2 research outputs found
Supervising Embedding Algorithms Using the Stress
While classical scaling, just like principal component analysis, is
parameter-free, most other methods for embedding multivariate data require the
selection of one or several parameters. This tuning can be difficult due to the
unsupervised nature of the situation. We propose a simple, almost obvious,
approach to supervise the choice of tuning parameter(s): minimize a notion of
stress. We substantiate this choice by reference to rigidity theory. We extend
a result by Aspnes et al. (IEEE Mobile Computing, 2006), showing that general
random geometric graphs are trilateration graphs with high probability. And we
provide a stability result \`a la Anderson et al. (SIAM Discrete Mathematics,
2010). We illustrate this approach in the context of the MDS-MAP(P) algorithm
of Shang and Ruml (IEEE INFOCOM, 2004). As a prototypical patch-stitching
method, it requires the choice of patch size, and we use the stress to make
that choice data-driven. In this context, we perform a number of experiments to
illustrate the validity of using the stress as the basis for tuning parameter
selection. In so doing, we uncover a bias-variance tradeoff, which is a
phenomenon which may have been overlooked in the multidimensional scaling
literature. By turning MDS-MAP(P) into a method for manifold learning, we
obtain a local version of Isomap for which the minimization of the stress may
also be used for parameter tuning
Minimax Estimation of Distances on a Surface and Minimax Manifold Learning in the Isometric-to-Convex Setting
We start by considering the problem of estimating intrinsic distances on a
smooth surface. We show that sharper estimates can be obtained via a
reconstruction of the surface, and discuss the use of the tangential Delaunay
complex for that purpose. We further show that the resulting approximation rate
is in fact optimal in an information-theoretic (minimax) sense. We then turn to
manifold learning and argue that a variant of Isomap where the distances are
instead computed on a reconstructed surface is minimax optimal for the problem
of isometric manifold embedding