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

[itex]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[/itex]

(Edit: maximizing this [itex]f[/itex] 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 [itex]\mathbf T[/itex] that has a small angle [itex]\theta[/itex] to each [itex]\mathbf X \in \mathbb X[/itex]. (Actually there must be infinitely many solutions, since any linear transformation of [itex]\mathbf T[/itex] 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 [itex]\mathbb X[/itex].

I can use numerical algorithms to optimize [itex]f[/itex], 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 [itex]T[/itex]. Is a closed-form solution possible? Or is this problem similar to an existing problem, which I can adapt?

If [itex]k=2[/itex], the solution should just be to rotate [itex]\mathbf X_1[/itex] halfway to [itex]\mathbf X_2[/itex], 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 [itex]\mathbb X[/itex], if that makes sense?

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

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# Vector with minimal angle to a set of vectors

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