Dimensionality Estimation and Manifold Learning using Tensor Voting

We address instance-based learning from a perceptual organization standpoint using tensor voting. The goal of instance-based learning is to learn the underlying relationships between observations, which are points in an N -D continuous space, under the assumption that they lie in a limited part of the N -D space, typically a manifold, the dimensionality of which is an indication of the degrees of freedom of the underlying system. We pose the problem of instance-based learning in an equivalent form: manifold learning from observations. Unlike traditional manifold learning approaches, we do not perform dimensionality reduction, which would limit the class of datasets we are able to process, but instead perform all operations in the original input space. To this end we apply tensor voting, a perceptual organization framework that has mostly been applied to computer vision problems, after modifying it to make its implementation in high-dimensional spaces practical. We are able to estimate local dimensionality and structure for datasets that cannot be handled by current state-of-the-art approaches. We first show that we can obtain reliable dimensionality estimates at each point, instead of a global average estimate for the entire dataset, which is a major advantage over competing methods. We then present a quantitative evaluation of our results in the estimation of local manifold structure using synthetic datasets with known ground truth. Our software is available to the scientific community at http://iris.usc.edu/home/iris/medioni/User.html.