Codes and Expansions (CodEx) Seminar
Nate Strawn (Georgetown University)
3D Filament Plots from Optimally Smooth 2D Andrew's Plots
We introduce filament plots for lossless visualization of high-dimensional datasets, where each data point in a dataset is mapped isometrically to a pair of \(L^2\) "symmetric curvature functions", thereby inducing a 3D "filament" using equations similar to the Frenet-Serret system. The isometric embedding into the space of symmetric curvature functions is chosen to minimize the mean quadratic variation over \(L^2\) curves, and we characterize the solutions of this minimization problem subject to some constraints which attempt to remove biases in the visualization. In particular, the solution set recovers a form of 2D Andrew's plots. This characterization indicates there are many degrees of freedom for the solution set, and (using recent theory on quadratic Gauss sums) we demonstrate that there is a particular minimizer which also satisfies an asymptotic projective "tour" property which is useful for removing visual biases. Metric comparisons are considered, and we discuss how the finite-dimensional analogue of our approach leads to "hyper frames" which enjoy properties of tight super frames and fusion frames.