Options for 3D coordinate systems?

I'm trying to solve biochemistry problems (think protein folding) with DNNs. Are there 2D / 3D coordinate systems that are particularly well suited for deep neural networks (DNNs) to process?

For example, if we were training a DNN to predict gravitational attraction between two objects, cartesian coordinates would presumably do worse than polar coordinates, because the key value to calculate gravitation attraction is distance, a value directly expressed in polar coordinates but would require a DNN to learn the Pythagorean rule when using cartesian coordinates.

As another example, arguably the positional embeddings based on sine and cosine used in early Transformers was an interesting "coordinate system" in 1D.

Another answer might be not to worry about it - that DNNs will figure it out no matter what coordinate system (within reason) you pick.

2 Answers

Along with the positions of the amino acids, protein folding can also be described by the angles in the peptide bonds such as used in a Ramachandran plot, and with global values such as the radius of gyration.

You could apply all those values together in a single model such as is being done with QSAR models to predict physicochemical properties of molecules.

• Thanks. It's sounds useful to include the radius of gyration between the residue to the left and right of a residue. I think it's probably too hard on a NN to ONLY have this info, since figuring out the relative locations of the first and last residue would require calculating relative radii of gyrations all the way down the protein. Overall, my suspicion right now is to include as many coordinates systems as possible and let the NN throw away any information it can't use. Commented Mar 27, 2023 at 15:54

If we know something about the problem we're trying to model, then representing that knowledge to the model via a good set of features is incredibly valuable. This is often called "," and it can yield dramatic improvements in model quality. Sextus's answer provides an example, in the context of protien folding.

But there's no single "best representation" of arbitrary data, because not all problems have the same underlying mechanics. In other words, even if you do an experiment and validate that polar coordinates are more useful than Cartesian on one task, there's surely a different task where you can demonstrate the opposite result.

Another layer of nuance is considering how many model parameters are required to achieve the desired result. It may be the case that having a more refined feature representation allows one to achieve similar results using a smaller network (measured by the number of parameters).