  We study the data-generating mechanism for reconstructive SSL to shed light
on its effectiveness. With an infinite amount of labeled samples, we provide a
sufficient and necessary condition for perfect linear approximation. The
condition reveals a full-rank component that preserves the label classes of Y,
along with a redundant component. Motivated by the condition, we propose to
approximate the redundant component by a low-rank factorization and measure the
approximation quality by introducing a new quantity $\epsilon_s$, parameterized
by the rank of factorization s. We incorporate $\epsilon_s$ into the excess
risk analysis under both linear regression and ridge regression settings, where
the latter regularization approach is to handle scenarios when the dimension of
the learned features is much larger than the number of labeled samples n for
downstream tasks. We design three stylized experiments to compare SSL with
supervised learning under different settings to support our theoretical
findings.
