Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

We consider a sequence of compositions of orientation preserving diffeomorphisms of the circle chosen randomly with a fixed distribution law. There is naturally associated a Lyapunov exponent, which measures the rate of exponential contractions of the sequence. It is known that under some assumptions, if this Lyapunov exponent is null then all the diffeomorphisms are simultaneously conjugated to rotations. If the Lyapunov exponent is not null but close to 0, we study how well we can approach rotations by a simultaneous conjugation. In particular, our results can apply to random products of matrices 2x2, giving quantitative versions of the classical Furstenberg theorem.

Numerical integration of given dynamic systems can be viewed as a forward problem with the learning of unknown dynamics from available state observations as an inverse problem. The latter has been around in various settings such as the model reduction of multiscale processes. It has received particular attention recently in the data-driven modeling via deep/machine learning. Indeed, solving both forward and inverse problems forms the loop of informative and intelligent scientific computing. A natural question is whether a good numerical integrator for discretizing prescribed dynamics is also good for discovering unknown dynamics. This lecture presents a study in the context of Linear multistep methods (LMMs).

The problem of group synchronization asks to recover states of objects associated with group elements given possibly corrupted relative state measurements (or group ratios) between pairs of objects. This problem arises in important data-related tasks, such as structure from motion, simultaneous localization and mapping, Cryo-EM, community detection and sensor network localization. Two common groups in these problems are the rotation and symmetric groups. We propose a general framework for group synchronization with compact groups. The main part of the talk discusses a novel message passing procedure that uses cycle consistency information in order to estimate the corruption levels of group ratios. Under our mathematical model of adversarial corruption, it can be used to infer the corrupted group ratios and thus to solve the synchronization problem. We first explain why the group cycle consistency information is essential for effectively solving group synchronization problems. We then establish exact recovery and linear convergence guarantees for the proposed message passing procedure under a deterministic setting with adversarial corruption. We also establish the stability of the proposed procedure to sub-Gaussian noise. We further establish competitive theoretical results under a uniform corruption model. Finally, we discuss the MPLS (Message Passing Least Squares) or Minneapolis framework for solving real scenarios with high levels of corruption and noise and with nontrivial scenarios of corruption. We demonstrate state-of-the-art results for two different problems that occur in structure from motion and involve the rotation and permutation groups.