arXiv subjects

Lizhen Lin

Publications and source records attributed to Lizhen Lin.

At least 37 records · Page 2Linked to original sources

arXiv2023

Machine Learning and the Future of Bayesian Computation

Bayesian models are a powerful tool for studying complex data, allowing the analyst to encode rich hierarchical dependencies and leverage prior information. Most importantly, they facilitate a complete characterization of uncertainty through the posterior distribution. Practical posterior computation is commonly performed via MCMC, which can be computationally infeasible for high dimensional models with many observations. In this article we discuss the potential to improve posterior computation using ideas from machine learning. Concrete future directions are explored in vignettes on normalizing flows, Bayesian coresets, distributed Bayesian inference, and variational inference.

stat.ML↗

arXiv2023

Bayesian community detection for networks with covariates

The increasing prevalence of network data in a vast variety of fields and the need to extract useful information out of them have spurred fast developments in related models and algorithms. Among the various learning tasks with network data, community detection, the discovery of node clusters or "communities," has arguably received the most attention in the scientific community. In many real-world applications, the network data often come with additional information in the form of node or edge covariates that should ideally be leveraged for inference. In this paper, we add to a limited literature on community detection for networks with covariates by proposing a Bayesian stochastic block model with a covariate-dependent random partition prior. Under our prior, the covariates are explicitly expressed in specifying the prior distribution on the cluster membership. Our model has the flexibility of modeling uncertainties of all the parameter estimates including the community membership. Importantly, and unlike the majority of existing methods, our model has the ability to learn the number of the communities via posterior inference without having to assume it to be known. Our model can be applied to community detection in both dense and sparse networks, with both categorical and continuous covariates, and our MCMC algorithm is very efficient with good mixing properties. We demonstrate the superior performance of our model over existing models in a comprehensive simulation study and an application to two real datasets.

stat.ME↗

arXiv2023

A likelihood approach to nonparametric estimation of a singular distribution using deep generative models

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.

stat.ML↗

arXiv2023

Intrinsic and extrinsic deep learning on manifolds

We propose extrinsic and intrinsic deep neural network architectures as general frameworks for deep learning on manifolds. Specifically, extrinsic deep neural networks (eDNNs) preserve geometric features on manifolds by utilizing an equivariant embedding from the manifold to its image in the Euclidean space. Moreover, intrinsic deep neural networks (iDNNs) incorporate the underlying intrinsic geometry of manifolds via exponential and log maps with respect to a Riemannian structure. Consequently, we prove that the empirical risk of the empirical risk minimizers (ERM) of eDNNs and iDNNs converge in optimal rates. Overall, The eDNNs framework is simple and easy to compute, while the iDNNs framework is accurate and fast converging. To demonstrate the utilities of our framework, various simulation studies, and real data analyses are presented with eDNNs and iDNNs.

stat.ML↗

arXiv2023

Hierarchical Stochastic Block Model for Community Detection in Multiplex Networks

Multiplex networks have become increasingly more prevalent in many fields, and have emerged as a powerful tool for modeling the complexity of real networks. There is a critical need for developing inference models for multiplex networks that can take into account potential dependencies across different layers, particularly when the aim is community detection. We add to a limited literature by proposing a novel and efficient Bayesian model for community detection in multiplex networks. A key feature of our approach is the ability to model varying communities at different network layers. In contrast, many existing models assume the same communities for all layers. Moreover, our model automatically picks up the necessary number of communities at each layer (as validated by real data examples). This is appealing, since deciding the number of communities is a challenging aspect of community detection, and especially so in the multiplex setting, if one allows the communities to change across layers. Borrowing ideas from hierarchical Bayesian modeling, we use a hierarchical Dirichlet prior to model community labels across layers, allowing dependency in their structure. Given the community labels, a stochastic block model (SBM) is assumed for each layer. We develop an efficient slice sampler for sampling the posterior distribution of the community labels as well as the link probabilities between communities. In doing so, we address some unique challenges posed by coupling the complex likelihood of SBM with the hierarchical nature of the prior on the labels. An extensive empirical validation is performed on simulated and real data, demonstrating the superior performance of the model over single-layer alternatives, as well as the ability to uncover interesting structures in real networks.

