Implement generalised decomposition

[davidelegacci]

Generalized PINV is already implemented in Homomorphism class. Decomposition takes place in Game class, inner products are defined in GameFull class.

• Merge classes so that inner products are available in basic Game class
• perform decomposition using the generalized PINV everywhere rather than the Euclidean one

🔥 Be careful to use it both in Game space, for the notion of normalization, and in flow space, for the exact/harmonic decomposition.

• once implemented, decompose random game, and check with generalized harmonic checker whether it is harmonic wrt prescribed measure

[davidelegacci]

Done some partial progress. Merged Game and GameFull class so that now, in the same class

1. inner products in C0, C1, C0N are defined given a harmonic measure $\mu$
2. these inner products are used to build the harmonic operator; the harmonic games are its kernel. The codes computes the kernel and generates a rando harmonic game
3. This is consistent: the produced game is $\mu$-harmonic, as can be checked feeding the game to the code itself, or to independent checkers based on the master harmonic equation (see jupyter files in this repo)
4. now, it would be nice that also the decomposition is consistent with the harmonic measure
5. the measure enters the decomposition not via the harmonic operator, but via the projection operators, more precisely via the generalised pseudo-inverse
6. currently the pseudo-inverse is defined wrt the generalized inner product, but the result is not consistent: given and decomposed random game into $u = u_p + u_h + u_k$, and feeding $u_h$ back to the code, it turns out that $u_h$ is not harmonic.

Possible issues to look into

• see exactly where $\mu$ enters decomposition. Only in pinv or also elsewhere? Abdou et al make $\mu$ enter also the deviation map itself. I don't like it and prefer to let $D$ be natural and have the measure enter only its pseudo-inverse. See if it's possible to tweak the $\mu$ dependence.
• there is probably confusion about the inner products being up or down. The inner product in C1 should be fine but the one in C0 and C0N probably need attention.