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We consider a multiple hypothesis testing problem in a sensor network over the joint spatial-time domain. The sensor network is modeled as a graph, with each vertex representing a sensor and a signal over time associated with each vertex. We assume a hypothesis test and an associated p-value for every sample point in the joint spatial-time domain. Our goal is to determine which points have true alternative hypotheses. By parameterizing the unknown alternative distribution of $p$-values and the prior probabilities of hypotheses being null with a bandlimited generalized graph signal, we can obtain consistent estimates for them. Consequently, we also obtain an estimate of the local false discovery rates (lfdr). We prove that by using a step-up procedure on the estimated lfdr, we can achieve asymptotic false discovery rate control at a pre-determined level. Numerical experiments validate the effectiveness of our approach compared to existing methods.

In Bayesian quantile regression, the most commonly used likelihood is the asymmetric Laplace (AL) likelihood. The reason for this choice is not that it is a plausible data-generating model but that the corresponding maximum likelihood estimator is identical to the classical estimator by Koenker and Bassett (1978), and in that sense, the AL likelihood can be thought of as a working likelihood. AL-based quantile regression has been shown to produce good finite-sample Bayesian point estimates and to be consistent. However, if the AL distribution does not correspond to the data-generating distribution, credible intervals based on posterior standard deviations can have poor coverage. Yang, Wang, and He (2016) proposed an adjustment to the posterior covariance matrix that produces asymptotically valid intervals. However, we show that this adjustment is sensitive to the choice of scale parameter for the AL likelihood and can lead to poor coverage when the sample size is small to moderate. We therefore propose using Infinitesimal Jackknife (IJ) standard errors (Giordano & Broderick, 2023). These standard errors do not require resampling but can be obtained from a single MCMC run. We also propose a version of IJ standard errors for clustered data. Simulations and applications to real data show that the IJ standard errors have good frequentist properties, both for independent and clustered data. We provide an R-package that computes IJ standard errors for clustered or independent data after estimation with the brms wrapper in R for Stan.

The paper investigates the distributed estimation problem under low bit rate communications. Based on the signal-comparison (SC) consensus protocol under binary-valued communications, a new consensus+innovations type distributed estimation algorithm is proposed. Firstly, the high-dimensional estimates are compressed into binary-valued messages by using a periodic compressive strategy, dithered noises and a sign function. Next, based on the dithered noises and expanding triggering thresholds, a new stochastic event-triggered mechanism is proposed to reduce the communication frequency. Then, a modified SC consensus protocol is applied to fuse the neighborhood information. Finally, a stochastic approximation estimation algorithm is used to process innovations. The proposed SC-based algorithm has the advantages of high effectiveness and low communication cost. For the effectiveness, the estimates of the SC-based algorithm converge to the true value in the almost sure and mean square sense. A polynomial almost sure convergence rate is also obtained. For the communication cost, the local and global average bit rates for communications decay to zero at a polynomial rate. The trade-off between the convergence rate and the communication cost is established through event-triggered coefficients. A better convergence rate can be achieved by decreasing event-triggered coefficients, while lower communication cost can be achieved by increasing event-triggered coefficients. A simulation example is given to demonstrate the theoretical results.

Analyzing time series of fluxes from stars, known as stellar light curves, can reveal valuable information about stellar properties. However, most current methods rely on extracting summary statistics, and studies using deep learning have been limited to supervised approaches. In this research, we investigate the scaling law properties that emerge when learning from astronomical time series data using self-supervised techniques. By employing the GPT-2 architecture, we show the learned representation improves as the number of parameters increases from $10^4$ to $10^9$, with no signs of performance plateauing. We demonstrate that a self-supervised Transformer model achieves 3-10 times the sample efficiency compared to the state-of-the-art supervised learning model when inferring the surface gravity of stars as a downstream task. Our research lays the groundwork for analyzing stellar light curves by examining them through large-scale auto-regressive generative models., Comment: 11 pages, 5 figures, ICML 2024 AI4Science workshop

