About the EventA classical problem in geometry goes as follows. Suppose we are given two sets of D dimensional data, that is, sets of points in Euclidean Dspace, where D≥1. The data sets are indexed by the same set, and we know that pairwise distances between corresponding points are equal in the two data sets. In other words, the sets are isometric. Can this correspondence be extended to an isometry of the ambient Euclidean space?
In this form the question is not terribly interesting; the answer has long known
to be yes (see [Wells and Williams 1975], for example). But a related question
is actually fundamental in data analysis: here the known points are samples from
larger, unknown sets—say, manifolds in RD—and we seek to know what can be
said about the manifolds themselves. A typical example might be a face recognition
problem, where all we have is multiple finite images of people’s faces from various
views.
An added complication is that in general we are not given exact distances. We
have noise and so we need to demand that instead of the pairwise distances being
equal, they should be close in some reasonable metric. Some results on almost
isometries in Euclidean spaces can be found in [John 1961; Alestalo et al. 2003].
In this talk, I will discuss:
1. Joint research with Charles Fefferman where we study this problem and solve it completely in several situations.
2. Joint research with Charles Fefferman and Willam Glover on applications of the results of (1) to music manifolds.
3. My ongoing program of research related to this problem related to understanding extensions of the work in (1) and its applications to various problems in signal processing, face recognition, shape spaces and various other computer vision and data analysis applications.
