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This paper presents a computational framework for the concise encoding of an ensemble of persistence diagrams, in the form of weighted Wasserstein barycenters [99], [101] of a dictionary of atom diagrams. We introduce a multi-scale gradient descent approach for the efficient resolution of the corresponding minimization problem, which interleaves the optimization of the barycenter weights with the optimization of the atom diagrams. Our approach leverages the analytic expressions for the gradient of both sub-problems to ensure fast iterations and it additionally exploits shared-memory parallelism. Extensive experiments on public ensembles demonstrate the efficiency of our approach, with Wasserstein dictionary computations in the orders of minutes for the largest examples. We show the utility of our contributions in two applications. First, we apply Wassserstein dictionaries to data reduction and reliably compress persistence diagrams by concisely representing them with their weights in the dictionary. Second, we present a dimensionality reduction framework based on a Wasserstein dictionary defined with a small number of atoms (typically three) and encode the dictionary as a low dimensional simplex embedded in a visual space (typically in 2D). In both applications, quantitative experiments assess the relevance of our framework. Finally, we provide a C++ implementation that can be used to reproduce our results.

Within the concept of physical human-robot interaction (pHRI), the most important criterion is the safety of the human operator interacting with a high degree of freedom (DoF) robot. Therefore, a robust control scheme is in high demand to establish safe pHRI and stabilize nonlinear, high DoF systems. In this paper, an adaptive decentralized control strategy is designed to accomplish the abovementioned objectives. To do so, a human upper limb model and an exoskeleton model are decentralized and augmented at the subsystem level to enable a decentralized control action design. Moreover, human exogenous force (HEF) that can resist exoskeleton motion is estimated using radial basis function neural networks (RBFNNs). Estimating both human upper limb and robot rigid body parameters, along with HEF estimation, makes the controller adaptable to different operators, ensuring their physical safety. The barrier Lyapunov function (BLF) is employed to guarantee that the robot can operate in a safe workspace while ensuring stability by adjusting the control law. Unknown actuator uncertainty and constraints are also considered in this study to ensure a smooth and safe pHRI. Then, the asymptotic stability of the whole system is established by means of the virtual stability concept and virtual power flows (VPFs) under the proposed robust controller. The experimental results are presented and compared to proportional-derivative (PD) and proportional-integral-derivative (PID) controllers. To show the robustness of the designed controller and its good performance, experiments are performed at different velocities, with different human users, and in the presence of unknown disturbances. The proposed controller showed perfect performance in controlling the robot, whereas PD and PID controllers could not even ensure stable motion in the wrist joints of the robot.

As human-robot interaction (HRI) systems advance, so does the difficulty of evaluating and understanding the strengths and limitations of these systems in different environments and with different users. To this end, previous methods have algorithmically generated diverse scenarios that reveal system failures in a shared control teleoperation task. However, these methods require directly evaluating generated scenarios by simulating robot policies and human actions. The computational cost of these evaluations limits their applicability in more complex domains. Thus, we propose augmenting scenario generation systems with surrogate models that predict both human and robot behaviors. In the shared control teleoperation domain and a more complex shared workspace collaboration task, we show that surrogate assisted scenario generation efficiently synthesizes diverse datasets of challenging scenarios. We demonstrate that these failures are reproducible in real-world interactions.

Spatial maps of extreme precipitation are crucial in flood protection. With the aim of producing maps of precipitation return levels, we propose a novel approach to model a collection of spatially distributed time series where the asymptotic assumption, typical of the traditional extreme value theory, is relaxed. We introduce a Bayesian hierarchical model that accounts for the possible underlying variability in the distribution of event magnitudes and occurrences, which are described through latent temporal and spatial processes. Spatial dependence is characterized by geographical covariates and effects not fully described by the covariates are captured by spatial structure in the hierarchies. The performance of the approach is illustrated through simulation studies and an application to daily rainfall extremes across North Carolina (USA). The results show that we significantly reduce the estimation uncertainty with respect to state of the art techniques.