We introduce the FRactional-Order graph Neural Dynamical network (FROND), a new continuous graph neural network (GNN) framework. Unlike traditional continuous GNNs that rely on integer-order differential equations, FROND employs the Caputo fractional derivative to leverage the non-local properties of fractional calculus. This approach enables the capture of long-term dependencies in feature updates, moving beyond the Markovian update mechanisms in conventional integer-order models and offering enhanced capabilities in graph representation learning. We offer an interpretation of the node feature updating process in FROND from a non-Markovian random walk perspective when the feature updating is particularly governed by a diffusion process. We demonstrate analytically that oversmoothing can be mitigated in this setting. Experimentally, we validate the FROND framework by comparing the fractional adaptations of various established integer-order continuous GNNs, demonstrating their consistently improved performance and underscoring the framework's potential as an effective extension to enhance traditional continuous GNNs. The code is available at \url{https://github.com/zknus/ICLR2024-FROND}., Comment: The Twelfth International Conference on Learning Representations

This paper proposes a new distributed nonconvex stochastic optimization algorithm that can achieve privacy protection, communication efficiency and convergence simultaneously. Specifically, each node adds time-varying privacy noises to its local state to avoid information leakage, and then quantizes its noise-perturbed state before transmitting to improve communication efficiency. By employing the subsampling method controlled through the sample-size parameter, the proposed algorithm reduces the impact of privacy noises, and enhances the differential privacy level. When the global cost function satisfies the Polyak-Lojasiewicz condition, the mean and high-probability convergence rate and the oracle complexity of the proposed algorithm are given. Importantly, the proposed algorithm achieves both the mean convergence and a finite cumulative differential privacy budget over infinite iterations as the sample-size goes to infinity. A numerical example of the distributed training on the "MNIST" dataset is given to show the effectiveness of the algorithm.

In the seminal paper (Alt, Caffarelli and Friedman, Trans. Amer. Math. Soc., 282, (1984).), the regularity of the free boundary of two-phase fluid in two dimensions via the so-called ACF energy functional was investigated. It was shown the $C^1$ regularity of the free boundaries and asserted that the two free boundaries coincide under some additional assumptions. Later on the standard technique of Harnack inequality could be applied to improve the regularity to $C^{1,\eta}$. A recent significant breakthrough in the regularity of two-phase fluid is due to De Philippis, Spolaor and Velichkov, who investigated the free boundary of the two-phase fluid with the two-phase functional (De Philippis, Spolaor and Velichkov, Invent. Math., 225, (2021).), and the $C^{1,\eta}$ regularity of the whole free boundaries was given in dimension two. Moreover, the free boundaries of the two-phase fluids do not coincide and the zero level set may process positive Lebesgue measure. In this paper, we consider the free boundaries for the two-phase axisymmetric fluid and show the free boundary is $C^{1,\eta}$ smooth. The Lebesgue measure of the zero level set of may also be positive, and the main difference lies in the degenerate elliptic operator and the free boundary conditions. More precisely, we use partial boundary Harnack inequalities and establish a linearized problem, whose regularity of the solutions implies the flatness decay of the two-phase free boundaries. Then the iteration argument gives the smoothness of the free boundaries., Comment: 52 pages. Any comments are welcome

The theory of sampling and recovery of bandlimited graph signals has been extensively studied. However, in many cases, the observation of a signal is quite coarse. For example, users only provide simple comments such as "like" or "dislike" for a product on an e-commerce platform. This is a particular scenario where only the sign information of a graph signal can be measured. In this paper, we are interested in how to sample based on sign information in an online manner, by which the direction of the original graph signal can be estimated. The online signed sampling problem of a graph signal can be formulated as a Markov decision process in a finite horizon. Unfortunately, it is intractable for large size graphs. We propose a low-complexity greedy signed sampling algorithm (GSS) as well as a stopping criterion. Meanwhile, we prove that the objective function is adaptive monotonic and adaptive submodular, so that the performance is close enough to the global optimum with a lower bound. Finally, we demonstrate the effectiveness of the GSS algorithm by both synthesis and realworld data.

