Let's add such note in the kernel presentation parameter. We can add something like:

.. note::
   Be aware that `kernel="linear"` is equivalent of using a
   :class:`~sklearn.decomposition.PCA`. We recommend using
   :class:`~sklearn.decomposition.PCA` in this case such that `inverse_transform`
   will be an exact reconstruction without data loss.

Do you mean moving this note to the description of the parameter kernel? I assume we do not have to warn users about kernel="linear" as long as they do not use inverse_transform so I would suggest leaving it as it is.

Do you mean moving this note to the description of the parameter kernel?

It was what I was thinking.

I assume we do not have to warn users about kernel="linear" as long as they do not use inverse_transform so I would suggest leaving it as it is.

I would have an apriori thinking that a user might look at the parameter documentation but necessarily to the inverse_transform when creating the object.

Yes, it makes sense, but at the same time advising the users not to use kernel="linear" while using kernel="linear" as the default parameter looks very weird. I would add such advice to the description of the kernel when the default is no longer "linear".

Point taken. We can postpone after the default changes.