Graph learning is the fundamental task of estimating unknown graph connectivity from available data. Typical approaches assume that not only is all information available simultaneously but also that all nodes can be observed. However, in many real-world scenarios, data can neither be known completely nor obtained all at once. We present a novel method for online graph estimation that accounts for the presence of hidden nodes. We consider signals that are stationary on the underlying graph, which provides a model for the unknown connections to hidden nodes. We then formulate a convex optimization problem for graph learning from streaming, incomplete graph signals. We solve the proposed problem through an efficient proximal gradient algorithm that can run in real-time as data arrives sequentially. Additionally, we provide theoretical conditions under which our online algorithm is similar to batch-wise solutions. Through experimental results on synthetic and real-world data, we demonstrate the viability of our approach for online graph learning in the presence of missing observations.

Covariance-based data processing is widespread across signal processing and machine learning applications due to its ability to model data interconnectivities and dependencies. However, harmful biases in the data may become encoded in the sample covariance matrix and cause data-driven methods to treat different subpopulations unfairly. Existing works such as fair principal component analysis (PCA) mitigate these effects, but remain unstable in low sample regimes, which in turn may jeopardize the fairness goal. To address both biases and instability, we propose Fair coVariance Neural Networks (FVNNs), which perform graph convolutions on the covariance matrix for both fair and accurate predictions. Our FVNNs provide a flexible model compatible with several existing bias mitigation techniques. In particular, FVNNs allow for mitigating the bias in two ways: first, they operate on fair covariance estimates that remove biases from their principal components; second, they are trained in an end-to-end fashion via a fairness regularizer in the loss function so that the model parameters are tailored to solve the task directly in a fair manner. We prove that FVNNs are intrinsically fairer than analogous PCA approaches thanks to their stability in low sample regimes. We validate the robustness and fairness of our model on synthetic and real-world data, showcasing the flexibility of FVNNs along with the tradeoff between fair and accurate performance.

Graph filters are ubiquitous for processing data over graphs. However, most filters obtained from data are point-estimates and may be sensitive to changes in topology or data distributions. Thus, modeling uncertainty in filters is critical to quantify confidence in analyses or improve downstream tasks in low-data regimes. We introduce a Bayesian framework for graph filter design, termed Bayesian graph filters. Given input-output realizations on a graph, we obtain the posterior filter and prior filter precision hyper-parameters via a constrained EM algorithm. The posterior filter leads to uncertainty in its frequency response, which has implications for stability. We study the stability via the integral Lipschitz (IL) property and derive a lower bound for the probability of Bayesian filters being IL. Results show that Bayesian filters can be more stable across the spectrum and under perturbations, provide uncertainty estimates and can outperform point filters on multiple tasks.

This work presents a low-rank tensor model for multidimensional Markov chains. A common approach to simplify the dynamical behavior of a Markov chain is to impose low-rankness on the transition probability matrix. Inspired by the success of these matrix techniques, we present low-rank tensors for representing transition probabilities on multi-dimensional state spaces. Through tensor decomposition, we provide a connection between our method and classical probabilistic models. Moreover, our proposed model yields a parsimonious representation with fewer parameters than matrix-based approaches. Unlike these methods, which impose low-rankness uniformly across all states, our tensor method accounts for the multi-dimensionality of the state space. We also propose an optimization-based approach to estimate a Markov model as a low-rank tensor. Our optimization problem can be solved by the alternating direction method of multipliers (ADMM), which enjoys convergence to a stationary solution. We empirically demonstrate that our tensor model estimates Markov chains more efficiently than conventional techniques, requiring both fewer samples and parameters. We perform numerical simulations for both a synthetic low-rank Markov chain and a real-world example with New York City taxi data, showcasing the advantages of multi-dimensionality for modeling state spaces.