cs.SI↗

arXiv2022

Extrinsic Bayesian Optimizations on Manifolds

We propose an extrinsic Bayesian optimization (eBO) framework for general optimization problems on manifolds. Bayesian optimization algorithms build a surrogate of the objective function by employing Gaussian processes and quantify the uncertainty in that surrogate by deriving an acquisition function. This acquisition function represents the probability of improvement based on the kernel of the Gaussian process, which guides the search in the optimization process. The critical challenge for designing Bayesian optimization algorithms on manifolds lies in the difficulty of constructing valid covariance kernels for Gaussian processes on general manifolds. Our approach is to employ extrinsic Gaussian processes by first embedding the manifold onto some higher dimensional Euclidean space via equivariant embeddings and then constructing a valid covariance kernel on the image manifold after the embedding. This leads to efficient and scalable algorithms for optimization over complex manifolds. Simulation study and real data analysis are carried out to demonstrate the utilities of our eBO framework by applying the eBO to various optimization problems over manifolds such as the sphere, the Grassmannian, and the manifold of positive definite matrices.

math.OC↗

arXiv2022

Optimal Bayesian estimation of Gaussian mixtures with growing number of components

We study Bayesian estimation of finite mixture models in a general setup where the number of components is unknown and allowed to grow with the sample size. An assumption on growing number of components is a natural one as the degree of heterogeneity present in the sample can grow and new components can arise as sample size increases, allowing full flexibility in modeling the complexity of data. This however will lead to a high-dimensional model which poses great challenges for estimation. We novelly employ the idea of a sample size dependent prior in a Bayesian model and establish a number of important theoretical results. We first show that under mild conditions on the prior, the posterior distribution concentrates around the true mixing distribution at a near optimal rate with respect to the Wasserstein distance. Under a separation condition on the true mixing distribution, we further show that a better and adaptive convergence rate can be achieved, and the number of components can be consistently estimated. Furthermore, we derive optimal convergence rates for the higher-order mixture models where the number of components diverges arbitrarily fast. In addition, we suggest a simple recipe for using Dirichlet process (DP) mixture prior for estimating the finite mixture models and provide theoretical guarantees. In particular, we provide a novel solution for adopting the number of clusters in a DP mixture model as an estimate of the number of components in a finite mixture model. Simulation study and real data applications are carried out demonstrating the utilities of our method.

math.ST↗

arXiv2022

Robustness against Adversarial Attacks in Neural Networks using Incremental Dissipativity

Adversarial examples can easily degrade the classification performance in neural networks. Empirical methods for promoting robustness to such examples have been proposed, but often lack both analytical insights and formal guarantees. Recently, some robustness certificates have appeared in the literature based on system theoretic notions. This work proposes an incremental dissipativity-based robustness certificate for neural networks in the form of a linear matrix inequality for each layer. We also propose an equivalent spectral norm bound for this certificate which is scalable to neural networks with multiple layers. We demonstrate the improved performance against adversarial attacks on a feed-forward neural network trained on MNIST and an Alexnet trained using CIFAR-10.

cs.LG↗

arXiv2022

Multilevel Network Item Response Modeling for Discovering Differences Between Innovation and Regular School Systems in Korea

The innovation school system in South Korea has been developed in response to the traditional high-pressure school system in South Korea, with a view to cultivating a bottom-up and student-centered educational culture. Despite its ambitious goals, questions have been raised about the success of the innovation school system. Leveraging data from the Gyeonggi Education Panel Study (GEPS) along with advances in the statistical analysis of network data and educational data, we compare the two school systems in more depth. We find that some schools are indeed different from others, and those differences are not detected by conventional multilevel models. Having said that, we do not find much evidence that the innovation school system differs from the regular school system in terms of self-reported mental well-being, although we do detect differences among some schools that appear to be unrelated to the school system.

