Ery Arias-Castro

Mathematical Sciences Research Institute

Berkeley, CA

Detection in Point Cloud:
A Statistical Viewpoint with Multiscale Insight

         We observe n points in the unit d-dimensional hypercube. We want to know whether these points are uniformly distributed or whether a small fraction of them are actually concentrated near an object, such as a curve or sheet, which is only known to belong to some regularity class.

We argue that this hypothesis testing problem is relevant for the task of detecting structures in galaxy distributions.

We consider classes of Holder immersions and study the asymptotic power of the Generalized Likelihood Ratio Test (GLRT), or Scan Statistic, in this setting.

We also address computational issues. In turns out that some exact calculations are feasible in some situations, via Dynamic Programming.

However, in general, exact computations are known to be NP-hard. Approximations are nevertheless possible, at least in theory. Via custom-built graphical structures, it is possible to translate this computational task into some variation of "The Traveling Salesman Problem", famous in Computer Science and Operations Research.

We extend this study to higher order contact, which models recent experiments in Perceptual Psychophysics.

Collaborators: David Donoho (Stanford), Xiaoming Huo (Georgia Tech) and Craig Tovey (Georgia Tech).

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