Functional / Linear fields

[opeltre]

Instead of storing the energy function as a tensor, make it possible to compute it via a function e.g. linear or affine functional

Necessary for true convolutional layers (translation invariant)

Necessary for quotients (observable on a class of vertices lies in a low dimensional subspace)

Example usecase: images and MNIST dataset

• each image may be viewed as a global system with observables restricted to sums of local magnetic fields (bias vector)
• each layer may be viewed as a pair of systems with observables restricted to pairwise magnetic interactions (weight matrix)

This approach would make it much closer to torch or keras interfaces when piling up layers.

[opeltre]

An observable is a function over a fiber, represented by its bitrange.
It can hence either be represented by:

• a floating tensor over the bitrange
• a functional from any point of the bitrange to a number

N.B. this is already the case when factorising global observables as sums of local observables

Microstate -> Product of local microstates -> Sum of local energy values

This implies to harmonise global observable interface with field instances, to some extent. (e.g. eval method, but no integration possible)

Grouping of local observables is very much related to system quotients

[opeltre]

Implement projection to interaction subspaces via fft, as representation of homology classes of potentials (cocycles / supplements of boundaries in higher degrees) first

[opeltre]

See also 666ad17 draft implementation of deep RBMs / deep NNs, best example where weight/bias representation is efficient