Geometric Characteristics of the Wasserstein Metric on SPD(n) and Its Applications on Data Processing

The Wasserstein distance, especially among symmetric positive-definite matrices, has broad and deep influences on the development of artificial intelligence (AI) and other branches of computer science. In this paper, by involving the Wasserstein metric on SPD(n), we obtain computationally feasible expressions for some geometric quantities, including geodesics, exponential maps, the Riemannian connection, Jacobi fields and curvatures, particularly the scalar curvature. Furthermore, we discuss the behavior of geodesics and prove that the manifold is globally geodesic convex. Finally, we design algorithms for point cloud denoising and edge detecting of a polluted image based on the Wasserstein curvature on SPD(n). The experimental results show the efficiency and robustness of our curvature-based methods.

Three-dimensional transient electromagnetic inversion with optimal transport

Transient electromagnetic method (TEM), as one of the essential time-domain electromagnetic prospecting approaches, has the advantage of expedition, efficiency and convenience. In this paper, we study the transient electromagnetic inversion problem of different geological anomalies. First, Maxwell’s differential equations are discretized by the staggered finite-difference (FD) method; then we propose to solve the TEM inversion problem by minimizing the Wasserstein metric, which is related to the optimal transport (OT). Experimental tests based on the layered model and a 3D model are performed to demonstrate the feasibility of our proposed method.

Differential semblance optimisation based on the adaptive quadratic Wasserstein distance

As the robustness for the wave equation-based inversion methods, wave equation migration velocity analysis (WEMVA) is stable for overcoming the multipathing problem and has become popular in recent years. As a rapidly developed method, differential semblance optimisation (DSO) is convenient to implement and can automatically detect the moveout existing in common image gathers (CIGs). However, by implementing in the image domain with the target of minimising moveouts and improving coherence of the CIGs, the DSO method often suffers from imaging artefacts caused by uneven illumination and irregular observation geometry, which may produce poor velocity updates with artefact contamination. To deal with this issue, in this paper, by introducing Wiener-like filters, we modify the conventional image matching-based objective function to a new one by introducing the quadratic Wasserstein metric technique. The new misfit function measures the distance of two distributions obtained by the convolutional filters and target functions. With the new misfit function, the adjoint sources and the corresponding gradients are improved. We apply the new method to two numerical examples and one field dataset. The corresponding results indicate that the new method is robust to compensate low frequency components of velocity models.

Decision Making Under Model Uncertainty: Fréchet–Wasserstein Mean Preferences

This paper contributes to the literature on decision making under multiple probability models by studying a class of variational preferences. These preferences are defined in terms of Fréchet mean utility functionals, which are based on the Wasserstein metric in the space of probability models. In order to produce a measure that is the “closest” to all probability models in the given set, we find the barycenter of the set. We derive explicit expressions for the Fréchet–Wasserstein mean utility functionals and show that they can be expressed in terms of an expansion that provides a tractable link between risk aversion and ambiguity aversion. The proposed utility functionals are illustrated in terms of two applications. The first application allows us to define the social discount rate under model uncertainty. In the second application, the functionals are used in risk securitization. The barycenter in this case can be interpreted as the model that maximizes the probability that different decision makers will agree on, which could be useful for designing and pricing a catastrophe bond. This paper was accepted by Manel Baucells, decision analysis.

Ensemble Riemannian data assimilation over the Wasserstein space

In this paper, we present an ensemble data assimilation paradigm over a Riemannian manifold equipped with the Wasserstein metric. Unlike the Euclidean distance used in classic data assimilation methodologies, the Wasserstein metric can capture the translation and difference between the shapes of square-integrable probability distributions of the background state and observations. This enables us to formally penalize geophysical biases in state space with non-Gaussian distributions. The new approach is applied to dissipative and chaotic evolutionary dynamics, and its potential advantages and limitations are highlighted compared to the classic ensemble data assimilation approaches under systematic errors.

Evaluation of four point cloud similarity measures for the use in autonomous driving

Measuring the similarity between point clouds is required in many areas. In autonomous driving, point clouds for 3D perception are estimated from camera images but these estimations are error-prone. Furthermore, there is a lack of measures for quality quantification using ground truth. In this paper, we derive conditions point cloud comparisons need to fulfill and accordingly evaluate the Chamfer distance, a lower bound of the Gromov Wasserstein metric, and the ratio measure. We show that the ratio measure is not affected by erroneous points and therefore introduce the new measure “average ratio”. All measures are evaluated and compared using exemplary point clouds. We discuss characteristics, advantages and drawbacks with respect to interpretability, noise resistance, environmental representation, and computation.

Wasserstein Metric-Based Location Spoofing Attack Detection in WiFi Positioning Systems

WiFi positioning systems (WPS) have been introduced as parts of 5G location services (LCS) to provide fast positioning results of user devices in urban areas. However, they are prominently threatened by location spoofing attacks. To end this, we present a Wasserstein metric-based attack detection scheme to counter the location spoofing attacks in the WPS. The Wasserstein metric is used to measure the similarity of each two hotspots by their signal’s frequency offset distribution features. Then, we apply the clustering method to find the fake hotspots which are generated by the same device. When applied with WPS, the proposed method can prevent location spoofing by filtering out the fake hotspots set by attackers. We set up experimental tests by commercial WiFi devices, which show that our method can detect fake devices with 99% accuracy. Finally, the real-world test shows our method can effectively secure the positioning results against location spoofing attacks.