We often seek to understand the structure of a set of data points.
Generally, we assume that the points are sampled from some underlying
space in which we are interested. Many disciplines focus on the geometry
of this space, analyzing local properties quantitatively. However, the
topology of the space often determines the effectiveness of such geometric
algorithms. Topological questions have emerged naturally in many areas of
computer science, giving rise to the area of computational topology.
In this talk, I discuss persistent homology, an algebraic method for a
multi-scale analysis of a set of points. After describing the theory, I
will give a recent application that looks at the local structure of
natural images, with possible implications for image compression.
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