A cascade over a network refers to the diffusion process where behavior changes occurring in one part of an interconnected population lead to a series of sequential changes throughout the entire population. In recent years, there has been a surge in interest and efforts to understand and model cascade mechanisms since they motivate many significant research topics across different disciplines. The propagation structure of cascades is governed by underlying diffusion networks that are often hidden. Inferring diffusion networks thus enables interventions in cascading process to maximize information propagation and provides insights into the Granger causality of interaction mechanisms among individuals. In this project, we propose a novel double network mixture model for inferring latent diffusion network in presence of strong cascade heterogeneity. The new model represents cascade pathways as a distributional mixture over diffusion networks that capture different cascading patterns at the population level. We develop a data-driven optimization method to infer diffusion networks using only visible temporal cascade records, avoiding the need to model complex and heterogeneous individual states. Both statistical and computational guarantees are established for the proposed method. We apply the proposed model to analyze research topic cascades in social sciences across U.S. universities and uncover the latent research topic diffusion network among top U.S. social science programs.

It is difficult to evaluate machine learning classifiers without large labeled datasets, which are often unavailable. In contrast, unlabeled data is plentiful, but not easily used for evaluation. Here, we introduce Semi-Supervised Model Evaluation (SSME), a method that uses both labeled and unlabeled data to evaluate machine learning classifiers. The key idea is to estimate the joint distribution of ground truth labels and classifier scores using a semi-supervised mixture model. The semisupervised mixture model allows SSME to learn from three sources of information: unlabeled data, multiple classifiers, and probabilistic classifier scores. Once fit, the mixture model enables estimation of any metric that is a function of classifier scores and ground truth labels (e.g., accuracy or AUC). We derive theoretical bounds on the error of these estimates, showing that estimation error decreases with the number of classifiers and the amount of unlabeled data. We present experiments in four domains where obtaining large labeled datasets is often impractical: healthcare, content moderation, molecular property prediction, and text classification. Our results demonstrate that SSME estimates performance more accurately than do competing methods, reducing error by 5.1 relative to using labeled data alone and 2.4 relative to the next best method.

Model-based clustering integrated with variable selection is a powerful tool for uncovering latent structures within complex data. However, its effectiveness is often hindered by challenges such as identifying relevant variables that define heterogeneous subgroups and handling data that are missing not at random, a prevalent issue in fields like transcriptomics. While several notable methods have been proposed to address these problems, they typically tackle each issue in isolation, thereby limiting their flexibility and adaptability. This paper introduces a unified framework designed to address these challenges simultaneously. Our approach incorporates a data-driven penalty matrix into penalized clustering to enable more flexible variable selection, along with a mechanism that explicitly models the relationship between missingness and latent class membership. We demonstrate that, under certain regularity conditions, the proposed framework achieves both asymptotic consistency and selection consistency, even in the presence of missing data. This unified strategy significantly enhances the capability and efficiency of model-based clustering, advancing methodologies for identifying informative variables that define homogeneous subgroups in the presence of complex missing data patterns. The performance of the framework, including its computational efficiency, is evaluated through simulations and demonstrated using both synthetic and real-world transcriptomic datasets.

Motivated by the privacy, sensitivity and sharing limitations of health data, we present a comprehensive pipeline for inference of Bayesian mixture models within a federated learning setting, i.e. when data cannot be fully shared or pooled across compute nodes. We adopt a Consensus Monte Carlo (CMC) approach, in which an MCMC algorithm is run independently within each data silo to estimate local posterior distributions, which are then aggregated to approximate the posterior over the full data. The variational CMC approach of Rabinovich, Angelino and Jordan (2015) [1] frames the aggregation step as a variational inference problem, but their application to mixtures assumes the number of clusters and key mixture parameters to be known. Our main methodological contributions are: (i) an extension of variational CMC to over-fitted Bayesian mixture models that infer the number of clusters and all model parameters, without requiring conjugacy; (ii) novel cluster-matching algorithms suitable for cross-silo settings in which not every cluster appears in each local dataset; (iii) a number of inference strategies for the aggregation step, matched to different federated learning constraints; and (iv) guidelines for choosing among these in practice. A comprehensive simulation study validates the framework and allows us to compare to state-of-the-art federated learning alternatives. Notably, we show that when the composition of local datasets reflects the underlying clustering structure in the data, our approach can recover small clusters with greater accuracy than standard MCMC applied to the pooled data. We illustrate the framework on large-scale electronic health record data, identifying multi-morbidity patterns in a British geriatric population.

