  We propose a novel theoretical framework to understand contrastive
self-supervised learning (SSL) methods that employ dual pairs of deep ReLU
networks (e.g., SimCLR). First, we prove that in each SGD update of SimCLR with
various loss functions, including simple contrastive loss, soft Triplet loss
and InfoNCE loss, the weights at each layer are updated by a \emph{covariance
operator} that specifically amplifies initial random selectivities that vary
across data samples but survive averages over data augmentations. To further
study what role the covariance operator plays and which features are learned in
such a process, we model data generation and augmentation processes through a
\emph{hierarchical latent tree model} (HLTM) and prove that the hidden neurons
of deep ReLU networks can learn the latent variables in HLTM, despite the fact
that the network receives \emph{no direct supervision} from these unobserved
latent variables. This leads to a provable emergence of hierarchical features
through the amplification of initially random selectivities through contrastive
SSL. Extensive numerical studies justify our theoretical findings. Code is
released in https://github.com/facebookresearch/luckmatters/tree/master/ssl.
