## Wednesday, May 4, 2011

### Manifold Learning vs Manifold Embedding

These thoughts stem from the ambiguity that I think persists in differentiating learning vs embedding when it comes to thinking about manifold data. Any comments are welcome.

Whenever one mentions "manifold learning" in machine learning, a number of algorithms tend to pop in ones head (assuming one is familiar with this particular area in machine learning) such as Isomap, Locally Linear Embedding (LLE), Multidimensional Scaling (MDS), t-SNE, and of course, Principal Component Analysis (PCA). Generally we associate manifold learning to techniques that essentially embed very high dimensional data onto a low dimensional space (usually 2D) for purposes of visualization.

As far as I understand, these techniques generally try to build a geodesic distance map of neighborhoods for each point, and based on the distribution of such distances, construct an embedding function that maps these high dimensional points to a low dimensional space that respects these distance distributions. I am grossly simplifying the theme, but each technique builds the distribution of distances of the high dimensional points in its own way (often using an optimization formulation).

But the terminology "manifold learning" seems somewhat misleading to me. The purpose of the aforementioned techniques is more like manifold embedding; though such embedding techniques are also effective in building a recognition system, for example, I cannot fully accept the usage of "manifold learning" in describing these broad class of techniques.

Learning a manifold should entail something different. For very high dimensional data, the number of modes of variability in the data is usually far less than the number of dimensions. In other words, if one can quantify the modes of variability of the data, then one has a parametrized model that captures or describes the very high dimensional data. The learning portion is to answer the question of how one learns these variabilities. Obviously, in learning to capture these variabilities from very high dimensional data one is likely to resort to one of the many embedding techniques to make learning more tractable. Once the variabilities are learned, it can be used to embed high dimensional data into a low dimensional manifold! Being able to compactly describe a manifold might be also advantageous for classification.

In my mind, there appears to be a demarcation between the learning and the embedding views. I have not seen much work from this "learning to describe manifold" perspective. But this alternate view of manifolds might have some interesting application potential, very likely in computer vision.

#### 1 comment:

1. Hi Afonsobandeira,

Sorry for the late reply. To answer your question, you can check out papers by Piotr Dollar (Caltech) or Olhausen (Berkeley) related to learning image manifolds that will give some insight. I think people generally represent it via some parametrizations that are learned from the data. I don't have a good sense of how well that works, for example, to identify manifold generated by a face vs a car. Zisserman's group has a paper that introduces classification through some manifold distance measures, you might be interested in that as well.