In this paper we introduce some new algebraic and geometric perspectives on networked space communications. Our main contribution is a novel definition of a time-varying graph (TVG), defined in terms of a matrix with values in subsets of the real line P(R). We leverage semi-ring properties of P(R) to model multi-hop communication in a TVG using matrix multiplication and a truncated Kleene star. This leads to novel statistics on the communication capacity of TVGs called lifetime curves, which we generate for large samples of randomly chosen STARLINK satellites, whose connectivity is modeled over day-long simulations. Determining when a large subsample of STARLINK is temporally strongly connected is further analyzed using novel metrics introduced here that are inspired by topological data analysis (TDA). To better model networking scenarios between the Earth and Mars, we introduce various semi-rings capable of modeling propagation delay as well as protocols common to Delay Tolerant Networking (DTN), such as store-and-forward. Finally, we illustrate the applicability of zigzag persistence for featurizing different space networks and demonstrate the efficacy of K-Nearest Neighbors (KNN) classification for distinguishing Earth-Mars and Earth-Moon satellite systems using time-varying topology alone.
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As a transformative general-purpose technology, AI has empowered various industries and will continue to shape our lives through ubiquitous applications. Despite the enormous benefits from wide-spread AI deployment, it is crucial to address associated downside risks and therefore ensure AI advances are safe, fair, responsible, and aligned with human values. To do so, we need to establish effective AI governance. In this work, we show that the strategic interaction between the regulatory agencies and AI firms has an intrinsic structure reminiscent of a Stackelberg game, which motivates us to propose a game-theoretic modeling framework for AI governance. In particular, we formulate such interaction as a Stackelberg game composed of a leader and a follower, which captures the underlying game structure compared to its simultaneous play counterparts. Furthermore, the choice of the leader naturally gives rise to two settings. And we demonstrate that our proposed model can serves as a unified AI governance framework from two aspects: firstly we can map one setting to the AI governance of civil domains and the other to the safety-critical and military domains, secondly, the two settings of governance could be chosen contingent on the capability of the intelligent systems. To the best of our knowledge, this work is the first to use game theory for analyzing and structuring AI governance. We also discuss promising directions and hope this can help stimulate research interest in this interdisciplinary area. On a high, we hope this work would contribute to develop a new paradigm for technology policy: the quantitative and AI-driven methods for the technology policy field, which holds significant promise for overcoming many shortcomings of existing qualitative approaches.
This paper presents a novel transceiver design aimed at enabling Direct-to-Satellite Internet of Things (DtS-IoT) systems based on long range-frequency hopping spread spectrum (LR-FHSS). Our focus lies in developing an accurate transmission method through the analysis of the frame structure and key parameters outlined in Long Range Wide-Area Network (LoRaWAN) [1]. To address the Doppler effect in DtS-IoT networks and simultaneously receive numerous frequency hopping signals, a robust signal detector for the receiver is proposed. We verify the performance of the proposed LR-FHSS transceiver design through simulations conducted in a realistic satellite channel environment, assessing metrics such as miss detection probability and packet error probability.
The Terahertz (0.1-10 THz) band has been envisioned as one of the promising spectrum bands to support ultra-broadband sixth-generation (6G) and beyond communications. In this paper, a wideband channel measurement campaign in a 500- square-meter indoor lobby at 306-321 GHz is presented. The measurement system consists of a vector network analyzer (VNA)-based channel sounder, and a directional antenna equipped at the receiver to resolve multi-path components (MPCs) in the angular domain. In particular, 21 positions and 3780 channel impulse responses (CIRs) are measured in the lobby, including the line-of-sight (LoS), non-line-of-sight (NLoS) and obstructed-line-of-sight (OLoS) cases. The multi-path characteristics are summarized as follows. First, the main scatterers in the lobby include the glass, the pillar, and the LED screen. Second, best direction and omni-directional path losses are analyzed. Compared with the close-in path loss model, the optimal path loss offset in the alpha-beta path loss model exceeds 86 dB in the LoS case, and accordingly, the exponent decreases to 1.57 and below. Third, more than 10 clusters are observed in OLoS and NLoS cases, compared to 2.17 clusters on average in the LoS case. Fourth, the average power dispersion of MPCs is smaller in both temporal and angular domains in the LoS case, compared with the NLoS and OLoS counterparts. Finally, in contrast to hallway scenarios measured in previous works at the same frequency band, the lobby which is larger in dimension and square in shape, features larger path losses and smaller delay and angular spreads.
