$xin{R}^{dn}$Principal Component Analysis $($PCA$)$ is a dimensionality reduction algorithm that can be thought of as projecting a dataset to a lower dimensionality using either of the following objective functions:

$1.$ Minimizing reconstruction error

$2.$ Maximizing variance of projected data

Assume we have a data set, $X $\backslash$in $\backslash$mathbb$\{$R$\}^\{$d $\backslash$times n$\}$$ that is centered with a mean of $0.$ It is comprised of $n$ data points each of dimension $d$$.$

We wish to project $x$ down to a size $k$ subspace, where $kxinR$^($d$\backslash$times n$)$ that is centered with a mean of $0.$ It is comprised of $n$ data

points each of dimension $d.$

We wish to project $x$ down to a size $k$ subspace, where $mi{n}_{Uin{R}^{dk},U^{T}U}=I{\displaystyle \underset{i=1}{\overset{n}{}}}||x_i-U{U}^T{x}_i|{|}^{2}ma{x}_{Uin{R}^{dk},U^{T}U}=IE[||U^{T}x|{|}^2]xxk.$

$We$ can specify the two objectives above $as$:

$mi{n}_{Uin{R}^{dk},U^{T}U}=I{\displaystyle \underset{i=1}{\overset{n}{}}}||x_i-U{U}^T{x}_i|{|}^2$

$ma{x}_{Uin{R}^{dk},U^{T}U}=IE[||U^{T}x|{|}^2]$

Note: $In(2)we$ treat $xas$ a random variable whose value $is$ drawn uniformly $at$ random from $x.$

Show mathematically that these two objectives are equivalent. That $is,$ show that minimizing the

reconstruction error $is$ equivalent $to$ maximizing the variance $of$ the projected data.