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This tutorial serves as an introduction to recently developed non-asymptotic methods in the theory of -- mainly linear -- system identification. We emphasize tools we deem particularly useful for a range of problems in this domain, such as the covering technique, the Hanson-Wright Inequality and the method of self-normalized martingales. We then employ these tools to give streamlined proofs of the performance of various least-squares based estimators for identifying the parameters in autoregressive models. We conclude by sketching out how the ideas presented herein can be extended to certain nonlinear identification problems.

We introduce an end-to-end model of participatory budgeting grounded in social choice theory. This model accounts for both the first stage, in which participants propose projects to be shortlisted, and the second stage, in which they vote on which of the shortlisted projects should be funded. We introduce several shortlisting rules for the first stage and we analyse them in both normative and algorithmic terms. Our main focus is on the incentives of participants to engage in strategic behaviour, especially in the first stage, in which they need to reason about how their proposals will impact the range of strategies available to everyone in the second stage.

Graph representations are the generalization of geometric graph drawings from the plane to higher dimensions. A method introduced by Tutte to optimize properties of graph drawings is to minimize their energy. We explore this minimization for spherical graph representations, where the vertices lie on a unit sphere such that the origin is their barycentre. We present a primal and dual semidefinite program which can be used to find such a spherical graph representation minimizing the energy. We denote the optimal value of this program by $\rho(G)$ for a given graph $G$. The value turns out to be related to the second largest eigenvalue of the adjacency matrix of $G$, which we denote by $\lambda_2$. We show that for $G$ regular, $\rho(G) \leq \frac{\lambda_{2}}{2} \cdot v(G)$, and that equality holds if and only if the $\lambda_{2}$ eigenspace contains a spherical 1-design. Moreover, if $G$ is a random $d$-regular graph, $\rho(G)=\left(\sqrt{(d-1)} +o(1)\right)\cdot v(G)$, asymptotically almost surely.

We extend classical methods of computational complexity to the setting of distributed computing, where they prove even more effective in some respects than in their original context. Instead of a single computer, several networked computers communicate via synchronous message-passing to collectively solve some decision problem related to the network topology. Their running time is limited in two ways: the number of communication rounds is bounded by a constant, and the number of computation steps of each computer is polynomially bounded by the size of its local input and the messages it receives. By letting two players take turns assigning certificates to the computers, we obtain a generalization of the polynomial hierarchy (and hence of the complexity classes $\mathbf{P}$ and $\mathbf{NP}$). We then extend some key results of complexity theory to this setting, in particular the Cook-Levin theorem (which identifies Boolean satisfiability as a complete problem for $\mathbf{NP}$), and Fagin's theorem (which characterizes $\mathbf{NP}$ as the problems expressible in existential second-order logic). The original results can be recovered as the special case where the network consists of a single computer. But perhaps more surprisingly, the task of separating complexity classes becomes easier in the general case: we can show that our hierarchy is infinite, while it remains notoriously open whether the same is true in the case of a single computer. (By contrast, a collapse of our hierarchy would have implied a collapse of the polynomial hierarchy.) As an application, we propose quantifier alternation as a new approach to measuring the locality of problems in distributed computing.

We investigate the complexity of several manipulation and control problems under numerous prevalent approval-based multiwinner voting rules. Particularly, the rules we study include approval voting (AV), satisfaction approval voting (SAV), net-satisfaction approval voting (NSAV), proportional approval voting (PAV), approval-based Chamberlin-Courant voting (ABCCV), minimax approval voting (MAV), etc. We show that these rules generally resist the strategic types scrutinized in the paper, with only a few exceptions. In addition, we also obtain many fixed-parameter tractability results for these problems with respect to several natural parameters, and derive polynomial-time algorithms for certain special cases.

We introduce the general notions of an index and a core of a relation. We postulate a limited form of the axiom of choice -- specifically that all partial equivalence relations have an index -- and explore the consequences of adding the axiom to standard axiom systems for point-free reasoning. Examples of the theorems we prove are that a core/index of a difunction is a bijection, and that the so-called ``all or nothing'' axiom used to facilitate pointwise reasoning is derivable from our axiom of choice.

Mesh degeneration is a bottleneck for fluid-structure interaction (FSI) simulations and for shape optimization via the method of mappings. In both cases, an appropriate mesh motion technique is required. The choice is typically based on heuristics, e.g., the solution operators of partial differential equations (PDE), such as the Laplace or biharmonic equation. Especially the latter, which shows good numerical performance for large displacements, is expensive. Moreover, from a continuous perspective, choosing the mesh motion technique is to a certain extent arbitrary and has no influence on the physically relevant quantities. Therefore, we consider approaches inspired by machine learning. We present a hybrid PDE-NN approach, where the neural network (NN) serves as parameterization of a coefficient in a second order nonlinear PDE. We ensure existence of solutions for the nonlinear PDE by the choice of the neural network architecture. Moreover, we present an approach where a neural network corrects the harmonic extension such that the boundary displacement is not changed. In order to avoid technical difficulties in coupling finite element and machine learning software, we work with a splitting of the monolithic FSI system into three smaller subsystems. This allows to solve the mesh motion equation in a separate step. We assess the quality of the learned mesh motion technique by applying it to a FSI benchmark problem.

We introduce a new methodology to conduct simultaneous inference of the nonparametric component in partially linear time series regression models where the nonparametric part is a multivariate unknown function. In particular, we construct a simultaneous confidence region (SCR) for the multivariate function by extending the high-dimensional Gaussian approximation to dependent processes with continuous index sets. Our results allow for a more general dependence structure compared to previous works and are widely applicable to a variety of linear and nonlinear autoregressive processes. We demonstrate the validity of our proposed methodology by examining the finite-sample performance in the simulation study. Finally, an application in time series, the forward premium regression, is presented, where we construct the SCR for the foreign exchange risk premium from the exchange rate and macroeconomic data.

We introduce a multi-task setup of identifying and classifying entities, relations, and coreference clusters in scientific articles. We create SciERC, a dataset that includes annotations for all three tasks and develop a unified framework called Scientific Information Extractor (SciIE) for with shared span representations. The multi-task setup reduces cascading errors between tasks and leverages cross-sentence relations through coreference links. Experiments show that our multi-task model outperforms previous models in scientific information extraction without using any domain-specific features. We further show that the framework supports construction of a scientific knowledge graph, which we use to analyze information in scientific literature.

Object detection typically assumes that training and test data are drawn from an identical distribution, which, however, does not always hold in practice. Such a distribution mismatch will lead to a significant performance drop. In this work, we aim to improve the cross-domain robustness of object detection. We tackle the domain shift on two levels: 1) the image-level shift, such as image style, illumination, etc, and 2) the instance-level shift, such as object appearance, size, etc. We build our approach based on the recent state-of-the-art Faster R-CNN model, and design two domain adaptation components, on image level and instance level, to reduce the domain discrepancy. The two domain adaptation components are based on H-divergence theory, and are implemented by learning a domain classifier in adversarial training manner. The domain classifiers on different levels are further reinforced with a consistency regularization to learn a domain-invariant region proposal network (RPN) in the Faster R-CNN model. We evaluate our newly proposed approach using multiple datasets including Cityscapes, KITTI, SIM10K, etc. The results demonstrate the effectiveness of our proposed approach for robust object detection in various domain shift scenarios.