Henri Schmidt

Henri Schmidt, Peter Halmos, Ben Raphael

Optimal transport (OT) finds a least cost transport plan between two probability distributions using a cost matrix defined on pairs of points. Unlike standard OT, which infers unstructured pointwise mappings, low-rank optimal transport explicitly constrains the rank of the transport plan to infer latent structure. This improves statistical stability and robustness, yields sharper parametric rates for estimating Wasserstein distances adaptive to the intrinsic rank, and generalizes $K$-means to co-clustering. These advantages, however, come at the cost of a non-convex and NP-hard optimization problem. We introduce transport clustering, an algorithm to compute a low-rank OT plan that reduces low-rank OT to a clustering problem on correspondences obtained from a full-rank $\textit{transport registration}$ step. We prove that this reduction yields polynomial-time, constant-factor approximation algorithms for low-rank OT: specifically, a $(1+γ)$ approximation for negative-type metrics and a $(1+γ+\sqrt{2γ}\,)$ approximation for kernel costs, where $γ\in [0,1]$ denotes the approximation ratio of the optimal full-rank solution relative to the low-rank optimal. Empirically, transport clustering outperforms existing low-rank OT solvers on synthetic benchmarks and large-scale, high-dimensional datasets.

Henri Schmidt, Christian Düll

We provide an implementation to compute the flat metric in any dimension. The flat metric, also called dual bounded Lipschitz distance, generalizes the well-known Wasserstein distance $W_1$ to the case that the distributions are of unequal total mass. Thus, our implementation adapts very well to mass differences and uses them to distinguish between different distributions. This is of particular interest for unbalanced optimal transport tasks and for the analysis of data distributions where the sample size is important or normalization is not possible. The core of the method is based on a neural network to determine an optimal test function realizing the distance between two given measures. Special focus was put on achieving comparability of pairwise computed distances from independently trained networks. We tested the quality of the output in several experiments where ground truth was available as well as with simulated data.

Jack Terwilliger, Michael Glazer, Henri Schmidt et al.

Humans, as both pedestrians and drivers, generally skillfully navigate traffic intersections. Despite the uncertainty, danger, and the non-verbal nature of communication commonly found in these interactions, there are surprisingly few collisions considering the total number of interactions. As the role of automation technology in vehicles grows, it becomes increasingly critical to understand the relationship between pedestrian and driver behavior: how pedestrians perceive the actions of a vehicle/driver and how pedestrians make crossing decisions. The relationship between time-to-arrival (TTA) and pedestrian gap acceptance (i.e., whether a pedestrian chooses to cross under a given window of time to cross) has been extensively investigated. However, the dynamic nature of vehicle trajectories in the context of non-verbal communication has not been systematically explored. Our work provides evidence that trajectory dynamics, such as changes in TTA, can be powerful signals in the non-verbal communication between drivers and pedestrians. Moreover, we investigate these effects in both simulated and real-world datasets, both larger than have previously been considered in literature to the best of our knowledge.

Dina AlAdawy, Michael Glazer, Jack Terwilliger et al.

When asked, a majority of people believe that, as pedestrians, they make eye contact with the driver of an approaching vehicle when making their crossing decisions. This work presents evidence that this widely held belief is false. We do so by showing that, in majority of cases where conflict is possible, pedestrians begin crossing long before they are able to see the driver through the windshield. In other words, we are able to circumvent the very difficult question of whether pedestrians choose to make eye contact with drivers, by showing that whether they think they do or not, they can't. Specifically, we show that over 90\% of people in representative lighting conditions cannot determine the gaze of the driver at 15m and see the driver at all at 30m. This means that, for example, that given the common city speed limit of 25mph, more than 99% of pedestrians would have begun crossing before being able to see either the driver or the driver's gaze. In other words, from the perspective of the pedestrian, in most situations involving an approaching vehicle, the crossing decision is made by the pedestrian solely based on the kinematics of the vehicle without needing to determine that eye contact was made by explicitly detecting the eyes of the driver.

Henri Schmidt, Jack Terwilliger, Dina AlAdawy et al.

We use an immersive virtual reality environment to explore the intricate social cues that underlie non-verbal communication involved in a pedestrian's crossing decision. We "hack" non-verbal communication between pedestrian and vehicle by engineering a set of 15 vehicle trajectories, some of which follow social conventions and some that break them. By subverting social expectations of vehicle behavior we show that pedestrians may use vehicle kinematics to infer social intentions and not merely as the state of a moving object. We investigate human behavior in this virtual world by conducting a study of 22 subjects, with each subject experiencing and responding to each of the trajectories by moving their body, legs, arms, and head in both the physical and the virtual world. Both quantitative and qualitative responses are collected and analyzed, showing that, in fact, social cues can be engineered through vehicle trajectory manipulation. In addition, we demonstrate that immersive virtual worlds which allow the pedestrian to move around freely, provide a powerful way to understand both the mechanisms of human perception and the social signaling involved in pedestrian-vehicle interaction.