GLOBAL REGISTRATION OF MULTIPLE POINT CLOUDS USING SEMIDEFINITE PROGRAMMING

Consider N points in R-d and M local coordinate systems that are related through unknown rigid transforms. For each point, we are given (possibly noisy) measurements of its local coordinates in some of the coordinate systems. Alternatively, for each coordinate system, we observe the coordinates of a subset of the points. The problem of estimating the global coordinates of the N points (up to a rigid transform) from such measurements comes up in distributed approaches to molecular conformation and sensor network localization, and also in computer vision and graphics. The least-squares formulation of this problem, although nonconvex, has a well-known closed-form solution when M = 2 (based on the singular value decomposition (SVD)). However, no closed-form solution is known for M >= 3. In this paper, we demonstrate how the least-squares formulation can be relaxed into a convex program, namely, a semidefinite program (SDP). By setting up connections between the uniqueness of this SDP and results from rigidity theory, we prove conditions for exact and stable recovery for the SDP relaxation. In particular, we prove that the SDP relaxation can guarantee recovery under more adversarial conditions compared to earlier proposed spectral relaxations, and we derive error bounds for the registration error incurred by the SDP relaxation. We also present results of numerical experiments on simulated data to confirm the theoretical findings. We empirically demonstrate that (a) unlike the spectral relaxation, the relaxation gap is mostly zero for the SDP (i.e., we are able to solve the original nonconvex least-squares problem) up to a certain noise threshold, and (b) the SDP performs significantly better than spectral and manifold-optimization methods, particularly at large noise levels.

Translation Induction on Indian Language Corpora Using Translingual Themes from Other Languages

Identifying translations from comparable corpora is a well-known problem with several applications, e.g. dictionary creation in resource-scarce languages. Scarcity of high quality corpora, especially in Indian languages, makes this problem hard, e.g. state-of-the-art techniques achieve a mean reciprocal rank (MRR) of 0.66 for English-Italian, and a mere 0.187 for Telugu-Kannada. There exist comparable corpora in many Indian languages with other ``auxiliary'' languages. We observe that translations have many topically related words in common in the auxiliary language. To model this, we define the notion of a translingual theme, a set of topically related words from auxiliary language corpora, and present a probabilistic framework for translation induction. Extensive experiments on 35 comparable corpora using English and French as auxiliary languages show that this approach can yield dramatic improvements in performance (e.g. MRR improves by 124% to 0.419 for Telugu-Kannada). A user study on WikiTSu, a system for cross-lingual Wikipedia title suggestion that uses our approach, shows a 20% improvement in the quality of titles suggested.

Lower Bounds for Sums of Powers of Low Degree Univariates

We consider the problem of representing a univariate polynomial f(x) as a sum of powers of low degree polynomials. We prove a lower bound of Omega(root d/t) for writing an explicit univariate degree-d polynomial f(x) as a sum of powers of degree-t polynomials.