Joint Network Topology Inference in the Presence of Hidden Nodes

We investigate the increasingly prominent task of jointly inferring multiple networks from nodal observations. While most joint inference methods assume that observations are available at all nodes, we consider the realistic and more difficult scenario where a subset of nodes are hidden and cannot be measured. Under the assumptions that the partially observed nodal signals are graph stationary and the networks have similar connectivity patterns, we derive structural characteristics of the connectivity between hidden and observed nodes. This allows us to formulate an optimization problem for estimating networks while accounting for the influence of hidden nodes. We identify conditions under which a convex relaxation yields the sparsest solution, and we formalize the performance of our proposed optimization problem with respect to the effect of the hidden nodes. Finally, synthetic and real-world simulations provide evaluations of our method in comparison with other baselines.

Learnable Digital Twin for Efficient Wireless Network Evaluation

Network digital twins (NDTs) facilitate the estimation of key performance indicators (KPIs) before physically implementing a network, thereby enabling efficient optimization of the network configuration. In this paper, we propose a learning-based NDT for network simulators. The proposed method offers a holistic representation of information flow in a wireless network by integrating node, edge, and path embeddings. Through this approach, the model is trained to map the network configuration to KPIs in a single forward pass. Hence, it offers a more efficient alternative to traditional simulation-based methods, thus allowing for rapid experimentation and optimization. Our proposed method has been extensively tested through comprehensive experimentation in various scenarios, including wired and wireless networks. Results show that it outperforms baseline learning models in terms of accuracy and robustness. Moreover, our approach achieves comparable performance to simulators but with significantly higher computational efficiency.

Deep Demixing: Reconstructing the Evolution of Network Epidemics

We propose the deep demixing (DDmix) model, a graph autoencoder that can reconstruct epidemics evolving over networks from partial or aggregated temporal information. Assuming knowledge of the network topology but not of the epidemic model, our goal is to estimate the complete propagation path of a disease spread. A data-driven approach is leveraged to overcome the lack of model awareness. To solve this inverse problem, DDmix is proposed as a graph conditional variational autoencoder that is trained from past epidemic spreads. DDmix seeks to capture key aspects of the underlying (unknown) spreading dynamics in its latent space. Using epidemic spreads simulated in synthetic and real-world networks, we demonstrate the accuracy of DDmix by comparing it with multiple (non-graph-aware) learning algorithms. The generalizability of DDmix is highlighted across different types of networks. Finally, we showcase that a simple post-processing extension of our proposed method can help identify super-spreaders in the reconstructed propagation path.

Solving Linear Inverse Problems using Higher-Order Annealed Langevin Diffusion

We propose a solution for linear inverse problems based on higher-order Langevin diffusion. More precisely, we propose pre-conditioned second-order and third-order Langevin dynamics that provably sample from the posterior distribution of our unknown variables of interest while being computationally more efficient than their first-order counterpart and the non-conditioned versions of both dynamics. Moreover, we prove that both pre-conditioned dynamics are well-defined and have the same unique invariant distributions as the non-conditioned cases. We also incorporate an annealing procedure that has the double benefit of further accelerating the convergence of the algorithm and allowing us to accommodate the case where the unknown variables are discrete. Numerical experiments in two different tasks (MIMO symbol detection and channel estimation) showcase the generality of our method and illustrate the high performance achieved relative to competing approaches (including learning-based ones) while having comparable or lower computational complexity.

Hypergraphs with Edge-Dependent Vertex Weights: Spectral Clustering based on the 1-Laplacian

We propose a flexible framework for defining the 1-Laplacian of a hypergraph that incorporates edge-dependent vertex weights. These weights are able to reflect varying importance of vertices within a hyperedge, thus conferring the hypergraph model higher expressivity than homogeneous hypergraphs. We then utilize the eigenvector associated with the second smallest eigenvalue of the hypergraph 1-Laplacian to cluster the vertices. From a theoretical standpoint based on an adequately defined normalized Cheeger cut, this procedure is expected to achieve higher clustering accuracy than that based on the traditional Laplacian. Indeed, we confirm that this is the case using real-world datasets to demonstrate the effectiveness of the proposed spectral clustering approach. Moreover, we show that for a special case within our framework, the corresponding hypergraph 1-Laplacian is equivalent to the 1-Laplacian of a related graph, whose eigenvectors can be computed more efficiently, facilitating the adoption on larger datasets.

