Subspace Fitting Meets Regression: The Effects of Supervision and Orthonormality Constraints on Double Descent of Generalization Errors
This work provides theoretical insights into generalization phenomena for subspace fitting, which is incremental but clarifies connections between unsupervised and supervised learning tasks.
The paper investigates linear subspace fitting in overparameterized regimes, analyzing how generalization errors exhibit double descent trends as supervision increases and orthonormality constraints soften.
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.