This paper proposes a node flux-linkage synchronizing control method (NFSCM) for power systems with 100% wind power generation based on a capacitor voltage balancing scheme (CVBS). Different from the conventional grid-forming controllers, NFSCM is designed to regulate inverters as virtual flux-linkage sources. Auto-synchronization of flux-linkage vectors is achieved through the CVBS-based NFSCM. The mismatch among the angular frequencies of flux-linkage vectors is eliminated by regulating the tracking errors of DC-link voltages, which establishes a negative feedback between the output frequency and active power of the inverter. NFSCM is adaptive to weak and strong grids. It avoids the excitation inrush currents in the step-up transformer of wind power generators. It also eliminates the DC components of the three-phase currents, and avoids low-frequency oscillations in active power. In order to limit the short-circuit current of inverters, a logic-based bang-bang funnel control (LBFC) is designed to control the switches of inverter bridges when over-current is detected. LBFC is able to restrict various fault currents within an acceptable range within the shortest time. LBFC and NFSCM are designed to operate in a switched manner according to a state-dependent switching strategy. Time-domain simulations were conducted on a 100% wind power generation test system, and the performance of NFSCM and LBFC were investigated., Comment: 9 pages, 9 figures

In this paper, we present a signal processing framework for directed graphs. Unlike undirected graphs, a graph shift operator such as the adjacency matrix associated with a directed graph usually does not admit an orthogonal eigenbasis. This makes it challenging to define the Fourier transform. Our methodology leverages the polar decomposition to define two distinct eigendecompositions, each associated with different matrices derived from this decomposition. We propose to extend the frequency domain and introduce a Fourier transform that jointly encodes the spectral response of a signal for the two eigenbases from the polar decomposition. This allows us to define convolution following a standard routine. Our approach has two features: it is lossless as the shift operator can be fully recovered from factors of the polar decomposition. Moreover, it subsumes the traditional graph signal processing if the graph is directed. We present numerical results to show how the framework can be applied.

Different types of phase transition from hadron to quark at high density near zero temperature may occur in the inner core of hybrid stars. We investigate the impacts of phase transition types and quark models on properties of hybrid star and quark cores. The quark-meson coupling (QMC) model is used to describe hadronic matter, and the MIT bag model as well as the Nambu-Jona-Lasinio (NJL) model are employed to describe quark matter for comparison. From the mass-radius curves obtained by using equations of state (EOS), we find that EOSs of hadron matter have a decisive influence on the maximum mass of a hybrid star in the first-order phase transition case, while quark matter EOSs have more impacts on results in the crossover transition case. It is also found in the present work that the thermodynamic correction arising from an interpolation scheme considerably stiffens the EOSs. Therefore the crossover type phase transition generally leads to hybrid stars with higher masses. In particular, by using the QMC model and the NJL model to construct crossover EOSs with thermodynamic correction, we discover that the maximum masses of hybrid stars can meet the recent observational constraint on mass from PSR J0952-0607, i.e., $2.35\pm0.17M_\odot$., Comment: 13pages, 10figures, 5 tables, revised version published in Physical Review D

Graphons are limit objects of sequences of graphs and are used to analyze the behavior of large graphs. Recently, graphon signal processing has been developed to study signal processing on large graphs. A major limitation of this approach is that any sparse sequence of graphs inevitably converges to the zero graphon, rendering the resulting signal processing theory trivial and inadequate for sparse graph sequences. To overcome this limitation, we propose a new signal processing framework that leverages the concept of generalized graphons and introduces the stretched cut distance as a measure to compare these graphons. Our framework focuses on the sampling of graph sequences from generalized graphons and explores the convergence properties of associated operators, spectra, and signals. Our signal processing framework provides a comprehensive approach to analyzing and processing signals on graph sequences, even if they are sparse. Finally, we discuss the practical implications of our theory for real-world large networks through numerical experiments.