Probabilistic graphical models have become an important unsupervised learning tool for detecting network structures for a variety of problems, including the estimation of functional neuronal connectivity from two-photon calcium imaging data. However, in the context of calcium imaging, technological limitations only allow for partially overlapping layers of neurons in a brain region of interest to be jointly recorded. In this case, graph estimation for the full data requires inference for edge selection when many pairs of neurons have no simultaneous observations. This leads to the Graph Quilting problem, which seeks to estimate a graph in the presence of block-missingness in the empirical covariance matrix. Solutions for the Graph Quilting problem have previously been studied for Gaussian graphical models; however, neural activity data from calcium imaging are often non-Gaussian, thereby requiring a more flexible modeling approach. Thus, in our work, we study two approaches for nonparanormal Graph Quilting based on the Gaussian copula graphical model, namely a maximum likelihood procedure and a low-rank based framework. We provide theoretical guarantees on edge recovery for the former approach under similar conditions to those previously developed for the Gaussian setting, and we investigate the empirical performance of both methods using simulations as well as real data calcium imaging data. Our approaches yield more scientifically meaningful functional connectivity estimates compared to existing Gaussian graph quilting methods for this calcium imaging data set.
Recent advancements in generative modeling have made it possible to generate high-quality content from context information, but a key question remains: how to teach models to know when to generate content? To answer this question, this study proposes a novel event generative model that draws its statistical intuition from marked temporal point processes, and offers a clean, flexible, and computationally efficient solution for a wide range of applications involving multi-dimensional marks. We aim to capture the distribution of the point process without explicitly specifying the conditional intensity or probability density. Instead, we use a conditional generator that takes the history of events as input and generates the high-quality subsequent event that is likely to occur given the prior observations. The proposed framework offers a host of benefits, including exceptional efficiency in learning the model and generating samples, as well as considerable representational power to capture intricate dynamics in multi- or even high-dimensional event space. Our numerical results demonstrate superior performance compared to other state-of-the-art baselines.
Communicating science and technology is essential for the public to understand and engage in a rapidly changing world. Tweetorials are an emerging phenomenon where experts explain STEM topics on social media in creative and engaging ways. However, STEM experts struggle to write an engaging "hook" in the first tweet that captures the reader's attention. We propose methods to use large language models (LLMs) to help users scaffold their process of writing a relatable hook for complex scientific topics. We demonstrate that LLMs can help writers find everyday experiences that are relatable and interesting to the public, avoid jargon, and spark curiosity. Our evaluation shows that the system reduces cognitive load and helps people write better hooks. Lastly, we discuss the importance of interactivity with LLMs to preserve the correctness, effectiveness, and authenticity of the writing.
The design of algorithms for political redistricting generally takes one of two approaches: optimize an objective such as compactness or, drawing on fair division, construct a protocol whose outcomes guarantee partisan fairness. We aim to have the best of both worlds by optimizing an objective subject to a binary fairness constraint. As the fairness constraint we adopt the geometric target, which requires the number of seats won by each party to be at least the average (rounded down) of its outcomes under the worst and best partitions of the state. To study the feasibility of this approach, we introduce a new model of redistricting that closely mirrors the classic model of cake-cutting. This model has two innovative features. First, in any part of the state there is an underlying 'density' of voters with political leanings toward any given party, making it impossible to finely separate voters for different parties into different districts. This captures a realistic constraint that previously existing theoretical models of redistricting tend to ignore. Second, parties may disagree on the distribution of voters - whether by genuine disagreement or attempted strategic behavior. In the absence of a 'ground truth' distribution, a redistricting algorithm must therefore aim to simultaneously be fair to each party with respect to its own reported data. Our main theoretical result is that, surprisingly, the geometric target is always feasible with respect to arbitrarily diverging data sets on how voters are distributed. Any standard for fairness is only useful if it can be readily satisfied in practice. Our empirical results, which use real election data and maps of six US states, demonstrate that the geometric target is always feasible, and that imposing it as a fairness constraint comes at almost no cost to three well-studied optimization objectives.
Many algorithms for determining properties of real algebraic or semi-algebraic sets rely upon the ability to compute smooth points. Existing methods to compute smooth points on semi-algebraic sets use symbolic quantifier elimination tools. In this paper, we present a simple algorithm based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected compact component of a real (semi)-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for the $n=4$ case. We also apply our method to design an efficient algorithm to compute the real dimension of (semi)-algebraic sets, the original motivation for this research.
The problem of reducing a Hidden Markov Model (HMM) to one of smaller dimension that exactly reproduces the same marginals is tackled by using a system-theoretic approach. Realization theory tools are extended to HMMs by leveraging suitable algebraic representations of probability spaces. We propose two algorithms that return coarse-grained equivalent HMMs obtained by stochastic projection operators: the first returns models that exactly reproduce the single-time distribution of a given output process, while in the second the full (multi-time) distribution is preserved. The reduction method exploits not only the structure of the observed output, but also its initial condition, whenever the latter is known or belongs to a given subclass. Optimal algorithms are derived for a class of HMM, namely observable ones.
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.
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