp(['Hom. projection (bisection method)= [', num2str(z(1),'%.8f'),';',... num2str(z(2),'%.8f'),']']); z=hproj(x,Gd,@(x)hnorm_newton(x,Gd,P)); % homogeneous projection of x % on the unit sphere x'*P*x=1 % computed by Newton method disp(['Hom. projection (Newton method)= [', num2str(z(1),'%.8f'),';',... num2str(z(2),'%.8f'),']']);