stat.AP↗

arXiv2022

Neural-PDE: A RNN based neural network for solving time dependent PDEs

Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional finite difference and finite elements methods and emerging advancements in machine learning, we propose a sequence deep learning framework called Neural-PDE, which allows to automatically learn governing rules of any time-dependent PDE system from existing data by using a bidirectional LSTM encoder, and predict the next n time steps data. One critical feature of our proposed framework is that the Neural-PDE is able to simultaneously learn and simulate the multiscale variables.We test the Neural-PDE by a range of examples from one-dimensional PDEs to a high-dimensional and nonlinear complex fluids model. The results show that the Neural-PDE is capable of learning the initial conditions, boundary conditions and differential operators without the knowledge of the specific form of a PDE system.In our experiments the Neural-PDE can efficiently extract the dynamics within 20 epochs training, and produces accurate predictions. Furthermore, unlike the traditional machine learning approaches in learning PDE such as CNN and MLP which require vast parameters for model precision, Neural-PDE shares parameters across all time steps, thus considerably reduces the computational complexity and leads to a fast learning algorithm.

math.NA↗

arXiv2021

Bayesian Optimal Two-sample Tests in High-dimension

We propose optimal Bayesian two-sample tests for testing equality of high-dimensional mean vectors and covariance matrices between two populations. In many applications including genomics and medical imaging, it is natural to assume that only a few entries of two mean vectors or covariance matrices are different. Many existing tests that rely on aggregating the difference between empirical means or covariance matrices are not optimal or yield low power under such setups. Motivated by this, we develop Bayesian two-sample tests employing a divide-and-conquer idea, which is powerful especially when the difference between two populations is sparse but large. The proposed two-sample tests manifest closed forms of Bayes factors and allow scalable computations even in high-dimensions. We prove that the proposed tests are consistent under relatively mild conditions compared to existing tests in the literature. Furthermore, the testable regions from the proposed tests turn out to be optimal in terms of rates. Simulation studies show clear advantages of the proposed tests over other state-of-the-art methods in various scenarios. Our tests are also applied to the analysis of the gene expression data of two cancer data sets.

stat.ME↗

arXiv2021

Scalable Bayesian high-dimensional local dependence learning

In this work, we propose a scalable Bayesian procedure for learning the local dependence structure in a high-dimensional model where the variables possess a natural ordering. The ordering of variables can be indexed by time, the vicinities of spatial locations, and so on, with the natural assumption that variables far apart tend to have weak correlations. Applications of such models abound in a variety of fields such as finance, genome associations analysis and spatial modeling. We adopt a flexible framework under which each variable is dependent on its neighbors or predecessors, and the neighborhood size can vary for each variable. It is of great interest to reveal this local dependence structure by estimating the covariance or precision matrix while yielding a consistent estimate of the varying neighborhood size for each variable. The existing literature on banded covariance matrix estimation, which assumes a fixed bandwidth cannot be adapted for this general setup. We employ the modified Cholesky decomposition for the precision matrix and design a flexible prior for this model through appropriate priors on the neighborhood sizes and Cholesky factors. The posterior contraction rates of the Cholesky factor are derived which are nearly or exactly minimax optimal, and our procedure leads to consistent estimates of the neighborhood size for all the variables. Another appealing feature of our procedure is its scalability to models with large numbers of variables due to efficient posterior inference without resorting to MCMC algorithms. Numerical comparisons are carried out with competitive methods, and applications are considered for some real datasets.

