We consider a class of conditional forward-backward diffusion models for conditional generative modeling, that is, generating new data given a covariate (or control variable). To formally study the theoretical properties of these conditional generative models, we adopt a statistical framework of distribution regression to characterize the large sample properties of the conditional distribution estimators induced by these conditional forward-backward diffusion models. Here, the conditional distribution of data is assumed to smoothly change over the covariate. In particular, our derived convergence rate is minimax-optimal under the total variation metric within the regimes covered by the existing literature. Additionally, we extend our theory by allowing both the data and the covariate variable to potentially admit a low-dimensional manifold structure. In this scenario, we demonstrate that the conditional forward-backward diffusion model can adapt to both manifold structures, meaning that the derived estimation error bound (under the Wasserstein metric) depends only on the intrinsic dimensionalities of the data and the covariate.

In this work, we explore the theoretical properties of conditional deep generative models under the statistical framework of distribution regression where the response variable lies in a high-dimensional ambient space but concentrates around a potentially lower-dimensional manifold. More specifically, we study the large-sample properties of a likelihood-based approach for estimating these models. Our results lead to the convergence rate of a sieve maximum likelihood estimator (MLE) for estimating the conditional distribution (and its devolved counterpart) of the response given predictors in the Hellinger (Wasserstein) metric. Our rates depend solely on the intrinsic dimension and smoothness of the true conditional distribution. These findings provide an explanation of why conditional deep generative models can circumvent the curse of dimensionality from the perspective of statistical foundations and demonstrate that they can learn a broader class of nearly singular conditional distributions. Our analysis also emphasizes the importance of introducing a small noise perturbation to the data when they are supported sufficiently close to a manifold. Finally, in our numerical studies, we demonstrate the effective implementation of the proposed approach using both synthetic and real-world datasets, which also provide complementary validation to our theoretical findings.

This work establishes that sparse Bayesian neural networks achieve optimal posterior contraction rates over anisotropic Besov spaces and their hierarchical compositions. These structures reflect the intrinsic dimensionality of the underlying function, thereby mitigating the curse of dimensionality. Our analysis shows that Bayesian neural networks equipped with either sparse or continuous shrinkage priors attain the optimal rates which are dependent on the intrinsic dimension of the true structures. Moreover, we show that these priors enable rate adaptation, allowing the posterior to contract at the optimal rate even when the smoothness level of the true function is unknown. The proposed framework accommodates a broad class of functions, including additive and multiplicative Besov functions as special cases. These results advance the theoretical foundations of Bayesian neural networks and provide rigorous justification for their practical effectiveness in high-dimensional, structured estimation problems.

We investigate statistical properties of a likelihood approach to nonparametric estimation of a singular distribution using deep generative models. More specifically, a deep generative model is used to model high-dimensional data that are assumed to concentrate around some low-dimensional structure. Estimating the distribution supported on this low-dimensional structure, such as a low-dimensional manifold, is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. In the considered model, a usual likelihood approach can fail to estimate the target distribution consistently due to the singularity. We prove that a novel and effective solution exists by perturbing the data with an instance noise, which leads to consistent estimation of the underlying distribution with desirable convergence rates. We also characterize the class of distributions that can be efficiently estimated via deep generative models. This class is sufficiently general to contain various structured distributions such as product distributions, classically smooth distributions and distributions supported on a low-dimensional manifold. Our analysis provides some insights on how deep generative models can avoid the curse of dimensionality for nonparametric distribution estimation. We conduct a thorough simulation study and real data analysis to empirically demonstrate that the proposed data perturbation technique improves the estimation performance significantly.

Distance plays a crucial role in determining the degree of resemblance between two objects of interest. A wide range of learning tasks in statistics and machine learning, including classification, shape matching, dimension reduction, and clustering, are directly dependent on the measure of dissimilarity between observations. Consequently, it is of utmost necessity to have a distance measure capable of accurately including the underlying data structure. In this paper, an unique distance metric for network-valued data is proposed. Our major breakthrough is to construct a distance that can represent the topological properties of the data based on the long-term diffusion behavior of the entire network. On the basis of the Laplacian flow dynamic system on graphs, a unique distance measure termed network flow distance is established. A second contribution consists in proposing a computational pipeline for the network flow distance that makes use of simple concepts from numerical linear algebra, so making it an easily available alternative to other distance measurements. Using simulated examples, we demonstrate the utility of the proposed distance measure and its advantages over other existing measures. We also employ the distance for an unsupervised learning task of clustering network-valued data.