This paper proposes a consensus-based distributed nonlinear filter with kernel mean embedding (KME). This fills with gap of posterior density approximation with KME for distributed nonlinear dynamic systems. To approximate the posterior density, the system state is embedded into a higher-dimensional reproducing kernel Hilbert space (RKHS), and then the nonlinear measurement function is linearly converted. As a result, an update rule of KME of posterior distribution is established in the RKHS. To show the proposed distributed filter being capable of achieving the centralized estimation accuracy, a centralized filter, serving as an extension of the standard Kalman filter in the state space to the RKHS, is developed first. Benefited from the KME, the proposed distributed filter converges to the centralized one while maintaining the distributed pattern. Two examples are introduced to demonstrate the effectiveness of the developed filters in target tracking scenarios including nearly constantly moving target and turning target, respectively, with bearing-only, range and bearing measurements.

This paper proposes to analyze the motion stability of synchro-nous generator power systems using a Lagrangian model derived in the configuration space of generalized position and speed. In the first place, a Lagrangian model of synchronous generators is derived based on Lagrangian mechanics. The generalized potential energy of inductors and generalized kinetic energy of capacitors are defined. The mechanical and electrical dynamics can be modelled in a unified manner through constructing a Lagrangian function. Taking the first benchmark model of subsynchronous oscillation as an example, a Lagragian model is construct-ed and numerical solution of the model is obtained to validate the accuracy and effectiveness of the model. Compared with the traditional EMTP model in PSCAD, the obtained Lagrangian model is able to accurately describe the electromagnetic transient process of the system. Moreover, the Lagrangian model is analytical, which enables to analyze the motion stability of the system using Lyapunov motion stability theory. The Lagrangian model can not only be used to discuss the power angle stability, but also used for analyzing the stability of node voltages and frequency. It provides the feasibility for studying the unified stability of power systems., Comment: 20 pages, 11 figures

Traditionally, graph neural networks have been trained using a single observed graph. However, the observed graph represents only one possible realization. In many applications, the graph may encounter uncertainties, such as having erroneous or missing edges, as well as edge weights that provide little informative value. To address these challenges and capture additional information previously absent in the observed graph, we introduce latent variables to parameterize and generate multiple graphs. We obtain the maximum likelihood estimate of the network parameters in an Expectation-Maximization (EM) framework based on the multiple graphs. Specifically, we iteratively determine the distribution of the graphs using a Markov Chain Monte Carlo (MCMC) method, incorporating the principles of PAC-Bayesian theory. Numerical experiments demonstrate improvements in performance against baseline models on node classification for heterogeneous graphs and graph regression on chemistry datasets.

In the short note, we describe a sampling construction that yields a sequence of graphons converging to a prescribed limit graphon in 1-norm. This convergence is stronger than the convergence in the cut norm, usually used to study graphon sequences. The note also contains errata of the previous version of the note.

Differentially private distributed stochastic optimization has become a hot topic due to the urgent need of privacy protection in distributed stochastic optimization. In this paper, two-time scale stochastic approximation-type algorithms for differentially private distributed stochastic optimization with time-varying sample sizes are proposed using gradient- and output-perturbation methods. For both gradient- and output-perturbation cases, the convergence of the algorithm and differential privacy with a finite cumulative privacy budget $\varepsilon$ for an infinite number of iterations are simultaneously established, which is substantially different from the existing works. By a time-varying sample sizes method, the privacy level is enhanced, and differential privacy with a finite cumulative privacy budget $\varepsilon$ for an infinite number of iterations is established. By properly choosing a Lyapunov function, the algorithm achieves almost-sure and mean-square convergence even when the added privacy noises have an increasing variance. Furthermore, we rigorously provide the mean-square convergence rates of the algorithm and show how the added privacy noise affects the convergence rate of the algorithm. Finally, numerical examples including distributed training on a benchmark machine learning dataset are presented to demonstrate the efficiency and advantages of the algorithms., Comment: accepted by IEEE Trans. on Automatic Control

