In many real-world applications, data is inherently decentralized, necessitating data analysis methods that prioritize privacy while delivering interpretable results. Federated Non-Negative Matrix Factorization (FedNMF) meets this requirement by factorizing latent components from distributed data that cannot be freely shared among clients. A significant challenge in FedNMF arises when clients converge on different solutions due to prolonged independent optimization, leading to drift and incoherent models. While Federated Learning (FL) typically mitigates drift through frequent synchronizations and strong regularization, it often overlooks critical properties of Non-Negative Matrix Factorization, such as permutation invariance. As a result, solutions from FedNMF clients may be misidentified by FL drift as distinct, despite being equivalent. Using an alignment-aware drift, we create coherence through proximal optimization and barycenter aggregation for FedNMF. We analyze the computational complexity of our approach, provide efficient heuristics, and ensure the convergence of our algorithms. On a diverse set of real-world and synthetic datasets, we demonstrate the effectiveness of our methods.

Identifying informative components in binary data is an essential task in many application areas, including life sciences, social sciences, and recommendation systems. Boolean matrix factorization (BMF) is a family of methods that performs this task by factorizing the data into dense factor matrices. In real-world settings, the data is often distributed across stakeholders and required to stay private, prohibiting the straightforward application of BMF. To adapt BMF to this context, we approach the problem from a federated-learning perspective, building on a state-of-the-art continuous binary matrix factorization relaxation to BMF that enables efficient gradient-based optimization. Our approach only needs to share the relaxed component matrices, which are aggregated centrally using a proximal operator that regularizes for binary outcomes. We show the convergence of our federated proximal gradient descent algorithm and provide differential privacy guarantees. Our extensive empirical evaluation shows that our algorithm outperforms, in quality and efficacy, federation schemes of state-of-the-art BMF methods on a diverse set of real-world and synthetic data.

Stochastic Block Models (SBMs) are a popular approach to modeling single real-world graphs. The key idea of SBMs is to partition the vertices of the graph into blocks with similar edge densities within, as well as between different blocks. However, what if we are given not one but multiple graphs that are unaligned and of different sizes? How can we find out if these graphs share blocks with similar connectivity structures? In this paper, we propose the shared stochastic block modeling (SSBM) problem, in which we model n graphs using SBMs that share parameters of s blocks. We show that fitting an SSBM is NP-hard, and consider two approaches to fit good models in practice. In the first, we directly maximize the likelihood of the shared model using a Markov chain Monte Carlo algorithm. In the second, we first fit an SBM for each graph and then select which blocks to share. We propose an integer linear program to find the optimal shared blocks and to scale to large numbers of blocks, we propose a fast greedy algorithm. Through extensive empirical evaluation on synthetic and real-world data, we show that our methods work well in practice.

How does neural connectivity in autistic children differ from neural connectivity in healthy children or autistic youths? What patterns in global trade networks are shared across classes of goods, and how do these patterns change over time? Answering questions like these requires us to differentially describe groups of graphs: Given a set of graphs and a partition of these graphs into groups, discover what graphs in one group have in common, how they systematically differ from graphs in other groups, and how multiple groups of graphs are related. We refer to this task as graph group analysis, which seeks to describe similarities and differences between graph groups by means of statistically significant subgraphs. To perform graph group analysis, we introduce Gragra, which uses maximum entropy modeling to identify a non-redundant set of subgraphs with statistically significant associations to one or more graph groups. Through an extensive set of experiments on a wide range of synthetic and real-world graph groups, we confirm that Gragra works well in practice.

We are interested in discovering those patterns from data with an empirical frequency that is significantly differently than expected. To avoid spurious results, yet achieve high statistical power, we propose to sequentially control for false discoveries during the search. To avoid redundancy, we propose to update our expectations whenever we discover a significant pattern. To efficiently consider the exponentially sized search space, we employ an easy-to-compute upper bound on significance, and propose an effective search strategy for sets of significant patterns. Through an extensive set of experiments on synthetic data, we show that our method, Spass, recovers the ground truth reliably, does so efficiently, and without redundancy. On real-world data we show it works well on both single and multiple classes, on low and high dimensional data, and through case studies that it discovers meaningful results.

Addressing the interpretability problem of NMF on Boolean data, Boolean Matrix Factorization (BMF) uses Boolean algebra to decompose the input into low-rank Boolean factor matrices. These matrices are highly interpretable and very useful in practice, but they come at the high computational cost of solving an NP-hard combinatorial optimization problem. To reduce the computational burden, we propose to relax BMF continuously using a novel elastic-binary regularizer, from which we derive a proximal gradient algorithm. Through an extensive set of experiments, we demonstrate that our method works well in practice: On synthetic data, we show that it converges quickly, recovers the ground truth precisely, and estimates the simulated rank exactly. On real-world data, we improve upon the state of the art in recall, loss, and runtime, and a case study from the medical domain confirms that our results are easily interpretable and semantically meaningful.

Our goal is to discover the components of a dataset, characterize why we deem these components, explain how these components are different from each other, as well as identify what properties they share among each other. As is usual, we consider regions in the data to be components if they show significantly different distributions. What is not usual, however, is that we parameterize these distributions with patterns that are informative for one or more components. We do so because these patterns allow us to characterize what is going on in our data as well as explain our decomposition.We define the problem in terms of a regularized maximum likelihood, in which we use the Maximum Entropy principle to model each data component with a set of patterns. As the search space is large and unstructured, we propose the deterministic DISC algorithm to efficiently discover high-quality decompositions via an alternating optimization approach. Empirical evaluation on synthetic and real-world data shows that DISC efficiently discovers meaningful components and accurately characterises these in easily understandable terms.

The maximum entropy principle uniquely identifies the distribution that models our knowledge about the data, but is otherwise maximally unbiased. As soon as we include non-trivial observations in our model, however, exact inference quickly becomes intractable. We propose a relaxation that permits efficient inference by dynamically factorizing the joint distribution into factors. In particular, we show that these factors are learnable from data and that it is consistent with standard maximum entropy distribution. Through an extensive set of experiments we show that the relaxation is scalable, approximates the vanilla distribution closely, allows for a classification that is as accurate, as well as results in a concise set of patterns.