Aidan Rocke

Taking a few days off Twitter to work out the solution to an interesting problem. Another day, another challenge.

I would include motor control researchers in 'control theory'. :)

cc: @GunnarBlohm, @harrison_ritz, @JonAMichaels, @OlivierCodol

There also appears to be very little overlap between the causal inference community(dominated by statisticians) and the internal modelling community dominated by control theorists. Should control theorists create their own subfield of counterfactual control? Does it exist?

I see many natural connections between internal models and counterfactual reasoning. However, while theories of internal modelling rely heavily on inverse models, I don’t see much in causal inference review papers on inverse models. Have I overlooked something?

2. This is also relevant to scientists that want stable internal models for deep neural networks since a deep network is an exponentially large ensemble of linear models with compact support. For details on this last point, my article on the 26th of January offers more details.

1. Here I present an elementary proof for a classical result in random matrix theory that applies to any random matrix sampled from a continuous distribution. One of its many important consequences is that almost all linear models with square Jacobian matrices are invertible.

One useful insight is that the dot product XY may be viewed as a random walk on the real line where nonzero step lengths have constant expected magnitude and zero step lengths are a measure zero event. Also, positive and negative steps occur with equal probability.

By solving this simple case, you have the main insight required to solve the case of vectors sampled from an isotropic Gaussian. Assuming that your students know the central limit theorem you can use clever insights instead of doing cumbersome calculations.

@KordingLab regarding your orthogonal random vectors question I find it useful to analyse an illustrative special case X ~ U([-1,+1])^n because you end up with an expression where you can easily apply the central limit theorem.

'Fractional Calculus and Variational Mechanics' //http://large.stanford.edu/courses/2007/ph210/noriega2/ … An explanation of how the fractional calculus allows an extension of the Lagrangian to non-conservative systems.

@ampanmdagaba, @NoahGuzman14 do you guys think this can work for problems in theoretical neuroscience? I mean interesting problems which won't be worked out in the next 10 years otherwise as they simultaneously require a combination of skills and solving a coordination problem.

Note: I think this might be one of the most interesting open problems in machine learning and neural information processing, unless a theoretical neuroscientist has already adequately addressed this question in a slightly different setting.

Finally, we are all on Twitter to exchange ideas and not one-up each other so I hope everyone feels free to share their perspective. :)

I haven't seen this question properly formulated anywhere so this represents my attempt. From my discussions with an applied topologist it has yet to be properly addressed. I also highly doubt that this is one of those problems where there will be a single ‘eureka’ moment. ;)