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Information Geometry (ucr.edu)
144 points by dil8 on Dec 24, 2016 | hide | past | web | favorite | 11 comments

Information geometry is a very interesting research topic. In essence it allows to define a metric (and therefore a measure of distance) between probability distributions.

As a results, it has a lot of practical application in machine learning and has been use successfully for classification in Neuroscience, Radar signal processing and computer vision.

we can also note that Information Geometry can be seen as a sub-field of Riemannian Geometry, with some equivalence between metric. For example, the cannonical metric for symetric and positive definite (SPD) matrices in Riemannian geometry is actually equivalent to the metric for multivariate normal distribution obtained with Information geometry.

For some application, IG is very efficient. it has been used for multivariate time-series for classification of EEG signal and was at the center of the winning solution of 3 kaggle challenges : https://github.com/alexandrebarachant

There's also an interesting set of applications in statistical physics: many systems can be modelled by Markov chains whose rate parameters can change through time. Changing these parameters non-quasistatically (i.e. so that the system does not relax to the appropriate steady state) gives rise to distances on parameter space that are related to the amount of energy dissipated along the path through parameter space [0,1].

What's nice about this is that the derivation of a suitable metric allows us to compute trajectories that minimise quantities we care about (e.g. minimise energy dissipated), so this has clear potential to be useful. Some cool examples are in spin systems [2] and a harmonic trap [3].

For any differential geometers reading this: it seems to me that a good geometric way to think about this is as a fibre bundle, with the parameter space being the base space and the simplex being a vector bundle over it (see [4] on the simplex being a vector space).

[0] https://arxiv.org/pdf/0706.0559v2.pdf [1] https://arxiv.org/pdf/1201.4166.pdf [2] https://arxiv.org/pdf/1607.07425v1.pdf [3] http://journals.aps.org/pre/abstract/10.1103/PhysRevE.86.041... [4] https://golem.ph.utexas.edu/category/2016/06/how_the_simplex...

Skilling wrote a nice critique of information geometry that's worth reading:


The main point is that KL-divergence is not a metric, so imagining it as a distance in a space may give you some wrong intuitions. Its matrix of 2nd derivatives, the Fischer Information, works as a local metric, but then many people want to draw global pictures that try to extend this back to a global metric, which doesn't actually work.

As a complete layman reading that paper, Skilling's arguments seem quite damning of IG. In particular, Amari's fundamental assumption seems to be refuted:

"For inference, the only acceptable value for the Rényi-Tsallis parameter is α = 1, which is the correct information. That negates the generalisation to α != 1 which underlies Amari’s “α-divergences” in information geometry."

Have any IG proponents responded to or refuted Skilling's critiques? This is interesting because Shun'ichi Amari is credited amongst others with advancing the field in the 80's[1].

[1] https://en.m.wikipedia.org/wiki/Information_geometry

I've been hoping to find a rebuttal to Skilling's paper from someone who understands information geometry. Anyone here able to provide one? Or are his critiques viewed as legitimate?

Not an expert in the field. But I liked Centsov's theorem (when a metric comes from IG in the discrete case). I have not found a similar theorem for the general case. Amari's book is hard to follow. There is a serious lack of pedagogical intro to the subject, something like a starter : The main ideas + achievements of the theory + how to use it. There is something in the theory very deep but Amari just scratches the surface.

Can you imagine if GR metrics come from IG?

Have you heard about recent proposals that spacetime may emerge from quantum information measures? https://www.quantamagazine.org/20150428-how-quantum-pairs-st...

In the article, entanglement is a precise mathematical notion that is like an information based metric measuring the distance of a quantum state tensor (wavefunction) from the manifold of rank 1 tensors.

Nope, thank you miles7

If you have more material, just put it for all.

Much of this is due to Amari, who got really into merging theoretical neuroscience and IG, e.g. http://www.mitpressjournals.org/doi/abs/10.1162/089976602602...

Some interesting discussion here, particularly the refs in (6): http://mathoverflow.net/questions/215984/research-situation-...

From out of left field, or thereabouts, see also pseudometric space and "Geometry of Logic": http://finitegeometry.org/sc/16/logic.html ...

While there seems like a potential well-balanced in-between to these complementing/contrasting philosophies and layers of view points, I feel partly well-suited to vent that from what I've seen, statistical data analysis seems to zealously want to gain understanding by using brute force, where self-ordering "shapes" simply want to flow which shows their Nature.

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