A Tensor is a mathematical object for symbolic manipulation of relationships between other objects that belong in conceptually similar universes or spaces. Literature on Deep Learning, like Goodfellow, call for the CS-minded reader to just assume it's a fancy word for a matrix of more than two dimensions. That makes matters confusing because mathematically you could have scalar or vectorial tensors. The classic mathematical definition puts more restrictions on the "shape" of the "matrix" by requiring certain properties so that it can create a symbolic language for a tensor calculus. Then you can study how the relationships change as variables on the universes or spaces change.
Understanding the inertia tensor on classical mechanics or the stress tensor may illustrate where tensors come from, and I understand that GR also makes use of a lot of tensor calculus that came to be as mathematics developed manipulating and talking about tensors. I have a kinda firm grasp on some very rudimentary tensor calculus from trying to leanr GR, and a pretty solid grasp on classical mechanics. I've had hour long conversations with deep learning popes and thought leaders in varying states of mind and after that my understanding is that they use the word tensor in an overreaching fashion as like you could call a solar panel a nuclear fission reactor power source. This thought leaders include people with books and 1M+ view Youtube videos on the subject that use the word tensor and I'm not saying their names because they off-the-record admitted that it's a poor choice of term but it harms the ego of many Google engineers to publicly admit that.
I'm not aware of tensor shape type safety (in general) being widely used in deep learning, let alone sum-types. I believe Pytorch and TensorFlow lack support for tensor shape type checking (looks like there is a Pytorch issue open: https://github.com/pytorch/pytorch/issues/26889).
Rust to the rescue, as it always is for safety.
(I don't know if this directly translates for tensors, I used it for manipulating point clouds and iirc it allows for arbitrary dimension containers)
I use this crate extensively and while I have gotten good results, it is not ergonomic and it is not equivalent to numpy ndarray. There are a lot of issues and work still to be done.
Just a word of caution for anyone interested in using it.
Things like windowing, slicing, views are all “supported” but have many limitations and are non intuitive. In particular, I needed an “axis shift” fn and I ended up spending hours scouring docs just to find a way to shift along an axis. When I mean by shift is a roll in numpy, ala:
https://github.com/rust-ndarray/ndarray/issues/281
Answer: you can’t with this crate. I implemented a dynamic n-dim solution myself but it uses views of integer indices that get copied to a new array, which have indexes to another flattened array in order to avoid duplication of possibly massive amounts of n-dimensional data; using the crate alone, copying all the array data would be unavoidable.
Ultimately I’ve had to make my own axis shifting and windowing mechanisms. But the crate is still a useful lib and continuing effort.
While I don’t mind getting into the weeds, these kinds of side efforts can really impact context focus so it’s just something to be aware of.
In my experience (CV) you typically want 4 dims - (height, width, channels, batch). In Julia you can do element-wise operations on two arrays if their lengths (i.e. size after flattening) are the same. Obviously dimension-wise that sort of thing won't work (what's the dot product of a 3-vector by a 6-vector? Undefined!)
But really, the whitepaper should include this sort of detail, and you should be following the white paper.
Understanding the inertia tensor on classical mechanics or the stress tensor may illustrate where tensors come from, and I understand that GR also makes use of a lot of tensor calculus that came to be as mathematics developed manipulating and talking about tensors. I have a kinda firm grasp on some very rudimentary tensor calculus from trying to leanr GR, and a pretty solid grasp on classical mechanics. I've had hour long conversations with deep learning popes and thought leaders in varying states of mind and after that my understanding is that they use the word tensor in an overreaching fashion as like you could call a solar panel a nuclear fission reactor power source. This thought leaders include people with books and 1M+ view Youtube videos on the subject that use the word tensor and I'm not saying their names because they off-the-record admitted that it's a poor choice of term but it harms the ego of many Google engineers to publicly admit that.