A system of coupled oscillators on an arbitrary graph is locally driven by the tendency to mutual synchronization between nearby oscillators, but can and often exhibit nonlinear behavior on the whole graph. Understanding such nonlinear behavior has been a key challenge in predicting whether all oscillators in such a system will eventually synchronize. In this paper, we demonstrate that, surprisingly, such nonlinear behavior of coupled oscillators can be effectively linearized in certain latent dynamic spaces. The key insight is that there is a small number of `latent dynamics filters', each with a specific association with synchronizing and non-synchronizing dynamics on subgraphs so that any observed dynamics on subgraphs can be approximated by a suitable linear combination of such elementary dynamic patterns. Taking an ensemble of subgraph-level predictions provides an interpretable predictor for whether the system on the whole graph reaches global synchronization. We propose algorithms based on supervised matrix factorization to learn such latent dynamics filters. We demonstrate that our method performs competitively in synchronization prediction tasks against baselines and black-box classification algorithms, despite its simple and interpretable architecture.
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The rise of AI in human contexts places new demands on automated systems to be transparent and explainable. We examine some anthropomorphic ideas and principles relevant to such accountablity in order to develop a theoretical framework for thinking about digital systems in complex human contexts and the problem of explaining their behaviour. Structurally, systems are made of modular and hierachical components, which we abstract in a new system model using notions of modes and mode transitions. A mode is an independent component of the system with its own objectives, monitoring data, and algorithms. The behaviour of a mode, including its transitions to other modes, is determined by functions that interpret each mode's monitoring data in the light of its objectives and algorithms. We show how these belief functions can help explain system behaviour by visualising their evaluation as trajectories in higher-dimensional geometric spaces. These ideas are formalised mathematically by abstract and concrete simplicial complexes. We offer three techniques: a framework for design heuristics, a general system theory based on modes, and a geometric visualisation, and apply them in three types of human-centred systems.
We develop a numerical method for computing with orthogonal polynomials that are orthogonal on multiple, disjoint intervals for which analytical formulae are currently unknown. Our approach exploits the Fokas--Its--Kitaev Riemann--Hilbert representation of the orthogonal polynomials to produce an $\mathrm{O}(N)$ method to compute the first $N$ recurrence coefficients. The method can also be used for pointwise evaluation of the polynomials and their Cauchy transforms throughout the complex plane. The method encodes the singularity behavior of weight functions using weighted Cauchy integrals of Chebyshev polynomials. This greatly improves the efficiency of the method, outperforming other available techniques. We demonstrate the fast convergence of our method and present applications to integrable systems and approximation theory.
It is well-known that a multilinear system with a nonsingular M-tensor and a positive right-hand side has a unique positive solution. Tensor splitting methods generalizing the classical iterative methods for linear systems have been proposed for finding the unique positive solution. The Alternating Anderson-Richardson (AAR) method is an effective method to accelerate the classical iterative methods. In this study, we apply the idea of AAR for finding the unique positive solution quickly. We first present a tensor Richardson method based on tensor regular splittings, then apply Anderson acceleration to the tensor Richardson method and derive a tensor Anderson-Richardson method, finally, we periodically employ the tensor Anderson-Richardson method within the tensor Richardson method and propose a tensor AAR method. Numerical experiments show that the proposed method is effective in accelerating tensor splitting methods.
In a network of reinforced stochastic processes, for certain values of the parameters, all the agents' inclinations synchronize and converge almost surely toward a certain random variable. The present work aims at clarifying when the agents can asymptotically polarize, i.e. when the common limit inclination can take the extreme values, 0 or 1, with probability zero, strictly positive, or equal to one. Moreover, we present a suitable technique to estimate this probability that, along with the theoretical results, has been framed in the more general setting of a class of martingales taking values in [0, 1] and following a specific dynamics.
We characterize structures such as monotonicity, convexity, and modality in smooth regression curves using persistent homology. Persistent homology is a key tool in topological data analysis that detects higher dimensional topological features such as connected components and holes (cycles or loops) in the data. In other words, persistent homology is a multiscale version of homology that characterizes sets based on the connected components and holes. We use super-level sets of functions to extract geometric features via persistent homology. In particular, we explore structures in regression curves via the persistent homology of super-level sets of a function, where the function of interest is - the first derivative of the regression function. In the course of this study, we extend an existing procedure of estimating the persistent homology for the first derivative of a regression function and establish its consistency. Moreover, as an application of the proposed methodology, we demonstrate that the persistent homology of the derivative of a function can reveal hidden structures in the function that are not visible from the persistent homology of the function itself. In addition, we also illustrate that the proposed procedure can be used to compare the shapes of two or more regression curves which is not possible merely from the persistent homology of the function itself.
