The Picasso Algorithm For Bayesian Localization Via Paired Comparisons In A Union Of Subspaces Model

We develop a framework for localizing an unknown point $\w$ using paired comparisons of the form ``$\w$ is closer to point $\x_i$ than to $\x_j$'' when the points lie in a union of known subspaces. This model, which extends a broad class of existing methods to exploit union of subspaces structure, provides a powerful framework for using the types of structure found in many practical applications. We divide the problem into two phases: (1) determining which subspace $\w$ lies in, and (2) localizing $\w$ within the identified subspace using existing techniques. We introduce two algorithms for determining the subspace in which an unknown point lies: the first admits a sample complexity guarantee demonstrating the advantage of the union of subspaces model, and the second improves performance in practice using an adaptive Bayesian strategy. We demonstrate the efficacy of our method with experiments on synthetic data and in an image search application.