An introduction to graphical tensor notation for mechanistic interpretability
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 exampleinduction head'' circuit in graphical tensor notation.