Github user iyounus commented on the pull request:
https://github.com/apache/spark/pull/10702#issuecomment-178134675
For the case (3), I'm assuming that the label and features are not
standardized. So, in that case, the solution exists. Here is my perspective on
this.
The normal equation `X^T X \beta = X^T y` has a unique solution if matrix
`X` is full rank. If its not full rank, the `X^T X` becomes singular and hence
not invertible. With regularization, the equation `(X^T X + \lambda I) \beta =
X^T y` still has unique solution if the matrix `(X^T X + \lambda I)` is
invertible. This is true even if elements of `y` are constant. For the sample
data I'm using above, I can solve this equation by hand and obtain `\beta`
(with and without intercept). If I'm using any software, it must reproduce
these results. Note that standardization is completely independent operation.
Its not part of normal equation. So, if the user demands no standardization,
then the any software should produce the analytical solution.
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