Let
$$
x_1, x_2, \dots x_n \in\mathbb{R}^m
$$
find
$\hat x \in \mathbb{R}^m$
such that
$$
\operatorname{min} \sum_{k=1}^n |\theta_k|
$$
where
$$
\operatorname{cos} \theta_k = \frac{x_k\cdot \hat x}{||x_k||\cdot||\hat x||}
$$
cos defines $\forall x_j,x_k \in\mathbb{R}^m$
$$
\operatorname{cos} \theta = \frac{x_j}{||x_j||}\cdot \frac{x_k}{||x_k||}
$$
Equivalent form
$$
\operatorname{min} -\sum_{k=1}^n \frac{x_k\cdot \hat x}{||x_k||\cdot||\hat x||}
= \operatorname{min} -\frac{1}{c}\sum_{k=1}^n \frac{1}{||x_k||} \left(\sum_{l=1}^m x_{kl}\cdot \hat x_l\right)
$$
under condition
$$
\sum_{l=1}^{m} (\hat x_l)^2 = c^2
$$
where $c\in\mathbb{R}, c>0.$
$$
\mathcal{L}(\hat x_1,\hat x_2,\dots,\hat x_m,\lambda)
= -\frac{1}{c}\sum_{k=1}^n \frac{1}{||x_k||} \left(\sum_{l=1}^m x_{kl}\cdot \hat x_l\right)
+\lambda\left(
\sum_{l=1}^{m} (\hat x_l)^2 - c^2
\right)
$$
For $1\le l \le n$, let
$$
\frac{\partial \mathcal{L}(\hat x_1,\hat x_2,\dots,\hat x_m,\lambda)}{\partial \hat x_l}
= -\frac{1}{c}\sum_{k=1}^n \frac{x_{kl}}{||x_k||}
+2\lambda \hat x_l = 0
$$
then
$$
\hat x_l = \frac{1}{2\lambda c}\sum_{k=1}^n \frac{x_{kl}}{||x_k||}
$$
replace $\hat x_l$ with above expression
$$
\sum_{l=1}^{m} \left(
\frac{1}{2\lambda c}\sum_{k=1}^n \frac{x_{kl}}{||x_k||}
\right)^2 = c^2
$$
then
$$
2\lambda c =
\frac{1}{c}
\left(
\sum_{l=1}^{m}
\left(
\sum_{k=1}^n \frac{x_{kl}}{||x_k||}
\right)^2
\right)^{1/2}
$$
$$
\hat x_l = c\cdot
\left(
\sum_{k=1}^n \frac{x_{kl}}{||x_k||}
\right)
\left(
\sum_{l=1}^{m}
\left(
\sum_{k=1}^n \frac{x_{kl}}{||x_k||}
\right)^2
\right)^{-1/2}
$$
For $||x_k|| = 1$ and $c= 1$,
$$
\hat x_l =
\left(
\sum_{k=1}^n {x_{kl}}
\right)
\left(
\sum_{l=1}^{m}
\left(
\sum_{k=1}^n {x_{kl}}
\right)^2
\right)^{-1/2} =
\bar x_l
\left(
\sum_{l=1}^{m}
{\bar x_l}^2
\right)^{-1/2}
= \frac{\bar x_l}{||\bar x||}
$$
$$
\hat x
= \frac{\bar x}{||\bar x||}
$$
$$
||x_j - x_k||^2 = x_j^Tx_j - 2 x_j^Tx_k + x_k^Tx_k
$$
$$
||x_j - x_k||^2 = 2(1 - x_j^Tx_k)
$$