symmetry-generalization II

1. In 'Learning deep architectures for AI', Bengio revolves around the notion of symmetry without ever uttering the word!

Geometry appears 7 times, manifold 14 times

Generalization 80 times

Bengio insists heavily on the limitations of 'local' approaches: 100 occurrences, and opposes 'distributed representation', 51 oc.

2. The Bach team at Cachan on Vision spent time on looking for good priors. In hindsight, it passed by the deep CNN

3. The ecological rationality of Gigerenzer masks the environments symmetries

4. Mallat and al. have sought to join sequential learning and 'groups': translation, rotation, weak deformations (diffeomorphisms), CF CNN : deep symmetries

5. Many groups are Lie groups: manifolds

6. Https://en.wikipedia.org/wiki/Symmetry_(physics)

7. Let S be a system endowed with certain articulations or degrees of freedom

Any transformation T whose result is known on S:

T * S = S

provides strong constraints on (a model of) S

When T is a group, T * S is called a group operation

8. For example, since the Hamiltonian H is spherically symmetric for the system of the electron around the hydrogen nucleus, H commutes with the 3 components of the angular momentum J on the proper space E of H, so that SO ( 3) [group of 3D space rotations] on the ket 𝜓 solution of the Schrödinger equation:

 R(𝜃) * 𝜓 (x) = 𝜓(x) * R(𝜃)

We obtain (more easily) the solution 𝜓= Ylm(𝜃, 𝜙) fn(r), l = 0,1, ..., n- 1 and  m = -1, ..., l \)

9. An example of non-spatial / temporal symmetry in physics:

 a. Isospin (associated group: SU (2))

 b. Gauge symmetry: the local invariance constraint of the Dirac action implies the existence of the EM field and its interaction with the charged particles

10. On the other hand, the statistical instability of a relation y ~ x can be seen as an unknown transformation law:

given a set D on which it seems that y = 𝛃x

y and x the returns to t and t-1 of two instruments

When a new set D' is presented, we find y = 𝛃' x

Our law is therefore not invariant

In reality, we lack a dimension, or variable z, which would allow us to see that

 y = e (z) x

The line y ~ x rotates along z

In other words, the transformation of z corresponds to a transformation of the ratio y / x:

 z → z' 

 y/x → y'/x

Where does z come from ?

In the example of Stoikov, there is no coupling between activities at bid and ask, simply because the world is reduced to one market maker, not to a market maker + insider system as in Kyle85

11. The symmetries impose strong constraints and considerably reduce the field of the possible

CF for physics Zee 'fearful symmetry' (eg p209)