This document summarizes research into generating a new type of matching algorithm for datasets in high-dimensional spaces. Existing algorithms work for datasets with equal pairwise distances, but a new approach is needed to match datasets with unequal distances. The research focuses on restricting pairwise distances to be equal within a reasonable value, moving from an isometric to a near-isometric case. The objectives were to create a working example of the new method and verify the underlying techniques. Methods included understanding the mathematics and implementing "slow twist" matrices to slowly rotate datasets. Future work will involve creating synthetic datasets, expanding the focus to non-Euclidean spaces, and generalizing current image mapping algorithms to higher dimensions.

1. Abstract
The purpose of this research is to generate a
new type of matching algorithm in R^D.
Common and established approaches deal
with datasets that have equal pairwise dis-
tances of points, and can superimpose them
by a series of Euclidean motions. By calculat-
ing some value, there is then some metric for
how similar two datasets are. A problem arises
in matching two datasets that have unequal
pairwise distances. The focus of this study is re-
stricting the pairwise distances to be equal
within some reasonable value. We look into
creating an actual example of this form, by
moving from the isometric case to the
near-isometric case.
Objectives & Hypothesis:
The focus of the project was to bring to life a
working example of our method. While we
were confident that the techniques and ideas
projected in the paper of focus were correct,
we also needed to verify that they worked as
expected.
By expanding the idea of image mapping to
account for small variations, our techniques
can let us gain information about a data set
from its corresponding map.
Methods:
First and foremost, a strong focus on under-
standing the underlying mathematics was
necessary. After that, the purpose became to
the machinery working. Dubbed “slow twists”,
implementation of these matrices would
rotate datasets at a rate much slower than ex-
pected.
We started with the Procrustes problem. With
two point sets with equal distances between
all points, there exists set of Euclidean mo-
tions(translations, slides, and rotations) that
can superimpose the two.
Project Sponsor: Dr. Steven Damelin, Department of Mathematics
Student Researchers: Brad Schwartz, Sean Kelly
Conclusions:
In the future, we will work to create ideal synthetic
datasets on which to experiment, and get a final im-
plementation. We will continue to expand our focus
to non-Euclidean analogues. We plan to generalize
current image mapping algorithms, and expand our
work to high dimensions.
References:
S. B. Damelin and C. Fefferman, Extension, interpolation and matching in R^D, arxiv:1411.2451.
S. B. Damelin and C. Fefferman, Extensions in R^D, arxiv:1411.2468.
S. B. Damelin and C. Fefferman, On Extensions of e Diffeomorphisms, preprint, arxiv:1505.06950
Charles Fefferman, Steven. B. Damelin and William Glover, BMO Theorems for epsilon distorted diffeomor-
phisms on R^D and an application to comparing manifolds of speech and sound, Involve, a Journal of
Mathematics 5-2 (2012), 159--172. DOI 10.2140/involve.2012.5.159
Epsilon-Distorted Diffeomorphisms for
Interpolation and Matching in R^D, D ≥ 2
Illustrating slow twist as original dataset is rotated 90 degrees in R^2