Regular Article: Randomized Nonlinear Projections Uncover High-Dimensional Structure

Authors:
Lenore J. Cowen;Carey E. Priebe
Affiliations:
Department of Mathematical Sciences, Johns Hopkins University, Baltimore, Maryland, 21218;Department of Mathematical Sciences, Johns Hopkins University, Baltimore, Maryland, 21218
Venue:
Advances in Applied Mathematics
Year:
1997

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Abstract

We consider the problem of investigating the ''structure'' of a set of points in high-dimensional space (npoints ind-dimensional Euclidean space) whenn@?d. The analysis of such data sets is a notoriously difficult problem in both combinatorial optimization and statistics due to an exponential explosion ind. A randomized nonlinear projection method is presented that maps these observations to a low-dimensional space, while approximately preserving salient features of the original data. Classical statistical analyses can then be applied, and results from the multiple lower-dimensional projected spaces are combined to yield information about the high-dimensional structure. We apply our dimension reduction techniques to a pattern recognition problem involving PET scan brain volumes.