# Group theory in machine learning

I'm a Machine Learning researcher who would like to research applications of group theory in ML.

There is a term "Partially Observed Groups" in machine learning theory which has been popularized by recent work to understand deep learning. The idea is simple, instead of learning a recognition function (image -> object class) , the brain is learning orbits (images -> object orbit under the action of a group).

For example, all images of a bottle are 2-d projections (thus partially observed) of a 3-d image in an orbit of a bottle under the action of a group (rotation or translation).

I'm having difficulty finding relevant literature about probability distributions on group elements or even the concept of partially observed groups. I was hoping perhaps this is a well developed concept (in maybe physics) by some other name. Any suggestions on relevant work? Or other literature I should read to guide me in this.

Thank you for your time.

clarification:

The hypothesis I'm exploring is that the representations learned in neural networks via gradient descent work as well as they do because they are group invariant. So literature on Lie groups is very relevant.

But the distinguishing setting is that since elements of the group act like transformations on 3-d images and since we only see a 2-d projection of the 3-d images, we must define probability distributions over the product (object x group element).

Therefore the problem is under-determined and distributions over group elements become necessary to describe what generated the observed image; Many objects can be transformed to produce very similar images. I thought there might be analogous work done in physics. I hope this clarifies things.

## 4 Answers

I suggest taking a look at the work of Ulf Grenander. His 1963 book laid out the basis for applying probability theory to groups (Chapter 4 is on stochastic Lie groups) and other algebraic structures. He continued to develop these ideas (see his later book) in the context of pattern recognition.

There is definitely newer work in this area (some of which is mentioned in other answers), but Grenander has been investigating these ideas for 40+ years and is worth looking into.

As an aside, Grenander's approach seems to be a bit more formal than a lot of contemporary machine learning research, which is a strength or weakness according to the reader's taste.

• I think his algebraic probability is more or less serve as a prelude to his later general pattern theory developed in his late years. Of course it is always better to be formal. Also his algebraic probability, as I can tell, is more or less influenced Diaconis' book.jdc.math.uwo.ca/M9140a-2014-summer/Diaconis-1988.pdf – Henry.L Dec 4 '16 at 19:01

Here are some of the group theoretical references within the machine learning literature:

1. Have a look at recent papers by Stéphane Mallat , or first look at 2.
2. This NIPS 2012 talk by Stéphane Mallat
3. A Group Theoretic perspective of Deep Learning
4. Some papers by Risi Kondor, and also his thesis ("Group theoretical methods in machine learning")
5. Directional Statistics --- among others, it also covers $O(n)$ (viewed as a manifold)

There are several more, but I think these links above should provide a good start.

Yet, some more relations of group theory to machine learning:

Recent work by Cohen \emph{et al.} has achieved state-of-the-art results for learning spherical images in a rotation invariant way by using ideas from group representation theory and noncommutative harmonic analysis. In this paper we propose a generalization of this work that generally exhibits improved performace, but from an implementation point of view is actually simpler. An unusual feature of the proposed architecture is that it uses the Clebsch--Gordan transform as its only source of nonlinearity, thus avoiding repeated forward and backward Fourier transforms. The underlying ideas of the paper generalize to constructing neural networks that are invariant to the action of other compact groups.

As far as I understand images here are captured by special 360-angle cameras. That explains the origin of rotationl symmetry group. So it might not be widely used technology at present, nevertheless, it is quite interesting that group theoretic methods can be applied here.

In R. Kondor's thesis mentioned in Suvrit's answer. One of the topics is "learning on the symmetric group" which is somehow related to "ranking problem" i.e. ordering some answers/candidates/whatever is certain order. Problem of finding appropriate ranking is the same as finding appropriate permutation, that is how permutation group arises.

The most useful and practically used measure of quality in binary classification problems is "AUC ROC ", which is closely related to Mann-Whitney statistical test. Which is in turn roughly speaking the same as inversion number of certain permutation, which can be seen in the broader context of metrics on symmetric group (see P. Diaconis "Group representations in probability and statistics, Chapter 6. Metrics on Groups, and Their Statistical Use" and some comments at MO).

Geoffrey Hinton's Capsule Networks may be relevant. Convolutional NNs are effective because each layer is invariant under 2D translations; Capsule NN are designed to be invariant under much larger group transformations.

Other relevant research in the last couple years was to generalize $$\mathbb{R}^2$$ invariant convolutions of $$\mathbb{R}^2$$ images to $$SO(3)$$ invariance for spherical images. I don't recall the reference though.