Search for: All records

Building on the recent work by Geshkovski et al. (2023) which provides an interacting particle system interpretation of Transformers with a continuous-time evolution, we study the controllability attributes of the corresponding continuity equation across the landscape of probability space curves. In particular, we consider the parameters of the Transformer’s continuous-time evolution as control inputs. We prove that given an absolutely continuous probability measure and a non-local Lipschitz velocity field that satisfy a continuity equation, there exist control inputs such that the measure and the non-local velocity field of the Transformer’s continuous-time evolution approximate them, respectively, in the p-Wasserstein and Lp-sense, where 1 ≤ p < ∞. 

In recent years, LiDAR sensors have become pervasive in the solutions to localization tasks for autonomous systems. One key step in using LiDAR data for localization is the alignment of two LiDAR scans taken from different poses, a process called scan-matching or point cloud registration. Most existing algorithms for this problem are heuristic in nature and local, meaning they may not produce accurate results under poor initialization. Moreover, existing methods give no guarantee on the quality of their output, which can be detrimental for safety-critical tasks. In this paper, we analyze a simple algorithm for point cloud registration, termed PASTA. This algorithm is global and does not rely on point-to-point correspondences, which are typically absent in LiDAR data. Moreover, and to the best of our knowledge, we offer the first point cloud registration algorithm with provable error bounds. Finally, we illustrate the proposed algorithm and error bounds in simulation on a simple trajectory tracking task.