  We study the linear subspace fitting problem in the overparameterized
setting, where the estimated subspace can perfectly interpolate the training
examples. Our scope includes the least-squares solutions to subspace fitting
tasks with varying levels of supervision in the training data (i.e., the
proportion of input-output examples of the desired low-dimensional mapping) and
orthonormality of the vectors defining the learned operator. This flexible
family of problems connects standard, unsupervised subspace fitting that
enforces strict orthonormality with a corresponding regression task that is
fully supervised and does not constrain the linear operator structure. This
class of problems is defined over a supervision-orthonormality plane, where
each coordinate induces a problem instance with a unique pair of supervision
level and softness of orthonormality constraints. We explore this plane and
show that the generalization errors of the corresponding subspace fitting
problems follow double descent trends as the settings become more supervised
and less orthonormally constrained.
