Why RidgeClassifier can be significantly faster than LogisticRegression with a high number of classes? In Scikit document, we can find this statement

The RidgeClassifier can be significantly faster than e.g. LogisticRegression with a high number of classes because it can compute the projection matrix $(1/(X^TX))X^T$ only once.

Whys is this so? Can it be explained intuitively ?

Having a Hessian that must be updated every time the guess at parameter estimates $\beta$ changes is a necessary requirement for achieving excellent calibration (i.e., the weighted in iteratively reweighted least squares).  Multinomial logistic regression is built for prediction, and I suspect that your stated goal of classification would be better replaced with a goal of prediction, i.e., getting predicted probabilities of class membership.  Using ordinary Newton-Raphson iteration to solve for the maximum likelihood estimate of $\beta$ converges very fast, often in only 5 iterations (5 passes at the data).  So I'm not sure how much it is worth to develop shortcuts.
