Graphs and networks play an important role in modeling and analyzing complex interconnected systems such as transportation networks, integrated circuits, power grids, citation graphs, and biological and artificial neural networks. Graph clustering algorithms can be used to detect groups of strongly connected vertices and to derive coarse-grained models. We define transfer operators such as the Koopman operator and the Perron-Frobenius operator on graphs, study their spectral properties, introduce Galerkin projections of these operators, and illustrate how reduced representations can be estimated from data. In particular, we show that spectral clustering of undirected graphs can be interpreted in terms of eigenfunctions of the Koopman operator and propose novel clustering algorithms for directed graphs based on generalized transfer operators. We demonstrate the efficacy of the resulting algorithms on several benchmark problems and provide different interpretations of clusters.
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In many communication contexts, the capabilities of the involved actors cannot be known beforehand, whether it is a cell, a plant, an insect, or even a life form unknown to Earth. Regardless of the recipient, the message space and time scale could be too fast, too slow, too large, or too small and may never be decoded. Therefore, it pays to devise a way to encode messages agnostic of space and time scales. We propose the use of fractal functions as self-executable infinite-frequency carriers for sending messages, given their properties of structural self-similarity and scale invariance. We call it `fractal messaging'. Starting from a spatial embedding, we introduce a framework for a space-time scale-free messaging approach to this challenge. When considering a space and time-agnostic framework for message transmission, it would be interesting to encode a message such that it could be decoded at several spatio-temporal scales. Hence, the core idea of the framework proposed herein is to encode a binary message as waves along infinitely many frequencies (in power-like distributions) and amplitudes, transmit such a message, and then decode and reproduce it. To do so, the components of the Weierstrass function, a known fractal, are used as carriers of the message. Each component will have its amplitude modulated to embed the binary stream, allowing for a space-time-agnostic approach to messaging.
Energy-efficient spikformer has been proposed by integrating the biologically plausible spiking neural network (SNN) and artificial Transformer, whereby the Spiking Self-Attention (SSA) is used to achieve both higher accuracy and lower computational cost. However, it seems that self-attention is not always necessary, especially in sparse spike-form calculation manners. In this paper, we innovatively replace vanilla SSA (using dynamic bases calculating from Query and Key) with spike-form Fourier Transform, Wavelet Transform, and their combinations (using fixed triangular or wavelets bases), based on a key hypothesis that both of them use a set of basis functions for information transformation. Hence, the Fourier-or-Wavelet-based spikformer (FWformer) is proposed and verified in visual classification tasks, including both static image and event-based video datasets. The FWformer can achieve comparable or even higher accuracies ($0.4\%$-$1.5\%$), higher running speed ($9\%$-$51\%$ for training and $19\%$-$70\%$ for inference), reduced theoretical energy consumption ($20\%$-$25\%$), and reduced GPU memory usage ($4\%$-$26\%$), compared to the standard spikformer. Our result indicates the continuous refinement of new Transformers, that are inspired either by biological discovery (spike-form), or information theory (Fourier or Wavelet Transform), is promising.
Functional linear and single-index models are core regression methods in functional data analysis and are widely used for performing regression in a wide range of applications when the covariates are random functions coupled with scalar responses. In the existing literature, however, the construction of associated estimators and the study of their theoretical properties is invariably carried out on a case-by-case basis for specific models under consideration. In this work, assuming the predictors are Gaussian processes, we provide a unified methodological and theoretical framework for estimating the index in functional linear, and its direction in single-index models. In the latter case, the proposed approach does not require the specification of the link function. In terms of methodology, we show that the reproducing kernel Hilbert space (RKHS) based functional linear least-squares estimator, when viewed through the lens of an infinite-dimensional Gaussian Stein's identity, also provides an estimator of the index of the single-index model. Theoretically, we characterize the convergence rates of the proposed estimators for both linear and single-index models. Our analysis has several key advantages: (i) it does not require restrictive commutativity assumptions for the covariance operator of the random covariates and the integral operator associated with the reproducing kernel; and (ii) the true index parameter can lie outside of the chosen RKHS, thereby allowing for index misspecification as well as for quantifying the degree of such index misspecification. Several existing results emerge as special cases of our analysis.
The paper presents a new approach of stability evaluation of the approximate Riemann solvers based on the direct Lyapunov method. The present methodology offers a detailed understanding of the origins of numerical shock instability in the approximate Riemann solvers. The pressure perturbation feeding the density and transverse momentum perturbations is identified as the cause of the numerical shock instabilities in the complete approximate Riemann solvers, while the magnitude of the numerical shock instabilities are found to be proportional to the magnitude of the pressure perturbations. A shock-stable HLLEM scheme is proposed based on the insights obtained from this analysis about the origins of numerical shock instability in the approximate Riemann solvers. A set of numerical test cases are solved to show that the proposed scheme is free from numerical shock instability problems of the original HLLEM scheme at high Mach numbers.
Vast amounts of (open) data are increasingly used to make arguments about crisis topics such as climate change and global pandemics. Data visualizations are central to bringing these viewpoints to broader publics. However, visualizations often conceal the many contexts involved in their production, ranging from decisions made in research labs about collecting and sharing data to choices made in editorial rooms about which data stories to tell. In this paper, we examine how data visualizations about climate change and COVID-19 are produced in popular science magazines, using Scientific American, an established English-language popular science magazine, as a case study. To do this, we apply the analytical concept of data journeys (Leonelli, 2020) in a mixed methods study that centers on interviews with Scientific American staff and is supplemented by a visualization analysis of selected charts. In particular, we discuss the affordances of working with open data, the role of collaborative data practices, and how the magazine works to counter misinformation and increase transparency. This work provides an empirical contribution by providing insight into the data (visualization) practices of science communicators and demonstrating how the concept of data journeys can be used as an analytical framework.