Topological Signal Processing (TSP) utilizes simplicial complexes to model structures with higher order than vertices and edges. In this paper, we study the transferability of TSP via a generalized higher-order version of graphon, known as complexon. We recall the notion of a complexon as the limit of a simplicial complex sequence [1]. Inspired by the graphon shift operator and message-passing neural network, we construct a marginal complexon and complexon shift operator (CSO) according to components of all possible dimensions from the complexon. We investigate the CSO's eigenvalues and eigenvectors and relate them to a new family of weighted adjacency matrices. We prove that when a simplicial complex signal sequence converges to a complexon signal, the eigenvalues, eigenspaces, and Fourier transform of the corresponding CSOs converge to that of the limit complexon signal. This conclusion is further verified by two numerical experiments. These results hint at learning transferability on large simplicial complexes or simplicial complex sequences, which generalize the graphon signal processing framework., Comment: 9 pages, 2 figures

Graphons have traditionally served as limit objects for dense graph sequences, with the cut distance serving as the metric for convergence. However, sparse graph sequences converge to the trivial graphon under the conventional definition of cut distance, which make this framework inadequate for many practical applications. In this paper, we utilize the concepts of generalized graphons and stretched cut distance to describe the convergence of sparse graph sequences. Specifically, we consider a random graph process generated from a generalized graphon. This random graph process converges to the generalized graphon in stretched cut distance. We use this random graph process to model the growing sparse graph, and prove the convergence of the adjacency matrices' eigenvalues. We supplement our findings with experimental validation. Our results indicate the possibility of transfer learning between sparse graphs.

This paper is concerned with the optimal identification problem of dynamical systems in which only quantized output observations are available under the assumption of fixed thresholds and bounded persistent excitations. Based on a time-varying projection, a weighted Quasi-Newton type projection (WQNP) algorithm is proposed. With some mild conditions on the weight coefficients, the algorithm is proved to be mean square and almost surely convergent, and the convergence rate can be the reciprocal of the number of observations, which is the same order as the optimal estimate under accurate measurements. Furthermore, inspired by the structure of the Cramer-Rao lower bound, an information-based identification (IBID) algorithm is constructed with adaptive design about weight coefficients of the WQNP algorithm, where the weight coefficients are related to the parameter estimates which leads to the essential difficulty of algorithm analysis. Beyond the convergence properties, this paper demonstrates that the IBID algorithm tends asymptotically to the Cramer-Rao lower bound, and hence is asymptotically efficient. Numerical examples are simulated to show the effectiveness of the information-based identification algorithm., Comment: 16 pages, 3 figures, submitted to Automatica

Due to inappropriate sample selection and limited training data, a distribution shift often exists between the training and test sets. This shift can adversely affect the test performance of Graph Neural Networks (GNNs). Existing approaches mitigate this issue by either enhancing the robustness of GNNs to distribution shift or reducing the shift itself. However, both approaches necessitate retraining the model, which becomes unfeasible when the model structure and parameters are inaccessible. To address this challenge, we propose FR-GNN, a general framework for GNNs to conduct feature reconstruction. FRGNN constructs a mapping relationship between the output and input of a well-trained GNN to obtain class representative embeddings and then uses these embeddings to reconstruct the features of labeled nodes. These reconstructed features are then incorporated into the message passing mechanism of GNNs to influence the predictions of unlabeled nodes at test time. Notably, the reconstructed node features can be directly utilized for testing the well-trained model, effectively reducing the distribution shift and leading to improved test performance. This remarkable achievement is attained without any modifications to the model structure or parameters. We provide theoretical guarantees for the effectiveness of our framework. Furthermore, we conduct comprehensive experiments on various public datasets. The experimental results demonstrate the superior performance of FRGNN in comparison to multiple categories of baseline methods.

