  We present a simple picture of the training process of joint embedding
self-supervised learning methods. We find that these methods learn their
high-dimensional embeddings one dimension at a time in a sequence of discrete,
well-separated steps. We arrive at this conclusion via the study of a
linearized model of Barlow Twins applicable to the case in which the trained
network is infinitely wide. We solve the training dynamics of this model from
small initialization, finding that the model learns the top eigenmodes of a
certain contrastive kernel in a stepwise fashion, and obtain a closed-form
expression for the final learned representations. Remarkably, we then see the
same stepwise learning phenomenon when training deep ResNets using the Barlow
Twins, SimCLR, and VICReg losses. Our theory suggests that, just as kernel
regression can be thought of as a model of supervised learning, kernel PCA may
serve as a useful model of self-supervised learning.