Navigating multi-robot systems in complex terrains has always been a challenging task. This is due to the inherent limitations of traditional robots in collision avoidance, adaptation to unknown environments, and sustained energy efficiency. In order to overcome these limitations, this research proposes a solution by integrating living insects with miniature electronic controllers to enable robotic-like programmable control, and proposing a novel control algorithm for swarming. Although these creatures, called cyborg insects, have the ability to instinctively avoid collisions with neighbors and obstacles while adapting to complex terrains, there is a lack of literature on the control of multi-cyborg systems. This research gap is due to the difficulty in coordinating the movements of a cyborg system under the presence of insects' inherent individual variability in their reactions to control input. In response to this issue, we propose a novel swarm navigation algorithm addressing these challenges. The effectiveness of the algorithm is demonstrated through an experimental validation in which a cyborg swarm was successfully navigated through an unknown sandy field with obstacles and hills. This research contributes to the domain of swarm robotics and showcases the potential of integrating biological organisms with robotics and control theory to create more intelligent autonomous systems with real-world applications.
Regularization of inverse problems is of paramount importance in computational imaging. The ability of neural networks to learn efficient image representations has been recently exploited to design powerful data-driven regularizers. While state-of-the-art plug-and-play methods rely on an implicit regularization provided by neural denoisers, alternative Bayesian approaches consider Maximum A Posteriori (MAP) estimation in the latent space of a generative model, thus with an explicit regularization. However, state-of-the-art deep generative models require a huge amount of training data compared to denoisers. Besides, their complexity hampers the optimization involved in latent MAP derivation. In this work, we first propose to use compressive autoencoders instead. These networks, which can be seen as variational autoencoders with a flexible latent prior, are smaller and easier to train than state-of-the-art generative models. As a second contribution, we introduce the Variational Bayes Latent Estimation (VBLE) algorithm, which performs latent estimation within the framework of variational inference. Thanks to a simple yet efficient parameterization of the variational posterior, VBLE allows for fast and easy (approximate) posterior sampling. Experimental results on image datasets BSD and FFHQ demonstrate that VBLE reaches similar performance than state-of-the-art plug-and-play methods, while being able to quantify uncertainties faster than other existing posterior sampling techniques.
For multivariate data, tandem clustering is a well-known technique aiming to improve cluster identification through initial dimension reduction. Nevertheless, the usual approach using principal component analysis (PCA) has been criticized for focusing solely on inertia so that the first components do not necessarily retain the structure of interest for clustering. To address this limitation, a new tandem clustering approach based on invariant coordinate selection (ICS) is proposed. By jointly diagonalizing two scatter matrices, ICS is designed to find structure in the data while providing affine invariant components. Certain theoretical results have been previously derived and guarantee that under some elliptical mixture models, the group structure can be highlighted on a subset of the first and/or last components. However, ICS has garnered minimal attention within the context of clustering. Two challenges associated with ICS include choosing the pair of scatter matrices and selecting the components to retain. For effective clustering purposes, it is demonstrated that the best scatter pairs consist of one scatter matrix capturing the within-cluster structure and another capturing the global structure. For the former, local shape or pairwise scatters are of great interest, as is the minimum covariance determinant (MCD) estimator based on a carefully chosen subset size that is smaller than usual. The performance of ICS as a dimension reduction method is evaluated in terms of preserving the cluster structure in the data. In an extensive simulation study and empirical applications with benchmark data sets, various combinations of scatter matrices as well as component selection criteria are compared in situations with and without outliers. Overall, the new approach of tandem clustering with ICS shows promising results and clearly outperforms the PCA-based approach.
This paper considers computational methods that split a vector field into three components in the case when both the vector field and the split components might be unbounded. We first employ classical Taylor expansion which, after some algebra, results in an expression for a second-order splitting which, strictly speaking, makes sense only for bounded operators. Next, using an alternative approach, we derive an error expression and an error bound in the same setting which are however valid in the presence of unbounded operators. While the paper itself is concerned with second-order splittings using three components, the method of proof in the presence of unboundedness remains valid (although significantly more complicated) in a more general scenario, which will be the subject of a forthcoming paper.
We study a multi-server queueing system with a periodic arrival rate and customers whose joining decision is based on their patience and a delay proxy. Specifically, each customer has a patience level sampled from a common distribution. Upon arrival, they receive an estimate of their delay before joining service and then join the system only if this delay is not more than their patience, otherwise they balk. The main objective is to estimate the parameters pertaining to the arrival rate and patience distribution. Here the complication factor is that this inference should be performed based on the observed process only, i.e., balking customers remain unobserved. We set up a likelihood function of the state dependent effective arrival process (i.e., corresponding to the customers who join), establish strong consistency of the MLE, and derive the asymptotic distribution of the estimation error. Due to the intrinsic non-stationarity of the Poisson arrival process, the proof techniques used in previous work become inapplicable. The novelty of the proving mechanism in this paper lies in the procedure of constructing i.i.d. objects from dependent samples by decomposing the sample path into i.i.d. regeneration cycles. The feasibility of the MLE-approach is discussed via a sequence of numerical experiments, for multiple choices of functions which provide delay estimates. In particular, it is observed that the arrival rate is best estimated at high service capacities, and the patience distribution is best estimated at lower service capacities.
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