One pillar of quantum magnonics is to explore the utilization of mediation role of magnons in different platforms to develop quantum technologies. The efficient coupling between magnons and various quantum entities is a prerequisite. Here, we propose a scheme to enhance the spin-magnon coupling by the magnonic Kerr nonlinearity in a YIG sphere. We find that the Kerr enhanced spin-magnon coupling invalidates the widely used single-Kittel-mode approximation to magnons. It is revealed that the spin decoherence induced by the multi-mode magnons in the strong-coupling regime becomes not severe, but suppressed manifesting as either population trapping or persistent Rabi-like oscillation. This anomalous effect is due to that the spin changes to be so hybridized with the magnons that one or two bound states are formed between them. Enriching the spin-magnon coupling physics, the result supplies a guideline to control the spin-magnon interface.

As one of the closest supernovae (SNe) in the last decade, SN 2023ixf is an unprecedented target to investigate the progenitor star that exploded. However, there is still significant uncertainty in the reported progenitor properties. In this work, we present a detailed study of the progenitor of SN 2023ixf with two independent analyses. We first modelled its spectral energy distribution (SED) based on Hubble Space Telescope optical, Spitzer mid-infrared (IR), and ground-based near-IR data. We find that stellar pulsation and circumstellar extinction have great impacts on SED fitting, and the result suggests a relatively massive red supergiant (RSG) surrounded by C-rich dust with an initial mass of 16.2--17.4 Msun. The corresponding rate of mass-loss occurring at least 3 years before the SN explosion is about $2 \times 10^{-4} M_\odot$yr$^{-1}$. We also derived the star formation history of the SN environment based on resolved stellar populations, and the most recent star-forming epoch corresponds to a progenitor initial mass of 17--19 Msun, in agreement with that from our SED fitting. Therefore, we conclude that the progenitor of SN 2023ixf is close to the high-mass end for Type II SN progenitors., Comment: 12 pages, 5 figures, accepted by ApJL

In this expository article, we provide a self-contained overview of the notion of convolution embedded in different theories: from the classical Fourier theory to the theory of algebraic signal processing. We discuss their relations and differences. Toward the end, we provide an opinion on whether there is a consistent approach to convolution that unifies seemingly different approaches by different theories.

We present precise photometric estimates of stellar parameters, including effective temperature, metallicity, luminosity classification, distance, and stellar age, for nearly 26 million stars using the methodology developed in the first paper of this series, based on the stellar colors from the Stellar Abundances and Galactic Evolution Survey (SAGES) DR1 and Gaia EDR3. The optimal design of stellar-parameter sensitive $uv$ filters by SAGES has enabled us to determine photometric-metallicity estimates down to $-3.5$, similar to our previous results with the SkyMapper Southern Survey (SMSS), yielding a large sample of over five million metal-poor (MP; [Fe/H]$\le -1.0$) stars and nearly one million very metal-poor (VMP; [Fe/H]$\le -2.0$) stars. The typical precision is around $0.1$ dex for both dwarf and giant stars with [Fe/H]$>-1.0$, and 0.15-0.25/0.3-0.4 dex for dwarf/giant stars with [Fe/H]$<-1.0$. Using the precise parallax measurements and stellar colors from Gaia, effective temperature, luminosity classification, distance and stellar age are further derived for our sample stars. This huge data set in the Northern sky from SAGES, together with similar data in the Southern sky from SMSS, will greatly advance our understanding of the Milky Way, in particular its formation and evolution., Comment: 14 pages, 14 figures, 3 tables, accepted by ApJ. arXiv admin note: text overlap with arXiv:2104.14154

Knowledge distillation is of key importance to launching multilingual pre-trained language models for real applications. To support cost-effective language inference in multilingual settings, we propose AMTSS, an adaptive multi-teacher single-student distillation framework, which allows distilling knowledge from multiple teachers to a single student. We first introduce an adaptive learning strategy and teacher importance weight, which enables a student to effectively learn from max-margin teachers and easily adapt to new languages. Moreover, we present a shared student encoder with different projection layers in support of multiple languages, which contributes to largely reducing development and machine cost. Experimental results show that AMTSS gains competitive results on the public XNLI dataset and the realistic industrial dataset AliExpress (AE) in the E-commerce scenario.

