  Understanding the mechanisms through which neural networks extract statistics
from input-label pairs through feature learning is one of the most important
unsolved problems in supervised learning. Prior works demonstrated that the
gram matrices of the weights (the neural feature matrices, NFM) and the average
gradient outer products (AGOP) become correlated during training, in a
statement known as the neural feature ansatz (NFA). Through the NFA, the
authors introduce mapping with the AGOP as a general mechanism for neural
feature learning. However, these works do not provide a theoretical explanation
for this correlation or its origins. In this work, we further clarify the
nature of this correlation, and explain its emergence. We show that this
correlation is equivalent to alignment between the left singular structure of
the weight matrices and the newly defined pre-activation tangent features at
each layer. We further establish that the alignment is driven by the
interaction of weight changes induced by SGD with the pre-activation features,
and analyze the resulting dynamics analytically at early times in terms of
simple statistics of the inputs and labels. Finally, motivated by the
observation that the NFA is driven by this centered correlation, we introduce a
simple optimization rule that dramatically increases the NFA correlations at
any given layer and improves the quality of features learned.
