As is known, the rank-1 PCA aims to solve the following optimization problem
$$\min\_{x\in\mathbb{R}^d}\quad -x^T \Sigma x\quad\quad\quad \text{s.t.}\quad \Vert x\Vert\_{2}=1,$$
where $\Sigma\in\mathbb{S}^{d}$ is the covariance matrix. Thus the optimum $x^\*$ of the PCA problem is the top unit eigenvector of $\Sigma$. Given an approximation $\tilde{x}$ (normalized), the error between the $\tilde{x}$ and $x^\*$ is measured by the sine function
$$\sin^{2}(\tilde{x}, x^\*) = 1-(\tilde{x}^T x^\*)^2.$$
I was wondering does there exist any analytical relationship between the objective function $\tilde{x}^\top\Sigma \tilde{x}$ and the error $\sin^2(\tilde{x}, x^\*)$?
Any help appreciated.



PCA, relation between the error and variance
$\newcommand{\Si}{\Sigma}\newcommand{\R}{\mathbb R}\newcommand{\la}{\lambda}$Let us work in an orthonormal eigenbasis of $\Si$. Then without loss of generality $\Si$ is the diagonal matrix with diagonal entries $\la\_1\ge\la\_2\ge\cdots\ge\la\_d\ge0$, $x\_\*:=x^\*=[1,0,\dots,0]^T$, $x:=\tilde x=[x\_1,x\_2,\dots,x\_d]\in\R^d$, $\|x\|\_2=1$, $x\_\*^T\Si x\_\*=\la\_1$, $x^T\Si x=\sum\_1^d\la\_j x\_j^2$,
\begin{equation}
 \sin^2(x, x\_\*)=1-(x^Tx\_\*)^2=1-x\_1^2, 
\end{equation}
and
\begin{equation}
 0\le x\_\*^T\Si x\_\*-x^T\Si x=\la\_1(1-x\_1^2)-\sum\_2^d\la\_j x\_j^2\le\la\_1(1-x\_1^2)=\|\Si\|\sin^2(x,x\_\*),
\end{equation}
where $\|\Si\|:=\la\_1$, the operator/spectral norm of $\Si$. Thus, the nonnegative error $x\_\*^T\Si x\_\*-x^T\Si x$ in the value of the objective function is bounded by the norm of $\Si$ times the error $\sin^2(x,x\_\*)$.


