## LESSWRONGLW

ML-related math trick: I find it easier to imagine a 4D tensor, say of dimensions , as a big matrix with dimensions within which are nested matrices of dimensions . The nice thing about this is, at least for me, it makes it easier to imagine applying operations over the matrices in parallel, which is something I've had to thing about a number of times doing ML-related programming, e.g. trying to figure out how write the code to apply a 1D convolution-like operation to an entire batch in parallel.

1crabman8moI've been studying tensor decompositions and approximate tensor formats for half a year. Since I've learned about tensor networks, I've noticed that I can draw them to figure out how to code some linear operations on tensors. Once I used this to figure out how to implement backward method of some simple neural network layer (not something novel, it was for the sake of learning how deep learning frameworks work). Another time I needed to figure out how to implement forward method for a Conv2d layer with weights tensor in CP format. After drawing its output as a tensor network diagram, it was clear that I could just do a sequence of 3 Conv2d layers: pointwise, depthwise, pointwise. I am not saying that you should learn tensor networks, it's probably a lot of buck for not too large bang unless you want to work with tensor decompositions and formats.
1NaiveTortoise8moFrom cursory Googling, it looks like tensor networks are mostly used for understanding quantum systems. I'm not opposed to learning about them, but is there a good resource you can point me to that introduces them independent of the physics concepts? Were you learning them for use in physics? For example, have you happened to read this Google AI paper [https://arxiv.org/abs/1905.01330] introducing their TensorNetworks library and giving an overview?

Unfortunately I don't know any quantum stuff. I learned them for machine learning purposes.

A monograph by Cichocki et al. (part 1, part 2) is an overview of how tensor decompositions, tensor formats, and tensor networks can be used in machine learning and signal processing. I think it lacks some applications, including acceleration and compression of neural networks by compression of weights of layers using tensor decompositions (this also sometimes improves accuracy, probably by reducing overfit).

Tensor decompositions and Applications by Kolda, Bader 2009