Graph neural networks (GNNs) have become a workhorse approach for learning from data defined over irregular domains, typically by implicitly assuming that the data structure is represented by a homophilic graph. However, recent works have revealed that many relevant applications involve heterophilic data where the performance of GNNs can be notably compromised. To address this challenge, we present a simple yet effective architecture designed to mitigate the limitations of the homophily assumption. The proposed architecture reinterprets the role of graph filters in convolutional GNNs, resulting in a more general architecture while incorporating a stronger inductive bias than GNNs based on filter banks. The proposed convolutional layer enhances the expressive capacity of the architecture enabling it to learn from both homophilic and heterophilic data and preventing the issue of oversmoothing. From a theoretical standpoint, we show that the proposed architecture is permutation equivariant. Finally, we show that the proposed GNNs compares favorably relative to several state-of-the-art baselines in both homophilic and heterophilic datasets, showcasing its promising potential.

In this work, we propose data augmentation via pairwise mixup across subgroups to improve group fairness. Many real-world applications of machine learning systems exhibit biases across certain groups due to underrepresentation or training data that reflects societal biases. Inspired by the successes of mixup for improving classification performance, we develop a pairwise mixup scheme to augment training data and encourage fair and accurate decision boundaries for all subgroups. Data augmentation for group fairness allows us to add new samples of underrepresented groups to balance subpopulations. Furthermore, our method allows us to use the generalization ability of mixup to improve both fairness and accuracy. We compare our proposed mixup to existing data augmentation and bias mitigation approaches on both synthetic simulations and real-world benchmark fair classification data, demonstrating that we are able to achieve fair outcomes with robust if not improved accuracy.

The myriad complex systems with multiway interactions motivate the extension of graph-based pairwise connections to higher-order relations. In particular, the simplicial complex has inspired generalizations of graph neural networks (GNNs) to simplicial complex-based models. Learning on such systems requires large amounts of data, which can be expensive or impossible to obtain. We propose data augmentation of simplicial complexes through both linear and nonlinear mixup mechanisms that return mixtures of existing labeled samples. In addition to traditional pairwise mixup, we present a convex clustering mixup approach for a data-driven relationship among several simplicial complexes. We theoretically demonstrate that the resultant synthetic simplicial complexes interpolate among existing data with respect to homomorphism densities. Our method is demonstrated on both synthetic and real-world datasets for simplicial complex classification.

Node features bolster graph-based learning when exploited jointly with network structure. However, a lack of nodal attributes is prevalent in graph data. We present a framework to recover completely missing node features for a set of graphs, where we only know the signals of a subset of graphs. Our approach incorporates prior information from both graph topology and existing nodal values. We demonstrate an example implementation of our framework where we assume that node features depend on local graph structure. Missing nodal values are estimated by aggregating known features from the most similar nodes. Similarity is measured through a node embedding space that preserves local topological features, which we train using a Graph AutoEncoder. We empirically show not only the accuracy of our feature estimation approach but also its value for downstream graph classification. Our success embarks on and implies the need to emphasize the relationship between node features and graph structure in graph-based learning.

We propose estimating Gaussian graphical models (GGMs) that are fair with respect to sensitive nodal attributes. Many real-world models exhibit unfair discriminatory behavior due to biases in data. Such discrimination is known to be exacerbated when data is equipped with pairwise relationships encoded in a graph. Additionally, the effect of biased data on graphical models is largely underexplored. We thus introduce fairness for graphical models in the form of two bias metrics to promote balance in statistical similarities across nodal groups with different sensitive attributes. Leveraging these metrics, we present Fair GLASSO, a regularized graphical lasso approach to obtain sparse Gaussian precision matrices with unbiased statistical dependencies across groups. We also propose an efficient proximal gradient algorithm to obtain the estimates. Theoretically, we express the tradeoff between fair and accurate estimated precision matrices. Critically, this includes demonstrating when accuracy can be preserved in the presence of a fairness regularizer. On top of this, we study the complexity of Fair GLASSO and demonstrate that our algorithm enjoys a fast convergence rate. Our empirical validation includes synthetic and real-world simulations that illustrate the value and effectiveness of our proposed optimization problem and iterative algorithm.

