Tuesday, December 18, 2012

Geometry, Machine Learning and Deep Learning


Over the years [..] the way we construct mental models of data has changed. And as I've argued before, understanding how we think about data, and what shape we give it, is key to the whole enterprise of finding patterns in data.

The model that one always starts with is Euclidean space. Data = points, features = dimensions, and so on. And as a first approximation of a data model, it isn't terrible.

There are many ways to modify this space. You can replace the ℓ2 norm by ℓ1. You can normalize the points (again with ℓ2 or ℓ1, sending you to the sphere or the simplex). You can weight the dimensions, or even do a wholesale scale-rotation.

But that's not all. Kernels take this to another level. You can encode weak nonlinearity in the data by assuming that it's flat once you lift it. In a sense, this is still an ℓ2 space, but a larger class of spaces that you can work with. The entire SVM enterprise was founded on this principle.

But that's not all either. The curse of dimensionality means that it's difficult to find patterns in such high dimensional data. Arguably, "real data" is in fact NOT high dimensional, or is not generated by a process with many parameters, and so sparsity-focused methods like compressed sensing start playing a role.

But it gets even more interesting. Maybe the data is low-dimensional, but doesn't actually lie in a subspace. This gets you into manifold learning and variants: the data lies on a low-dimensional curved sheet of some kind, and you need to learn
on that space.

While the challenge for geometry (and algorithms) is to keep up with the new data models, the challenge for data analysts is to design data models that are realistic and workable.

So what does this have to do with deep learning ?

Deep learning networks "work" in that they appear to be able to identify interesting semantic structures in data that can be quite noisy. But to me it's not entirely clear why that is [..].

A central idea of [Deep Learning] work is that deep belief networks can be trained "layer by layer", where each layer uses features identified from the previous layer.

If you stare at these things long enough, you begin to see a picture not of sparse data, or low-rank data, or even manifold data. What you see is a certain hierarchical collection of subspaces, where low-dimensional spaces interact in a low dimensional way to form higher level spaces, and so on. So you might have a low-level "lip" feature described by a collection of 2-3 dimensional noisy subspaces in an image space. These "lip" features in turn combine with "eye" features and so on.