David B. Dunson

Lizhen Lin, Bayan Saparbayeva, Michael Minyi Zhang et al.

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.

Austin Talbot, David Dunson, Kafui Dzirasa et al.

Targeted stimulation of the brain has the potential to treat mental illnesses. We propose an approach to help design the stimulation protocol by identifying electrical dynamics across many brain regions that relate to illness states. We model multi-region electrical activity as a superposition of activity from latent networks, where the weights on the latent networks relate to an outcome of interest. In order to improve on drawbacks of latent factor modeling in this context, we focus on supervised autoencoders (SAEs), which can improve predictive performance while maintaining a generative model. We explain why SAEs yield improved predictions, describe the distributional assumptions under which SAEs are an appropriate modeling choice, and provide modeling constraints to ensure biological relevance of the learned network. We use the analysis strategy to find a network associated with stress that characterizes a genotype associated with bipolar disorder. This discovered network aligns with a previously used stimulation technique, providing experimental validation of our approach.

Meimei Liu, Zhengwu Zhang, David B. Dunson

There has been huge interest in studying human brain connectomes inferred from different imaging modalities and exploring their relationship with human traits, such as cognition. Brain connectomes are usually represented as networks, with nodes corresponding to different regions of interest (ROIs) and edges to connection strengths between ROIs. Due to the high-dimensionality and non-Euclidean nature of networks, it is challenging to depict their population distribution and relate them to human traits. Current approaches focus on summarizing the network using either pre-specified topological features or principal components analysis (PCA). In this paper, building on recent advances in deep learning, we develop a nonlinear latent factor model to characterize the population distribution of brain graphs and infer the relationships between brain structural connectomes and human traits. We refer to our method as Graph AuTo-Encoding (GATE). We applied GATE to two large-scale brain imaging datasets, the Adolescent Brain Cognitive Development (ABCD) study and the Human Connectome Project (HCP) for adults, to understand the structural brain connectome and its relationship with cognition. Numerical results demonstrate huge advantages of GATE over competitors in terms of prediction accuracy, statistical inference and computing efficiency. We found that structural connectomes have a stronger association with a wide range of human cognitive traits than was apparent using previous approaches.

Didong Li, David B Dunson

Many statistical and machine learning approaches rely on pairwise distances between data points. The choice of distance metric has a fundamental impact on performance of these procedures, raising questions about how to appropriately calculate distances. When data points are real-valued vectors, by far the most common choice is the Euclidean distance. This article is focused on the problem of how to better calculate distances taking into account the intrinsic geometry of the data, assuming data are concentrated near an unknown subspace or manifold. The appropriate geometric distance corresponds to the length of the shortest path along the manifold, which is the geodesic distance. When the manifold is unknown, it is challenging to accurately approximate the geodesic distance. Current algorithms are either highly complex, and hence often impractical to implement, or based on simple local linear approximations and shortest path algorithms that may have inadequate accuracy. We propose a simple and general alternative, which uses pieces of spheres, or spherelets, to locally approximate the unknown subspace and thereby estimate the geodesic distance through paths over spheres. Theory is developed showing lower error for many manifolds, with applications in clustering, conditional density estimation and mean regression. The conclusion is supported through multiple simulation examples and real data sets.

Jieren Xu, Yitong Li, Haizhao Yang et al.

This paper proposes a novel kernel-based optimization scheme to handle tasks in the analysis, e.g., signal spectral estimation and single-channel source separation of 1D non-stationary oscillatory data. The key insight of our optimization scheme for reconstructing the time-frequency information is that when a nonparametric regression is applied on some input values, the output regressed points would lie near the oscillatory pattern of the oscillatory 1D signal only if these input values are a good approximation of the ground-truth phase function. In this work, Gaussian Process (GP) is chosen to conduct this nonparametric regression: the oscillatory pattern is encoded as the Pattern-inducing Points (PiPs) which act as the training data points in the GP regression; while the targeted phase function is fed in to compute the correlation kernels, acting as the testing input. Better approximated phase function generates more precise kernels, thus resulting in smaller optimization loss error when comparing the kernel-based regression output with the original signals. To the best of our knowledge, this is the first algorithm that can satisfactorily handle fully non-stationary oscillatory data, close and crossover frequencies, and general oscillatory patterns. Even in the example of a signal {produced by slow variation in the parameters of a trigonometric expansion}, we show that PiPs admits competitive or better performance in terms of accuracy and robustness than existing state-of-the-art algorithms.

Xiangyu Wang, David Dunson, Chenlei Leng

Ordinary least squares (OLS) is the default method for fitting linear models, but is not applicable for problems with dimensionality larger than the sample size. For these problems, we advocate the use of a generalized version of OLS motivated by ridge regression, and propose two novel three-step algorithms involving least squares fitting and hard thresholding. The algorithms are methodologically simple to understand intuitively, computationally easy to implement efficiently, and theoretically appealing for choosing models consistently. Numerical exercises comparing our methods with penalization-based approaches in simulations and data analyses illustrate the great potential of the proposed algorithms.

Xiangyu Wang, Chenlei Leng, David B. Dunson

Variable screening is a fast dimension reduction technique for assisting high dimensional feature selection. As a preselection method, it selects a moderate size subset of candidate variables for further refining via feature selection to produce the final model. The performance of variable screening depends on both computational efficiency and the ability to dramatically reduce the number of variables without discarding the important ones. When the data dimension $p$ is substantially larger than the sample size $n$, variable screening becomes crucial as 1) Faster feature selection algorithms are needed; 2) Conditions guaranteeing selection consistency might fail to hold. This article studies a class of linear screening methods and establishes consistency theory for this special class. In particular, we prove the restricted diagonally dominant (RDD) condition is a necessary and sufficient condition for strong screening consistency. As concrete examples, we show two screening methods $SIS$ and $HOLP$ are both strong screening consistent (subject to additional constraints) with large probability if $n > O((ρs + σ/τ)^2\log p)$ under random designs. In addition, we relate the RDD condition to the irrepresentable condition, and highlight limitations of $SIS$.

Rajarshi Guhaniyogi, David B. Dunson

As an alternative to variable selection or shrinkage in high dimensional regression, we propose to randomly compress the predictors prior to analysis. This dramatically reduces storage and computational bottlenecks, performing well when the predictors can be projected to a low dimensional linear subspace with minimal loss of information about the response. As opposed to existing Bayesian dimensionality reduction approaches, the exact posterior distribution conditional on the compressed data is available analytically, speeding up computation by many orders of magnitude while also bypassing robustness issues due to convergence and mixing problems with MCMC. Model averaging is used to reduce sensitivity to the random projection matrix, while accommodating uncertainty in the subspace dimension. Strong theoretical support is provided for the approach by showing near parametric convergence rates for the predictive density in the large p small n asymptotic paradigm. Practical performance relative to competitors is illustrated in simulations and real data applications.