After learning about tensor diagrams a few months ago, they're my default notation for tensors. I liked your chart and also Jordan Taylor's diagram for multi-he

After learning about tensor diagrams a few months ago, they're my default notation for tensors. I liked your chart and also Jordan Taylor's diagram for multi-head attention. Some notes for other readers seeing this for the first time: My favorite property of that these diagrams is that they make it easy to re-interpret a multilinear expression as a multilinear function of any of its variables. For example, in standard matrix notation you'd write x^T A x to get a quadratic form with respect to the variable x. I think most people read this either left to right or right to left: take a matrix-vector product, and then take an inner product between vectors. Tensor notation is more like prolog: the diagram x - A - x involves these two indices/variables (the lines) "bound" by three tensors/relations (A and two copies of x.) That framing makes it easier to think about the expression as a function of A: it's just a "Frobenius inner product" between -A- and the tensor product -x x-. The same thing happens with the inner product between a signal and a convolution of two other signals. In standard notation it might take a little thought to remember how to differentiate <x, y * z> with respect to y (<x, y * z> = <y, x * z'> where x' is a time-reversal), but thinking with a tensor diagram reminds you to focus on the relation x = y + z (a 3-dimensional tensor) constraining the indices x, y and z of your three signals. All of this becomes increasingly critical when you have more indices involved. For example, how can you write the flattened matrix vec(AX + XB) as a matrix-vector product of vec(X) so we can solve the equation AX + XB = C? (Example stolen from your book.) I still have to get a hold of all the rules for dealing with non-linearities ("bubbles") though. I'll have to take a look at your tensor cookbook :) I'm also sad that I can't write tensor diagrams easily in my digital notes. Tensor diagrams are algebraically the same thing as factor graphs in probability theory. (Tensors correspond to factors and indices correspond to variables.) The only difference is that factors in probability theory need to be non-negative. You can define a contraction over indices for tensors taking values in any semiring though. The max-plus semiring gives you maximum log-likelihood problems, and so on.