Topological signal processing (TSP) over simplicial complexes typically assumes observations associated with the simplicial complexes are real scalars. In this paper, we develop TSP theories for the case where observations belong to general abelian groups, including function spaces that are commonly used to represent time-varying signals. Our approach generalizes the Hodge decomposition and allows for signal processing tasks to be performed on these more complex observations. We propose a unified and flexible framework for TSP that expands its applicability to a wider range of signal processing applications. Numerical results demonstrate the effectiveness of this approach and provide a foundation for future research in this area.

In node classification using graph neural networks (GNNs), a typical model generates logits for different class labels at each node. A softmax layer often outputs a label prediction based on the largest logit. We demonstrate that it is possible to infer hidden graph structural information from the dataset using these logits. We introduce the key notion of label non-uniformity, which is derived from the Wasserstein distance between the softmax distribution of the logits and the uniform distribution. We demonstrate that nodes with small label non-uniformity are harder to classify correctly. We theoretically analyze how the label non-uniformity varies across the graph, which provides insights into boosting the model performance: increasing training samples with high non-uniformity or dropping edges to reduce the maximal cut size of the node set of small non-uniformity. These mechanisms can be easily added to a base GNN model. Experimental results demonstrate that our approach improves the performance of many benchmark base models.

Synchronous generator system is a complicated dynamical system for energy transmission, which plays an important role in modern industrial production. In this article, we propose some predictor-corrector methods and structure-preserving methods for a generator system based on the first benchmark model of subsynchronous resonance, among which the structure-preserving methods preserve a Dirac structure associated with the so-called port-Hamiltonian descriptor systems. To illustrate this, the simplified generator system in the form of index-1 differential-algebraic equations has been derived. Our analyses provide the global error estimates for a special class of structure-preserving methods called Gauss methods, which guarantee their superior performance over the PSCAD/EMTDC and the predictor-corrector methods in terms of computational stability. Numerical simulations are implemented to verify the effectiveness and advantages of our methods.

In graph neural networks (GNNs), both node features and labels are examples of graph signals, a key notion in graph signal processing (GSP). While it is common in GSP to impose signal smoothness constraints in learning and estimation tasks, it is unclear how this can be done for discrete node labels. We bridge this gap by introducing the concept of distributional graph signals. In our framework, we work with the distributions of node labels instead of their values and propose notions of smoothness and non-uniformity of such distributional graph signals. We then propose a general regularization method for GNNs that allows us to encode distributional smoothness and non-uniformity of the model output in semi-supervised node classification tasks. Numerical experiments demonstrate that our method can significantly improve the performance of most base GNN models in different problem settings.

Many graph neural networks have been developed to learn graph representations in either Euclidean or hyperbolic space, with all nodes' representations embedded in a single space. However, a graph can have hyperbolic and Euclidean geometries at different regions of the graph. Thus, it is sub-optimal to indifferently embed an entire graph into a single space. In this paper, we explore and analyze two notions of local hyperbolicity, describing the underlying local geometry: geometric (Gromov) and model-based, to determine the preferred space of embedding for each node. The two hyperbolicities' distributions are aligned using the Wasserstein metric such that the calculated geometric hyperbolicity guides the choice of the learned model hyperbolicity. As such our model Joint Space Graph Neural Network (JSGNN) can leverage both Euclidean and hyperbolic spaces during learning by allowing node-specific geometry space selection. We evaluate our model on both node classification and link prediction tasks and observe promising performance compared to baseline models.