Graphs are ubiquitous in nature and can therefore serve as models for many practical but also theoretical problems. For this purpose, they can be defined as many different types which suitably reflect the individual contexts of the represented problem. To address cutting-edge problems based on graph data, the research field of Graph Neural Networks (GNNs) has emerged. Despite the field's youth and the speed at which new models are developed, many recent surveys have been published to keep track of them. Nevertheless, it has not yet been gathered which GNN can process what kind of graph types. In this survey, we give a detailed overview of already existing GNNs and, unlike previous surveys, categorize them according to their ability to handle different graph types and properties. We consider GNNs operating on static and dynamic graphs of different structural constitutions, with or without node or edge attributes. Moreover, we distinguish between GNN models for discrete-time or continuous-time dynamic graphs and group the models according to their architecture. We find that there are still graph types that are not or only rarely covered by existing GNN models. We point out where models are missing and give potential reasons for their absence.

We consider a causal inference model in which individuals interact in a social network and they may not comply with the assigned treatments. In particular, we suppose that the form of network interference is unknown to researchers. To estimate meaningful causal parameters in this situation, we introduce a new concept of exposure mapping, which summarizes potentially complicated spillover effects into a fixed dimensional statistic of instrumental variables. We investigate identification conditions for the intention-to-treat effects and the average treatment effects for compliers, while explicitly considering the possibility of misspecification of exposure mapping. Based on our identification results, we develop nonparametric estimation procedures via inverse probability weighting. Their asymptotic properties, including consistency and asymptotic normality, are investigated using an approximate neighborhood interference framework. For an empirical illustration, we apply our method to experimental data on the anti-conflict intervention school program. The proposed methods are readily available with a companion R package.

This paper considers the Westervelt equation, one of the most widely used models in nonlinear acoustics, and seeks to recover two spatially-dependent parameters of physical importance from time-trace boundary measurements. Specifically, these are the nonlinearity parameter $\kappa(x)$ often referred to as $B/A$ in the acoustics literature and the wave speed $c_0(x)$. The determination of the spatial change in these quantities can be used as a means of imaging. We consider identifiability from one or two boundary measurements as relevant in these applications. For a reformulation of the problem in terms of the squared slowness $\mathfrak{s}=1/c_0^2$ and the combined coefficient $\eta=\frac{B/A+2}{\varrho_0 c_0^4}$ we devise a frozen Newton method and prove its convergence. The effectiveness (and limitations) of this iterative scheme are demonstrated by numerical examples.

Dependence is undoubtedly a central concept in statistics. Though, it proves difficult to locate in the literature a formal definition which goes beyond the self-evident 'dependence = non-independence'. This absence has allowed the term 'dependence' and its declination to be used vaguely and indiscriminately for qualifying a variety of disparate notions, leading to numerous incongruities. For example, the classical Pearson's, Spearman's or Kendall's correlations are widely regarded as 'dependence measures' of major interest, in spite of returning 0 in some cases of deterministic relationships between the variables at play, evidently not measuring dependence at all. Arguing that research on such a fundamental topic would benefit from a slightly more rigid framework, this paper suggests a general definition of the dependence between two random variables defined on the same probability space. Natural enough for aligning with intuition, that definition is still sufficiently precise for allowing unequivocal identification of a 'universal' representation of the dependence structure of any bivariate distribution. Links between this representation and familiar concepts are highlighted, and ultimately, the idea of a dependence measure based on that universal representation is explored and shown to satisfy Renyi's postulates.

This manuscript portrays optimization as a process. In many practical applications the environment is so complex that it is infeasible to lay out a comprehensive theoretical model and use classical algorithmic theory and mathematical optimization. It is necessary as well as beneficial to take a robust approach, by applying an optimization method that learns as one goes along, learning from experience as more aspects of the problem are observed. This view of optimization as a process has become prominent in varied fields and has led to some spectacular success in modeling and systems that are now part of our daily lives.

We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.