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# Structure Discovery in Sampled Spaces

We discuss a number of tools from computational topology and geometry

for structure discovery and visualization in large distributed data

sets. Topology studies the connectivity of spaces, so it is global by

its very nature. It is able to determine certain connectivity

invariants in a way that is unaffected by deformations of an object and

does not require explicit parameterizations of the object geometry. Its

strength lies, in a sense, in its relative insensitivity to geometric

properties, which permits it to discern underlying combinatorial

information about how the geometric object is constructed, and

therefore detect some qualitative properties. This type of global

analysis can be quite important in understanding the overall structure

of data sets. Geometry, though more local by nature, can also be used

to study global structure by discovering how parts of an object relate

to another, or how parts of different objects can be similar. For

example, the Erlanger program of Felix Klein has fueled for over a

century mathematicians' interest in invariance under certain group

actions as a key principle for understanding geometric spaces. Such

invariances or symmetries can also be key to understanding and

reasoning about data sets. We aim to study how such tools can lead to

useful segmentations of data sets, and the discovery of repeated

structure, symmetries, and other global patterns.