Vector with minimal angle to a set of vectors

[Obelus]

Let $\mathbb X$ be a set of mean-shifted vectors $\mathbf X_1, \mathbf X_2, \ldots, \mathbf X_k$ in $\mathbb R^n$. By mean-shifted, I mean that $\text{Mean}(\mathbf X_i) = 0$ $\forall i \in {0, \ldots, k}$. I want to find a vector $\mathbf T \in \mathbb R^n$ that maximizes the function

$f(\mathbf T) = \sum \limits_{\mathbf X \in \mathbb X} \big ( cos(\theta_{\mathbf X, \mathbf T}) \big )^2 = \sum \limits_{\mathbf X \in \mathbb X} \big ( \frac{\mathbf X \cdot \mathbf T}{||\mathbf X|| \cdot ||\mathbf T||} \big )^2$

(Edit: maximizing this $f$ does not actually get what I want. Instead, I want to maximize the geometric mean of the correlations - see my followup posts).

In other words, I want to find a vector $\mathbf T$ that has a small angle $\theta$ to each $\mathbf X \in \mathbb X$. (Actually there must be infinitely many solutions, since any linear transformation of $\mathbf T$ is also a solution).

My real motivation is statistical: this is equivalent to finding a vector that has a high Pearson correlation coefficient with all the vectors in $\mathbb X$.

I can use numerical algorithms to optimize $f$, but I would prefer to find a closed-form solution. This problem is part of a larger algorithm that I am trying to speed up, and it spends most of its time trying to find $T$. Is a closed-form solution possible? Or is this problem similar to an existing problem, which I can adapt?

If $k=2$, the solution should just be to rotate $\mathbf X_1$ halfway to $\mathbf X_2$, using some multidimensional rotation as discussed in http://forums.xkcd.com/viewtopic.php?f=17&t=29603". Perhaps this rotation concept could be extended to find the "rotational average" of $\mathbb X$, if that makes sense?

Any help towards finding a closed-form solution to this problem would be appreciated.

 

[Obelus]

Actually, I just realized that $f$ in my previous post is not exactly what I want to maximize. I want $T$ to be equally and maximally close to all vectors in $\mathbb X$, so maximizing the minimum correlation with $T$ would be better:

$f = \displaystyle{\mathop{\mbox{min}}_{\mathbf X \in \mathbb X}} \big ( \frac{\mathbf X \cdot \<br /> \mathbf T}{||\mathbf X|| \cdot ||\mathbf T||} \big )^2$

However, this makes $f$ non-differentiable, so perhaps using the http://mathoverflow.net/questions/35191/a-differentiable-approximation-to-the-minimum-function" would work:

$f = \big ( \frac{1}{|\mathbb X|}\sum\limits_{\mathbf X \in \mathbb X} \frac{\mathbf X \cdot \<br /> \mathbf T}{||\mathbf X|| \cdot ||\mathbf T||} \big )^{\frac{1}{k}}$

for $k \rightarrow - \infty$.

 

[Obelus]

Argh, actually that's not what I want either. If one vector in $\mathbb X$ is very different from the others, it will pull $\mathbf T$ towards it too much. Instead, I think I want to maximize the geometric mean, or the limit as $k \rightarrow 0$ of the generalized mean.

Sorry I keep changing my mind. Basically, I want $\mathbf T$ to be a good angular approximation of all the vectors in $\mathbb X$.