In some work I am doing right now, I am trying to solve a variation of the standard least squares linear regression. Suppose I have a set [; Y ;] of data vectors [; y ;]. Instead of minimizing the residual of some linear estimator, [; \sum_{y \in Y}||y - A x||^2 ;], I'm trying to minimize the squared difference between each data vector and it's projection onto the span of a single vector, [; \sum_{y \in Y}||y - dd^t y||^2 ;], where [; d ;] is a unit vector. That is, I need to solve
[; d^* = argmin_d \sum_{y \in Y}||y - dd^t y||^2 ;]
subject to [; ||d||^2 = 1 ;]
I would be very surprised if this wasn't already solved, but for some reason I am having trouble finding it and I can't quite seem to figure it out myself. Can anyone point me in the right direction?
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