Neurocat
Neurocat is an experimental toy library studying 2 things:

The link between category theory and supervised learning algorithm and neural networks through the concepts described in this amazing paper Backprop as Functor: A compositional perspective on supervised learning which tries to unify (partly at least) category theory & supervised learning concepts & simple neural networks/

how to represent matrices (and thus neural networks) which dimensions are checked at compiletime using the new feature singletontypes described in SIP23 which allows to manipulate the integer
3
as the value3
but also the type3
. My matrix experimentations use the little & really cool library singletonops by Frank S. Thomas the author of the greatrefined
library too. ShapelessNat
is a nice idea but not good for big naturals because this is a recursive structure (down to 0) checked at compiletime so100000
as
Very superficially, the idea of the paper is quite simple (minus a few details):
 A supervised learning algorithm can be seen as a structure able to approximate a function
A > B
relying on parametersP
which are updated through an optimization/training process using a set of training samples.  This paper shows that the set of supervised learning algorithms equipped with 3 functions (implement, updateparams, requestinput) forms a symmetric monoidal category
Learn
and then demonstrates that supervised learning algorithms can be composed  It also shows that there exists a Functor from the category
ParaFn
of parametrised functionsP > A > B
toLearn
category.
ParaFn > Learn
 Then it shows that a (trained) neural network can be seen as an approximation of a function
InputLayer > OutputLayer
parametrised by the weightsW
.  Thus it demonstrates there is also a Functor from the category of neural network (W, InputLayer, OutputLayer) to the category of parametrised functions (W > InputLayer > OutputLayer)
I: NNet > ParaFn
 By simple functor composition, you have then a Functor from neural networks to supervised learning algorithms:
NNet > Learn : (ParaFn > Learn) ∘ (NNet > ParaFn)
I'll stop there for now but my work has just started and there are more concepts about the bimonoidal aspects of neural networks under euclidean space constraints and pending studies about recurrent networks and more.
Discovering that formulation, I just said: "Whoaaa that's cool, exactly what I had in mind without being able to put words on it".
Why? Because everything I've seen about neural networks looks like programming from the 70s, not like I program nowadays with Functional Programming, types & categories.
This starts unifying concepts and is exactly the reason of being of category theory in maths. I think programming learning algorithms will change a lot in the future exactly as programming backends changed a lot those last 10 years.
I'm just scratching the surface of all of those concepts. I'm not a NeuralNetwork expert at all neither a good mathematician so I just want to open this field of study in a language which now has singletontypes allowing really cool new ways of manipulating data structures
So first, have a look at this sample:
 Basic Compiletime Matrix calculus: https://github.com/mandubian/neurocat/blob/master/src/test/scala/MatTest.scala#L15L47
 Neural network layers transformed into Learn instances and then composed and trained: https://github.com/mandubian/neurocat/blob/master/src/test/scala/NNetTest.scala#L40L129
 Neural network + Huge Matrices that compiles in human time: https://github.com/mandubian/neurocat/blob/master/src/test/scala/NNetTest.scala#L131L163
For info, to manipulate matrices, I used ND4J to have an array abstraction to test both in CPU or GPU mode but any library doing this could be used naturally.