The numerical solution of continuum damage mechanics (CDM) problems suffers from convergence-related challenges during the material softening stage, and consequently existing iterative solvers are subject to a trade-off between computational expense and solution accuracy. In this work, we present a novel unified arc-length (UAL) method, and we derive the formulation of the analytical tangent matrix and governing system of equations for both local and non-local gradient damage problems. Unlike existing versions of arc-length solvers that monolithically scale the external force vector, the proposed method treats the latter as an independent variable and determines the position of the system on the equilibrium path based on all the nodal variations of the external force vector. This approach renders the proposed solver substantially more efficient and robust than existing solvers used in CDM problems. We demonstrate the considerable advantages of the proposed algorithm through several benchmark 1D problems with sharp snap-backs and 2D examples under various boundary conditions and loading scenarios. The proposed UAL approach exhibits a superior ability of overcoming critical increments along the equilibrium path. Moreover, in the presented examples, the proposed UAL method is 1-2 orders of magnitude faster than force-controlled arc-length and monolithic Newton-Raphson solvers.
Navigating dynamic environments requires the robot to generate collision-free trajectories and actively avoid moving obstacles. Most previous works designed path planning algorithms based on one single map representation, such as the geometric, occupancy, or ESDF map. Although they have shown success in static environments, due to the limitation of map representation, those methods cannot reliably handle static and dynamic obstacles simultaneously. To address the problem, this paper proposes a gradient-based B-spline trajectory optimization algorithm utilizing the robot's onboard vision. The depth vision enables the robot to track and represent dynamic objects geometrically based on the voxel map. The proposed optimization first adopts the circle-based guide-point algorithm to approximate the costs and gradients for avoiding static obstacles. Then, with the vision-detected moving objects, our receding-horizon distance field is simultaneously used to prevent dynamic collisions. Finally, the iterative re-guide strategy is applied to generate the collision-free trajectory. The simulation and physical experiments prove that our method can run in real-time to navigate dynamic environments safely. Our software is available on GitHub as an open-source package.
Covariance matrices of random vectors contain information that is crucial for modelling. Certain structures and patterns of the covariances (or correlations) may be used to justify parametric models, e.g., autoregressive models. Until now, there have been only few approaches for testing such covariance structures systematically and in a unified way. In the present paper, we propose such a unified testing procedure, and we will exemplify the approach with a large variety of covariance structure models. This includes common structures such as diagonal matrices, Toeplitz matrices, and compound symmetry but also the more involved autoregressive matrices. We propose hypothesis tests for these structures, and we use bootstrap techniques for better small-sample approximation. The structures of the proposed tests invite for adaptations to other covariance patterns by choosing the hypothesis matrix appropriately. We prove their correctness for large sample sizes. The proposed methods require only weak assumptions. With the help of a simulation study, we assess the small sample properties of the tests. We also analyze a real data set to illustrate the application of the procedure.
We consider scalar semilinear elliptic PDEs, where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. To linearize the arising discrete nonlinear problem, we employ a damped Zarantonello iteration, which leads to a linear Poisson-type equation that is symmetric and positive definite. The resulting system is solved by a contractive algebraic solver such as a multigrid method with local smoothing. We formulate a fully adaptive algorithm that equibalances the various error components coming from mesh refinement, iterative linearization, and algebraic solver. We prove that the proposed adaptive iteratively linearized finite element method (AILFEM) guarantees convergence with optimal complexity, where the rates are understood with respect to the overall computational cost (i.e., the computational time). Numerical experiments investigate the involved adaptivity parameters.
Categorical compositional distributional semantics is an approach to modelling language that combines the success of vector-based models of meaning with the compositional power of formal semantics. However, this approach was developed without an eye to cognitive plausibility. Vector representations of concepts and concept binding are also of interest in cognitive science, and have been proposed as a way of representing concepts within a biologically plausible spiking neural network. This work proposes a way for compositional distributional semantics to be implemented within a spiking neural network architecture, with the potential to address problems in concept binding, and give a small implementation. We also describe a means of training word representations using labelled images.
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