  Self-supervised learning (SSL) has empirically shown its data representation
learnability in many downstream tasks. There are only a few theoretical works
on data representation learnability, and many of those focus on final data
representation, treating the nonlinear neural network as a ``black box".
However, the accurate learning results of neural networks are crucial for
describing the data distribution features learned by SSL models. Our paper is
the first to analyze the learning results of the nonlinear SSL model
accurately. We consider a toy data distribution that contains two features: the
label-related feature and the hidden feature. Unlike previous linear setting
work that depends on closed-form solutions, we use the gradient descent
algorithm to train a 1-layer nonlinear SSL model with a certain initialization
region and prove that the model converges to a local minimum. Furthermore,
different from the complex iterative analysis, we propose a new analysis
process which uses the exact version of Inverse Function Theorem to accurately
describe the features learned by the local minimum. With this local minimum, we
prove that the nonlinear SSL model can capture the label-related feature and
hidden feature at the same time. In contrast, the nonlinear supervised learning
(SL) model can only learn the label-related feature. We also present the
learning processes and results of the nonlinear SSL and SL model via simulation
experiments.
