We introduce a novel method for clustering using a semidefinite programming (SDP) relaxation of the Max k-Cut problem. The approach is based on a new methodology for rounding the solution of an SDP relaxation using iterated linear optimization. We show the vertices of the Max k-Cut relaxation correspond to partitions of the data into at most k sets. We also show the vertices are attractive fixed points of iterated linear optimization. Each step of this iterative process solves a relaxation of the closest vertex problem and leads to a new clustering problem where the underlying clusters are more clearly defined. Our experiments show that using fixed point iteration for rounding the Max k-Cut SDP relaxation leads to significantly better results when compared to randomized rounding.

We propose a generalization of the graphical chip-firing model allowing for the redistribution dynamics to be governed by any invertible integer matrix while maintaining the long term critical, superstable, and energy minimizing behavior of the classical model.

To recognize three-dimensional objects it is important to model how their appearances can change due to changes in viewpoint. A key aspect of this involves understanding which object features can be simultaneously visible under different viewpoints. We address this problem in an image-based framework, in which we use a limited number of images of an object taken from unknown viewpoints to determine which subsets of features might be simultaneously visible in other views. This leads to the problem of determining whether a set of images, each containing a set of features, is consistent with a single 3D object. We assume that each feature is visible from a disk of viewpoints on the viewing sphere. In this case we show the problem is NP-hard in general, but can be solved efficiently when all views come from a circle on the viewing sphere. We also give iterative algorithms that can handle noisy data and converge to locally optimal solutions in the general case. Our techniques can also be used to recover viewpoint information from the set of features that are visible in different images. We show that these algorithms perform well both on synthetic data and images from the COIL dataset.