Sensors and Samples: A Homological Approach

Presented at the Institute for Advanced Study in Princeton, NJ. Workshop on Topology: Identifying Order in Complex Systems

In their seminal work on homological sensor networks, de Silva and Ghrist showed the surprising fact that its possible to certify the coverage of a coordinate free sensor network even with very minimal knowledge of the space to be covered.
We give a new, simpler proof of the de Silva-Ghrist Topological Coverage Criterion that eliminates any assumptions about the smoothness of the boundary of the underlying space, allowing the results to be applied to much more general problems.
The new proof factors the geometric, topological, and combinatorial aspects of this approach.
This factoring reveals an interesting new connection between the topological coverage condition and the notion of weak feature size in geometric sampling theory.
We then apply this connection to the problem of showing that for a given scale, if one knows the number of connected components and the distance to the boundary, one can also infer the higher betti numbers or provide strong evidence that more samples are needed.
This is in contrast to previous work which merely assumed a good sample and gives no guarantees if the sampling condition is not met.