We develop a novel data-driven nonlinear mixup mechanism for graph data augmentation and present different mixup functions for sample pairs and their labels. Mixup is a data augmentation method to create new training data by linearly interpolating between pairs of data samples and their labels. Mixup of graph data is challenging since the interpolation between graphs of potentially different sizes is an ill-posed operation. Hence, a promising approach for graph mixup is to first project the graphs onto a common latent feature space and then explore linear and nonlinear mixup strategies in this latent space. In this context, we propose to (i) project graphs onto the latent space of continuous random graph models known as graphons, (ii) leverage convex clustering in this latent space to generate nonlinear data-driven mixup functions, and (iii) investigate the use of different mixup functions for labels and data samples. We evaluate our graph data augmentation performance on benchmark datasets and demonstrate that nonlinear data-driven mixup functions can significantly improve graph classification.

We consider the problem of estimating the topology of multiple networks from nodal observations, where these networks are assumed to be drawn from the same (unknown) random graph model. We adopt a graphon as our random graph model, which is a nonparametric model from which graphs of potentially different sizes can be drawn. The versatility of graphons allows us to tackle the joint inference problem even for the cases where the graphs to be recovered contain different number of nodes and lack precise alignment across the graphs. Our solution is based on combining a maximum likelihood penalty with graphon estimation schemes and can be used to augment existing network inference methods. We validate our proposed approach by comparing its performance against competing methods in synthetic and real-world datasets.

Learning graphs from sets of nodal observations represents a prominent problem formally known as graph topology inference. However, current approaches are limited by typically focusing on inferring single networks, and they assume that observations from all nodes are available. First, many contemporary setups involve multiple related networks, and second, it is often the case that only a subset of nodes is observed while the rest remain hidden. Motivated by these facts, we introduce a joint graph topology inference method that models the influence of the hidden variables. Under the assumptions that the observed signals are stationary on the sought graphs and the graphs are closely related, the joint estimation of multiple networks allows us to exploit such relationships to improve the quality of the learned graphs. Moreover, we confront the challenging problem of modeling the influence of the hidden nodes to minimize their detrimental effect. To obtain an amenable approach, we take advantage of the particular structure of the setup at hand and leverage the similarity between the different graphs, which affects both the observed and the hidden nodes. To test the proposed method, numerical simulations over synthetic and real-world graphs are provided.

Inferring graph structure from observations on the nodes is an important and popular network science task. Departing from the more common inference of a single graph, we study the problem of jointly inferring multiple graphs from the observation of signals at their nodes (graph signals), which are assumed to be stationary in the sought graphs. Graph stationarity implies that the mapping between the covariance of the signals and the sparse matrix representing the underlying graph is given by a matrix polynomial. A prominent example is that of Markov random fields, where the inverse of the covariance yields the sparse matrix of interest. From a modeling perspective, stationary graph signals can be used to model linear network processes evolving on a set of (not necessarily known) networks. Leveraging that matrix polynomials commute, a convex optimization method along with sufficient conditions that guarantee the recovery of the true graphs are provided when perfect covariance information is available. Particularly important from an empirical viewpoint, we provide high-probability bounds on the recovery error as a function of the number of signals observed and other key problem parameters. Numerical experiments demonstrate the effectiveness of the proposed method with perfect covariance information as well as its robustness in the noisy regime.

We consider the problem of inferring the unobserved edges of a graph from data supported on its nodes. In line with existing approaches, we propose a convex program for recovering a graph Laplacian that is approximately diagonalizable by a set of eigenvectors obtained from the second-order moment of the observed data. Unlike existing work, we incorporate prior knowledge about the distribution from where the underlying graph was drawn. In particular, we consider the case where the graph was drawn from a graphon model, and we supplement our convex optimization problem with a provably-valid regularizer on the spectrum of the graph to be recovered. We present the cases where the graphon model is assumed to be known and the more practical setting where the relevant features of the model are inferred from auxiliary network observations. Numerical experiments on synthetic and real-world data illustrate the advantage of leveraging the proposed graphon prior, even when the prior is imperfect.