Learning to Transmit with Provable Guarantees in Wireless Federated Learning

We propose a novel data-driven approach to allocate transmit power for federated learning (FL) over interference-limited wireless networks. The proposed method is useful in challenging scenarios where the wireless channel is changing during the FL training process and when the training data are not independent and identically distributed (non-i.i.d.) on the local devices. Intuitively, the power policy is designed to optimize the information received at the server end during the FL process under communication constraints. Ultimately, our goal is to improve the accuracy and efficiency of the global FL model being trained. The proposed power allocation policy is parameterized using a graph convolutional network and the associated constrained optimization problem is solved through a primal-dual (PD) algorithm. Theoretically, we show that the formulated problem has zero duality gap and, once the power policy is parameterized, optimality depends on how expressive this parameterization is. Numerically, we demonstrate that the proposed method outperforms existing baselines under different wireless channel settings and varying degrees of data heterogeneity.

Deep Graph Unfolding for Beamforming in MU-MIMO Interference Networks

We develop an efficient and near-optimal solution for beamforming in multi-user multiple-input-multiple-output single-hop wireless ad-hoc interference networks. Inspired by the weighted minimum mean squared error (WMMSE) method, a classical approach to solving this problem, and the principle of algorithm unfolding, we present unfolded WMMSE (UWMMSE) for MU-MIMO. This method learns a parameterized functional transformation of key WMMSE parameters using graph neural networks (GNNs), where the channel and interference components of a wireless network constitute the underlying graph. These GNNs are trained through gradient descent on a network utility metric using multiple instances of the beamforming problem. Comprehensive experimental analyses illustrate the superiority of UWMMSE over the classical WMMSE and state-of-the-art learning-based methods in terms of performance, generalizability, and robustness.

Signal Processing on Product Spaces

We establish a framework for signal processing on product spaces of simplicial and cellular complexes. For simplicity, we focus on the product of two complexes representing time and space, although our results generalize naturally to products of simplicial complexes of arbitrary dimension. Our framework leverages the structure of the eigenmodes of the Hodge Laplacian of the product space to jointly filter along time and space. To this end, we provide a decomposition theorem of the Hodge Laplacian of the product space, which highlights how the product structure induces a decomposition of each eigenmode into a spatial and temporal component. Finally, we apply our method to real world data, specifically for interpolating trajectories of buoys in the ocean from a limited set of observed trajectories.

Windowed Fourier Analysis for Signal Processing on Graph Bundles

We consider the task of representing signals supported on graph bundles, which are generalizations of product graphs that allow for "twists" in the product structure. Leveraging the localized product structure of a graph bundle, we demonstrate how a suitable partition of unity over the base graph can be used to lift the signal on the graph into a space where a product factorization can be readily applied. Motivated by the locality of this procedure, we demonstrate that bases for the signal spaces of the components of the graph bundle can be lifted in the same way, yielding a basis for the signal space of the total graph. We demonstrate this construction on synthetic graphs, as well as with an analysis of the energy landscape of conformational manifolds in stereochemistry.

Unsupervised Learning of Sampling Distributions for Particle Filters

Accurate estimation of the states of a nonlinear dynamical system is crucial for their design, synthesis, and analysis. Particle filters are estimators constructed by simulating trajectories from a sampling distribution and averaging them based on their importance weight. For particle filters to be computationally tractable, it must be feasible to simulate the trajectories by drawing from the sampling distribution. Simultaneously, these trajectories need to reflect the reality of the nonlinear dynamical system so that the resulting estimators are accurate. Thus, the crux of particle filters lies in designing sampling distributions that are both easy to sample from and lead to accurate estimators. In this work, we propose to learn the sampling distributions. We put forward four methods for learning sampling distributions from observed measurements. Three of the methods are parametric methods in which we learn the mean and covariance matrix of a multivariate Gaussian distribution; each methods exploits a different aspect of the data (generic, time structure, graph structure). The fourth method is a nonparametric alternative in which we directly learn a transform of a uniform random variable. All four methods are trained in an unsupervised manner by maximizing the likelihood that the states may have produced the observed measurements. Our computational experiments demonstrate that learned sampling distributions exhibit better performance than designed, minimum-degeneracy sampling distributions.