  Graphical tensor notation is a simple way of denoting linear operations on
tensors, originating from physics. Modern deep learning consists almost
entirely of operations on or between tensors, so easily understanding tensor
operations is quite important for understanding these systems. This is
especially true when attempting to reverse-engineer the algorithms learned by a
neural network in order to understand its behavior: a field known as
mechanistic interpretability. It's often easy to get confused about which
operations are happening between tensors and lose sight of the overall
structure, but graphical tensor notation makes it easier to parse things at a
glance and see interesting equivalences. The first half of this document
introduces the notation and applies it to some decompositions (SVD, CP, Tucker,
and tensor network decompositions), while the second half applies it to some
existing some foundational approaches for mechanistically understanding
language models, loosely following ``A Mathematical Framework for Transformer
Circuits'', then constructing an example ``induction head'' circuit in
graphical tensor notation.
