  Dimensionality-reduction techniques are a fundamental tool for extracting
useful information from high-dimensional data sets. Because secant sets encode
manifold geometry, they are a useful tool for designing meaningful
data-reduction algorithms. In one such approach, the goal is to construct a
projection that maximally avoids secant directions and hence ensures that
distinct data points are not mapped too close together in the reduced space.
This type of algorithm is based on a mathematical framework inspired by the
constructive proof of Whitney's embedding theorem from differential topology.
Computing all (unit) secants for a set of points is by nature computationally
expensive, thus opening the door for exploitation of GPU architecture for
achieving fast versions of these algorithms. We present a polynomial-time
data-reduction algorithm that produces a meaningful low-dimensional
representation of a data set by iteratively constructing improved projections
within the framework described above. Key to our algorithm design and
implementation is the use of GPUs which, among other things, minimizes the
computational time required for the calculation of all secant lines. One goal
of this report is to share ideas with GPU experts and to discuss a class of
mathematical algorithms that may be of interest to the broader GPU community.
