Question regarding the projection view of PCA

[Jiaji Zhou]

In class, we are taught that PCA can be formulated as minimizing projection residual. 

Here are two formulations:

1) Without centering (zero mean shift)

   min sum_i (|| W*W^T*x_i - x_i ||^2) 

   s.t. W^T*W = I

,where x_i is the ith feature vector, W is d*k, where k is the number of PCs we want. 

2) With mean shift, I guess the optimization could be formulated as

   min sum_i (|| W*W^T*(x_i - m) + m - x_i ||^2 

  s.t. W^T*W = I

,where both W and m is the optimization variable. If we formulate the lagrangian and stationary condition would give us that m is the mean of x_i, and then it's basically

a PCA with centered data (mean shift to zero)

But I think the second formulation is weird, think of the case that most data comes from the same vector v with tiny little pertubations (v + eps), then doing PCA will give us useless PCs which acts as a characterization of noise. On the other hand, the 

first formulation gives the first PC as v.

Thanks!