Transformers have emerged as a powerful neural network architecture capable of tackling a wide range of learning tasks. In this work, we provide a theoretical analysis of their ability to automatically extract structure from data in an unsupervised setting. In particular, we demonstrate their suitability for clustering when the input data is generated from a Gaussian mixture model. To this end, we study a simplified two-head attention layer and define a population risk whose minimization with unlabeled data drives the head parameters to align with the true mixture centroids. This phenomenon highlights the ability of attention-based layers to capture underlying distributional structure. We further examine an attention layer with key, query, and value matrices fixed to the identity, and show that, even without any trainable parameters, it can perform in-context quantization, revealing the surprising capacity of transformer-based methods to adapt dynamically to input-specific distributions.

We revisit the classical problem of estimating an unknown distribution from its samples by fitting a mixture model that minimizes cross-entropy loss. Framing the task as a stochastic convex optimization problem over the space of $ M $-component mixture distributions, we propose a family of estimators derived from the stochastic mirror descent (SMD) algorithm. This optimization-based approach provides a principled and flexible framework that generalizes traditional estimators and proposes a variety of novel estimators through the choice of Bregman divergences. A key advantage of our method is that it scales efficiently with the number of candidate components $ f_i $; that is, one can employ a large set of basis distributions in the mixture model without incurring significant computational overhead. This enables richer approximations and improved estimation accuracy. Moreover, in the case of categorical distribution (discrete outcomes) our estimators do not require a strict lower bound, in other words our framework does not require the precise knowledge of the support of the distribution. We demonstrate that, under mild conditions, the proposed $ φ$-SMD estimators achieve near-optimal convergence rates in both Kullback-Leibler (KL) divergence and $ \ell_2 $-norm and offer practical benefits when computation is expensive. Our numerical analysis highlights improved performance guaranties over classical estimators, particularly in terms of sample efficiency and scalability.

Large-scale hypothesis testing is central to modern science, where controlling the False Discovery Rate (FDR) has become the standard approach to managing false positives across many simultaneous tests. Hypotheses rarely exist in isolation; they often exhibit structure through proximity, connectivity, or hierarchy. This structure represents both a challenge and an opportunity: while classical methods treat these dependencies as obstacles requiring conservative correction, leveraging them can substantially increase discovery power. Here, we reframe structured FDR control as a regularized learning problem. By optimizing within a suitable Reproducing Kernel Hilbert Space (RKHS), we introduce a framework that unifies continuous domains, graphs, and hierarchies under a single algorithm through kernel choice alone. This formulation enables smooth solutions in place of the piecewise-constant fits of prior methods, principled likelihood-based hyperparameter selection rather than heuristic tuning, and inference at unobserved locations which in turn supports sample-efficient experimental design. Building on this estimator, we provide two decision rules which we prove to control the FDR. We validate our method on two sources: spatial locations derived from high-dimensional real-world datasets, and a differential gene expression task utilizing protein-protein interaction graphs.

The Gaussian mixture model is widely used in unsupervised learning, owing to its simplicity and interpretability. However, a fundamental limitation of the classical Gaussian mixture model is that it forces each observation to belong to exactly one component. In many practical applications, such as genetics, social network analysis, and text mining, an observation may naturally belong to multiple components or exhibit partial membership in several latent components. To overcome this limitation, we propose the mixed membership sub-Gaussian model, which extends the classical Gaussian mixture framework by allowing each observation to belong to multiple components. This model inherits the interpretability of the classical Gaussian mixture model while offering greater flexibility for capturing complex overlapping structures. We develop an efficient spectral algorithm to estimate the mixed membership of each individual observation, and under mild separation conditions on the component centres, we prove that the estimation error of the per-individual membership vector can be made arbitrarily small with high probability. To our knowledge, this is the first work to provide a computationally efficient estimator with such a vanishing-error guarantee for a mixed-membership extension of the Gaussian mixture model. Extensive experimental studies demonstrate that our method outperforms existing approaches that ignore mixed memberships.