Shuntuo Xu

Xinyi Hong, Shuntuo Xu, Zhou Yu

Hypergraphs provide a principled framework for modeling polyadic interactions, with applications in recommendation systems, social networks, and molecular modeling. Hypergraph generation remains challenging because incidence structures are discrete, sparse, and governed by heterogeneous higher-order interactions. Existing generators often rely on implicit latent spaces or continuous incidence decoders, which provide limited mechanistic interpretation of how node-hyperedge incidences arise. To address these limitations, we propose HYVINT, an intensity-driven hypergraph generative framework. Our key innovations are twofold: (i) we develop an intensity-driven incidence formation mechanism for hypergraphs that links latent interaction strength to binary incidence, and (ii) we derive a tractable lower-bound variational estimator for learning latent representations. We provide generation error bounds with asymptotic convergence rates and empirically show that HYVINT achieves strong fidelity while maintaining substantial novelty and diversity on synthetic and real-world hypergraphs.

Kenji Fukumizu, Wei Huang, Han Bao et al.

We reformulate Optimal Transport Conditional Flow Matching (OT-CFM), a class of dynamical generative models, showing that it admits an exact proximal formulation via an extended Brenier potential, without assuming that the target distribution has a density. In particular, the mapping to recover the target point is exactly given by a proximal operator, which yields an explicit proximal expression of the vector field. We also discuss the convergence of minibatch OT-CFM to the population formulation as the batch size increases. Finally, using second epi-derivatives of convex potentials, we prove that, for manifold-supported targets, OT-CFM is terminally normally hyperbolic: after time rescaling, the dynamics contracts exponentially in directions normal to the data manifold while remaining neutral along tangential directions.

Shuntuo Xu, Zhou Yu, Jian Huang

Identifying low-dimensional sufficient structures in nonlinear sufficient dimension reduction (SDR) has long been a fundamental yet challenging problem. Most existing methods lack theoretical guarantees of exhaustiveness in identifying lower dimensional structures, either at the population level or at the sample level. We tackle this issue by proposing a new method, generative sufficient dimension reduction (GenSDR), which leverages modern generative models. We show that GenSDR is able to fully recover the information contained in the central $σ$-field at both the population and sample levels. In particular, at the sample level, we establish a consistency property for the GenSDR estimator from the perspective of conditional distributions, capitalizing on the distributional learning capabilities of deep generative models. Moreover, by incorporating an ensemble technique, we extend GenSDR to accommodate scenarios with non-Euclidean responses, thereby substantially broadening its applicability. Extensive numerical results demonstrate the outstanding empirical performance of GenSDR and highlight its strong potential for addressing a wide range of complex, real-world tasks.

Shuntuo Xu, Zhou Yu

This paper investigates the connection between neural networks and sufficient dimension reduction (SDR), demonstrating that neural networks inherently perform SDR in regression tasks under appropriate rank regularizations. Specifically, the weights in the first layer span the central mean subspace. We establish the statistical consistency of the neural network-based estimator for the central mean subspace, underscoring the suitability of neural networks in addressing SDR-related challenges. Numerical experiments further validate our theoretical findings, and highlight the underlying capability of neural networks to facilitate SDR compared to the existing methods. Additionally, we discuss an extension to unravel the central subspace, broadening the scope of our investigation.

Shuntuo Xu, Zhou Yu, Jian Huang

The density ratio is an important metric for evaluating the relative likelihood of two probability distributions, with extensive applications in statistics and machine learning. However, existing estimation theories for density ratios often depend on stringent regularity conditions, mainly focusing on density ratio functions with bounded domains and ranges. In this paper, we study density ratio estimators using loss functions based on least squares and logistic regression. We establish upper bounds on estimation errors with standard minimax optimal rates, up to logarithmic factors. Our results accommodate density ratio functions with unbounded domains and ranges. We apply our results to nonparametric regression and conditional flow models under covariate shift and identify the tail properties of the density ratio as crucial for error control across domains affected by covariate shift. We provide sufficient conditions under which loss correction is unnecessary and demonstrate effective generalization capabilities of a source estimator to any suitable target domain. Our simulation experiments support these theoretical findings, indicating that the source estimator can outperform those derived from loss correction methods, even when the true density ratio is known.