Let A any matrix. If we eigen-decompose $A^TA=HDH^T$, where $H$ is unitary and $D$ diagonal, then the columns $H_i$ of $H$ satisfy
$$\|AH_1\|^2=\max \frac{\|Ax\|^2}{\|x\|^2}$$
$$\|AH_{i+1}\|^2=\max_{x \perp span(H_1, \dots H_i)}\frac{\|Ax\|^2}{\|x\|^2},\quad i=1,\dots,n-1.$$ (see http://en.wikipedia.org/wiki/Principal_components_analysis)
This suggests that for every $1\leq k\leq n$
$$\sum_{i=1}^{k}\|AH_i\|^2=\max_{(x_i)_{i=1\dots k} ONB}\sum_{i=1}^{k}\|Ax_i\|^2$$
However, I have difficulties finding a proof.