Persistent homology (and geometry?) June 13, 2010

Posted by Sarah in Uncategorized.
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I read this AMS article by Robert Ghrist about persistent homology and I am intrigued.

This is a method for using topological invariants to define data. In particular, we want to define the homology of a data set representing a point cloud in a high-dimensional Euclidean space. A topologist would replace the point cloud with a simplicial complex — one of the easiest to compute is the Rips Complex, whose k-simplices correspond to unordered k+1-tuples of points within pairwise Euclidean distance $\epsilon$.

However, the resulting simplicial complex depends on the choice of $\epsilon$. A very small $\epsilon$ leaves the complex a discrete set, while a very large $\epsilon$ results in the complex being one big simplex. As $\epsilon$ moves, holes in the simplicial complex are born, grow, and die; the picture over time provides a description of the data’s topology.

The “persistence complex” is a sequence of chain complexes $C_*^i$ together with chain maps $C_*^i \to C_*^{i+1}$, which are inclusions. This is motivated by an increasing sequence of $\epsilon$s and the inclusions from one complex to the next. (The precision here goes from fine to coarse.)

Ghrist introduces the notion of a “barcode” — each “bar” is a generator of the homology group, and the length of the bar is the range of values of $\epsilon$ for which this particular element is a generator of the homology group. A barcode is the persistence analogue of a Betti number.

Now, what I always wondered here is what this has to do with geometry. Consider a finger-shaped projection sticking out of a 2-dimensional surface. At different resolutions, the projection can appear to break off into an island. (This is a practical problem for protein visualization.) This would be an example of a feature that could be captured with persistence homology; but it could also be explained directly by noticing that the tip of the projection is a region of high curvature. Could other persistence homology features be explained by geometrical properties?

This paper by Gunnar Carlsson et al, seems to provide an answer. The authors define a tangent complex of a space, which is the closure of the set of all tangents to points in the space. Then, they define the filtered tangent complex, which is the set of tangent vectors for which the osculating circle has a radius larger than some $\delta$. We have an inclusion between filtered tangent complexes of different $\delta$s. (For curves, there is only one osculating circle; for surfaces, there is one in each direction, so the tangent space is defined based on the maximum between them.)

Then we look at the homology groups of the filtered tangent spaces. This provides a barcode. Such barcodes can, for instance, distinguish a bottle from a glass. (The relationship of the tangent-space barcode to the Rips-complex barcode remains mysterious to me.)