\[\newcommand{\mx}[1]{\mathbf{#1}}
\def\pone{\mathbf{\mathbf{\phi}}_{1}}
\def\ponet{\mathbf{\mathbf{\phi}}_{1}^{T}}
\def\phat{\hat{\phi}_{1}}
\def\xi{\mathbf{x}_{i}}
\def\xit{\mathbf{x_{i}}^{T}}
\def\X{\mathbf{X}}
\def\Xt{\mathbf{X}^{T}}
\def\Q{Q(\mathbf{\phi}_{1})}
\def\A{\mathbf{A}}
\def\U{\mathbf{U}}
\def\V{\mathbf{V}}
\def\Vt{\mathbf{V}^{T}}
\def\I{\mathbf{I}}
\def\L{\mathbf{\Lambda}}
\def\l{\mathbf{\lambda}}
\def\v{\mathbf{v}}
\def\sumn{\sum_{i=1}^{n}}
\def\sumj{\sum_{j=1}^{p}}
\def\summ{\sum_{m=1}^{p}}
\def\argmax{\mathrm{arg\, max}}
\def\argmin{\mathrm{arg\, min}}
\def\argminphii{\underset{\left \| \pone \right \|=1}{\argmin}}
\def\argmaxphii{\underset{\left \| \pone \right \|=1}{\argmax}}
\def\argmaxvphii{\underset{\left \| \V \pone \right \|=1}{\argmax}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% Diagnoal matrix Lambda
\def\Lmat{\begin{bmatrix}
\l_{1} & & & \\ & \l_{2} & & \\ & & \ddots & \\ & & & \l_{p}
\end{bmatrix}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% Transposed phi_1 * v
\def\phiv{\begin{bmatrix}
\pone \cdot\v_{1} \\ \pone \cdot\v_{2} \\ \vdots \\ \pone \cdot\v_{p} \\ \end{bmatrix}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% phi_1 * v
\def\phivc{\begin{bmatrix} \pone\cdot\v_{1} & \pone\cdot\v_{2} & \cdots & \pone\cdot\v_{p} \end{bmatrix}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% lambda * (phi_1 * v)
\def\phivl{\begin{bmatrix}
\l_{1} \pone \cdot\v_{1} & \l_{2} \pone \cdot\v_{2} & \cdots & \l_{p} \pone \cdot\v_{p} \end{bmatrix}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% lambda * (phi_1 * v)^2
\def\phivls{\l_{1} (\pone \cdot \v_{1})^{2} + \l_{2} (\pone \cdot \v_{2})^{2} + \cdots \l_{p} (\pone \cdot \v_{p})^{2}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% Transposed V
\def\Vmatt{\begin{bmatrix}
\v_{1}^{T} \\ \v_{2}^{T} \\ \vdots \\ \v_{p}^{T} \\ \end{bmatrix}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% V
\def\Vmat{\begin{bmatrix} \v_{1} & \v_{2} & \cdots & \v_{p} \end{bmatrix}}\]
Last updated on
Jan 17, 2021
16 min read