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.
To each regularity class we associate a graphical structure designed to approximate
the GLRT. Computing such approximations is challenging and some approaches are
surveyed in the Computer Science and Operations Research literatures.
The detection performance of the Longest Significant Run Test is also investigated.
We extend this study to higher order contact, which models recent experiments
in Perceptual Psychophysics; and to detection in graphs, which models networks of
sensors.
Collaborators: David Donoho (Stanford), Xiaoming Huo and Craig Tovey (Georgia
Tech).
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