stat.ME↗

arXiv2021

Hypothesis testing for populations of networks

It has become an increasingly common practice for scientists in modern science and engineering to collect samples of multiple network data in which a network serves as a basic data object. The increasing prevalence of multiple network data calls for developments of models and theory that can deal with inference problems for populations of networks. In this work, we propose a general procedure for hypothesis testing of networks and in particular, for differentiating distributions of two samples of networks. We consider a very general framework which allows us to perform tests on large and sparse networks. Our contribution is two-fold: (1) We propose a test statistics based on the singular value of a generalized Wigner matrix. The asymptotic null distribution of the statistics is shown to follow the Tracy--Widom distribution as the number of nodes tends to infinity. The test also yields asymptotic power guarantee with the power tending to one under the alternative; (2) The test procedure is adapted for change-point detection in dynamic networks which is proven to be consistent in detecting the change-points. In addition to theoretical guarantees, another appealing feature of this adapted procedure is that it provides a principled and simple method for selecting the threshold that is also allowed to vary with time. Extensive simulation studies and real data analyses demonstrate the superior performance of our procedure with competitors.

stat.ME↗

arXiv2021

Training Graph Neural Networks by Graphon Estimation

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.

cs.LG↗

arXiv2021

Bayesian classification, anomaly detection, and survival analysis using network inputs with application to the microbiome

While the study of a single network is well-established, technological advances now allow for the collection of multiple networks with relative ease. Increasingly, anywhere from several to thousands of networks can be created from brain imaging, gene co-expression data, or microbiome measurements. And these networks, in turn, are being looked to as potentially powerful features to be used in modeling. However, with networks being non-Euclidean in nature, how best to incorporate them into standard modeling tasks is not obvious. In this paper, we propose a Bayesian modeling framework that provides a unified approach to binary classification, anomaly detection, and survival analysis with network inputs. We encode the networks in the kernel of a Gaussian process prior via their pairwise differences and we discuss several choices of provably positive definite kernel that can be plugged into our models. Although our methods are widely applicable, we are motivated here in particular by microbiome research (where network analysis is emerging as the standard approach for capturing the interconnectedness of microbial taxa across both time and space) and its potential for reducing preterm delivery and improving personalization of prenatal care.

stat.AP↗

arXiv2020

A spectral-based framework for hypothesis testing in populations of networks

In this paper, we propose a new spectral-based approach to hypothesis testing for populations of networks. The primary goal is to develop a test to determine whether two given samples of networks come from the same random model or distribution. Our test statistic is based on the trace of the third order for a centered and scaled adjacency matrix, which we prove converges to the standard normal distribution as the number of nodes tends to infinity. The asymptotic power guarantee of the test is also provided. The proper interplay between the number of networks and the number of nodes for each network is explored in characterizing the theoretical properties of the proposed testing statistics. Our tests are applicable to both binary and weighted networks, operate under a very general framework where the networks are allowed to be large and sparse, and can be extended to multiple-sample testing. We provide an extensive simulation study to demonstrate the superior performance of our test over existing methods and apply our test to three real datasets.

stat.ME↗

arXiv2020

Accelerated Algorithms for Convex and Non-Convex Optimization on Manifolds

We propose a general scheme for solving convex and non-convex optimization problems on manifolds. The central idea is that, by adding a multiple of the squared retraction distance to the objective function in question, we "convexify" the objective function and solve a series of convex sub-problems in the optimization procedure. One of the key challenges for optimization on manifolds is the difficulty of verifying the complexity of the objective function, e.g., whether the objective function is convex or non-convex, and the degree of non-convexity. Our proposed algorithm adapts to the level of complexity in the objective function. We show that when the objective function is convex, the algorithm provably converges to the optimum and leads to accelerated convergence. When the objective function is non-convex, the algorithm will converge to a stationary point. Our proposed method unifies insights from Nesterov's original idea for accelerating gradient descent algorithms with recent developments in optimization algorithms in Euclidean space. We demonstrate the utility of our algorithms on several manifold optimization tasks such as estimating intrinsic and extrinsic Fréchet means on spheres and low-rank matrix factorization with Grassmann manifolds applied to the Netflix rating data set.

stat.CO↗

arXiv2020

Weight Prediction for Variants of Weighted Directed Networks

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 \textit{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 $k$NN 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.

cs.SI↗