In this work, we propose to train a graph neural network via resampling from a graphon estimate obtained from the underlying network data. More specifically, the graphon or the link probability matrix of the underlying network is first obtained from which a new network will be resampled and used during the training process at each layer. Due to the uncertainty induced from the resampling, it helps mitigate the well-known issue of over-smoothing in a graph neural network (GNN) model. Our framework is general, computationally efficient, and conceptually simple. Another appealing feature of our method is that it requires minimal additional tuning during the training process. Extensive numerical results show that our approach is competitive with and in many cases outperform the other over-smoothing reducing GNN training methods.

In this work, we propose to employ information-geometric tools to optimize a graph neural network architecture such as the graph convolutional networks. More specifically, we develop optimization algorithms for the graph-based semi-supervised learning by employing the natural gradient information in the optimization process. This allows us to efficiently exploit the geometry of the underlying statistical model or parameter space for optimization and inference. To the best of our knowledge, this is the first work that has utilized the natural gradient for the optimization of graph neural networks that can be extended to other semi-supervised problems. Efficient computations algorithms are developed and extensive numerical studies are conducted to demonstrate the superior performance of our algorithms over existing algorithms such as ADAM and SGD.

A weighted directed network (WDN) is a directed graph in which each edge is associated to a unique value called weight. These networks are very suitable for modeling real-world social networks in which there is an assessment of one vertex toward other vertices. One of the main problems studied in this paper is prediction of edge weights in such networks. We introduce, for the first time, a metric geometry approach to studying edge weight prediction in WDNs. We modify a usual notion of WDNs, and introduce a new type of WDNs which we coin the term almost-weighted directed networks (AWDNs). AWDNs can capture the weight information of a network from a given training set. We then construct a class of metrics (or distances) for AWDNs which equips such networks with a metric space structure. Using the metric geometry structure of AWDNs, we propose modified k nearest neighbors (kNN) methods and modified support-vector machine (SVM) methods which will then be used to predict edge weights in AWDNs. In many real-world datasets, in addition to edge weights, one can also associate weights to vertices which capture information of vertices; association of weights to vertices especially plays an important role in graph embedding problems. Adopting a similar approach, we introduce two new types of directed networks in which weights are associated to either a subset of origin vertices or a subset of terminal vertices. We, for the first time, construct novel classes of metrics on such networks, and based on these new metrics propose modified kNN and SVM methods for predicting weights of origins and terminals in these networks. We provide experimental results on several real-world datasets, using our geometric methodologies.

The last decade has witnessed an explosion in the development of models, theory and computational algorithms for ``big data'' analysis. In particular, distributed inference has served as a natural and dominating paradigm for statistical inference. However, the existing literature on parallel inference almost exclusively focuses on Euclidean data and parameters. While this assumption is valid for many applications, it is increasingly more common to encounter problems where the data or the parameters lie on a non-Euclidean space, like a manifold for example. Our work aims to fill a critical gap in the literature by generalizing parallel inference algorithms to optimization on manifolds. We show that our proposed algorithm is both communication efficient and carries theoretical convergence guarantees. In addition, we demonstrate the performance of our algorithm to the estimation of Fr\'echet means on simulated spherical data and the low-rank matrix completion problem over Grassmann manifolds applied to the Netflix prize data set.

Community detection, which focuses on clustering nodes or detecting communities in (mostly) a single network, is a problem of considerable practical interest and has received a great deal of attention in the research community. While being able to cluster within a network is important, there are emerging needs to be able to \emph{cluster multiple networks}. This is largely motivated by the routine collection of network data that are generated from potentially different populations. These networks may or may not have node correspondence. When node correspondence is present, we cluster networks by summarizing a network by its graphon estimate, whereas when node correspondence is not present, we propose a novel solution for clustering such networks by associating a computationally feasible feature vector to each network based on trace of powers of the adjacency matrix. We illustrate our methods using both simulated and real data sets, and theoretical justifications are provided in terms of consistency.

We propose a novel approach to Bayesian analysis that is provably robust to outliers in the data and often has computational advantages over standard methods. Our technique is based on splitting the data into non-overlapping subgroups, evaluating the posterior distribution given each independent subgroup, and then combining the resulting measures. The main novelty of our approach is the proposed aggregation step, which is based on the evaluation of a median in the space of probability measures equipped with a suitable collection of distances that can be quickly and efficiently evaluated in practice. We present both theoretical and numerical evidence illustrating the improvements achieved by our method.

Many Bayesian learning methods for massive data benefit from working with small subsets of observations. In particular, significant progress has been made in scalable Bayesian learning via stochastic approximation. However, Bayesian learning methods in distributed computing environments are often problem- or distribution-specific and use ad hoc techniques. We propose a novel general approach to Bayesian inference that is scalable and robust to corruption in the data. Our technique is based on the idea of splitting the data into several non-overlapping subgroups, evaluating the posterior distribution given each independent subgroup, and then combining the results. The main novelty is the proposed aggregation step which is based on finding the geometric median of posterior distributions. We present both theoretical and numerical results illustrating the advantages of our approach.