You cannot hope for anything like a closed form solution, or even an exact efficient algorithm for this problem, because it is [NP-hard][1]. The reduction is from the [max-cut][2] problem. Let's look at the special case $m=1$. Let $G = (V, E)$ be a graph, and orient the edges in some arbitrary way. Let $A$ be the edge-by-vertex incidence matrix of $G$, i.e. the rows of $A$ are indexed by edges and the columns by vertices, and $a_{e, v}$ is $1$ if $v$ is the tail of $e$, $-1$ if $v$ is the head of $e$, and $0$ otherwise. Then it's easy to check that for any $x \in \mathbb{R}^V$, $\|Ax\|^2 = \sum_{(u,v) \in E}{(x_u - x_v)^2}$. If $x \in \{-1, 1\}^V$ as in the $m=1$ case of your problem, then $\|Ax\|^2 = 4e(S, \bar{S})$, where $e(S, \bar{S})$ is the number of edges cut by the set $S = \{v: x_v = 1\}$. 

You can find a constant factor approximation algorithm (due to Nesterov) in Chapter 6.3. of the [Williamson and Shmoys book][3].


  [1]: https://en.wikipedia.org/wiki/NP-hardness
  [2]: https://en.wikipedia.org/wiki/Maximum_cut
  [3]: http://www.designofapproxalgs.com/download.php