We investigate a general matrix factorization for deviance-based losses, extending the ubiquitous singular value decomposition beyond squared error loss. While similar approaches have been explored before, here we propose an efficient algorithm that is flexible enough to allow for structural zeros and entry weights. Moreover, we provide theoretical support for these decompositions by (i) showing strong consistency under a generalized linear model setup, (ii) checking the adequacy of a chosen exponential family via a generalized Hosmer-Lemeshow test, and (iii) determining the rank of the decomposition via a maximum eigenvalue gap method. To further support our findings, we conduct simulation studies to assess robustness to decomposition assumptions and extensive case studies using benchmark datasets from image face recognition, natural language processing, network analysis, and biomedical studies. Our theoretical and empirical results indicate that the proposed decomposition is more flexible, general, and can provide improved performance when compared to traditional methods.

Implicit deep learning has received increasing attention recently due to the fact that it generalizes the recursive prediction rules of many commonly used neural network architectures. Its prediction rule is provided implicitly based on the solution of an equilibrium equation. Although a line of recent empirical studies has demonstrated its superior performances, the theoretical understanding of implicit neural networks is limited. In general, the equilibrium equation may not be well-posed during the training. As a result, there is no guarantee that a vanilla (stochastic) gradient descent (SGD) training nonlinear implicit neural networks can converge. This paper fills the gap by analyzing the gradient flow of Rectified Linear Unit (ReLU) activated implicit neural networks. For an $m$-width implicit neural network with ReLU activation and $n$ training samples, we show that a randomly initialized gradient descent converges to a global minimum at a linear rate for the square loss function if the implicit neural network is \textit{over-parameterized}. It is worth noting that, unlike existing works on the convergence of (S)GD on finite-layer over-parameterized neural networks, our convergence results hold for implicit neural networks, where the number of layers is \textit{infinite}.

Graphical models, used to express conditional dependence between random variables observed at various nodes, are used extensively in many fields such as genetics, neuroscience, and social network analysis. While most current statistical methods for estimating graphical models focus on scalar data, there is interest in estimating analogous dependence structures when the data observed at each node are functional, such as signals or images. In this paper, we propose a fully Bayesian regularization scheme for estimating functional graphical models. We first consider a direct Bayesian analog of the functional graphical lasso proposed by Qiao et al. (2019). We then propose a regularization strategy via the graphical horseshoe. We compare these approaches via simulation study and apply our proposed functional graphical horseshoe to two motivating applications, electroencephalography data for comparing brain activation between an alcoholic group and controls, as well as changes in structural connectivity in the presence of traumatic brain injury (TBI). Our results yield insight into how the brain attempts to compensate for disconnected networks after injury.

Surface roughness is primary measure of pavement performance that has been associated with ride quality and vehicle operating costs. Of all the surface roughness indicators, the International Roughness Index (IRI) is the most widely used. However, it is costly to measure IRI, and for this reason, certain road classes are excluded from IRI measurements at a network level. Higher levels of distresses are generally associated with higher roughness. However, for a given roughness level, pavement data typically exhibits a great deal of variability in the distress types, density, and severity. It is hypothesized that it is feasible to estimate the IRI of a pavement section given its distress types and their respective densities and severities. To investigate this hypothesis, this paper uses data from in-service pavements and machine learning methods to ascertain the extent to which IRI can be predicted given a set of pavement attributes. The results suggest that machine learning can be used reliably to estimate IRI based on the measured distress types and their respective densities and severities. The analysis also showed that IRI estimated this way depends on the pavement type and functional class. The paper also includes an exploratory section that addresses the reverse situation, that is, estimating the probability of pavement distress type distribution and occurrence severity/extent based on a given roughness level.

Modern data often take the form of a multiway array. However, most classification methods are designed for vectors, i.e., 1-way arrays. Distance weighted discrimination (DWD) is a popular high-dimensional classification method that has been extended to the multiway context, with dramatic improvements in performance when data have multiway structure. However, the previous implementation of multiway DWD was restricted to classification of matrices, and did not account for sparsity. In this paper, we develop a general framework for multiway classification which is applicable to any number of dimensions and any degree of sparsity. We conducted extensive simulation studies, showing that our model is robust to the degree of sparsity and improves classification accuracy when the data have multiway structure. For our motivating application, magnetic resonance spectroscopy (MRS) was used to measure the abundance of several metabolites across multiple neurological regions and across multiple time points in a mouse model of Friedreich's ataxia, yielding a four-way data array. Our method reveals a robust and interpretable multi-region metabolomic signal that discriminates the groups of interest. We also successfully apply our method to gene expression time course data for multiple sclerosis treatment. An R implementation is available in the package MultiwayClassification at http://github.com/lockEF/MultiwayClassification .

The increasing complexity of today's software requires the contribution of thousands of developers. This complex collaboration structure makes developers more likely to introduce defect-prone changes that lead to software faults. Determining when these defect-prone changes are introduced has proven challenging, and using traditional machine learning (ML) methods to make these determinations seems to have reached a plateau. In this work, we build contribution graphs consisting of developers and source files to capture the nuanced complexity of changes required to build software. By leveraging these contribution graphs, our research shows the potential of using graph-based ML to improve Just-In-Time (JIT) defect prediction. We hypothesize that features extracted from the contribution graphs may be better predictors of defect-prone changes than intrinsic features derived from software characteristics. We corroborate our hypothesis using graph-based ML for classifying edges that represent defect-prone changes. This new framing of the JIT defect prediction problem leads to remarkably better results. We test our approach on 14 open-source projects and show that our best model can predict whether or not a code change will lead to a defect with an F1 score as high as 86.25$\%$. This represents an increase of as much as 55.4$\%$ over the state-of-the-art in JIT defect prediction. We describe limitations, open challenges, and how this method can be used for operational JIT defect prediction.

Modern datasets often contain large subsets of correlated features and nuisance features, which are not or loosely related to the main underlying structures of the data. Nuisance features can be identified using the Laplacian score criterion, which evaluates the importance of a given feature via its consistency with the Graph Laplacians' leading eigenvectors. We demonstrate that in the presence of large numbers of nuisance features, the Laplacian must be computed on the subset of selected features rather than on the complete feature set. To do this, we propose a fully differentiable approach for unsupervised feature selection, utilizing the Laplacian score criterion to avoid the selection of nuisance features. We employ an autoencoder architecture to cope with correlated features, trained to reconstruct the data from the subset of selected features. Building on the recently proposed concrete layer that allows controlling for the number of selected features via architectural design, simplifying the optimization process. Experimenting on several real-world datasets, we demonstrate that our proposed approach outperforms similar approaches designed to avoid only correlated or nuisance features, but not both. Several state-of-the-art clustering results are reported.

This article introduces a novel nonparametric methodology for Generalized Linear Models which combines the strengths of the binary regression and latent variable formulations for categorical data, while overcoming their disadvantages. Requiring minimal assumptions, it extends recently published parametric versions of the methodology and generalizes it. If the underlying data generating process is asymmetric, it gives uniformly better prediction and inference performance over the parametric formulation. Furthermore, it introduces a new classification statistic utilizing which I show that overall, it has better model fit, inference and classification performance than the parametric version, and the difference in performance is statistically significant especially if the data generating process is asymmetric. In addition, the methodology can be used to perform model diagnostics for any model specification. This is a highly useful result, and it extends existing work for categorical model diagnostics broadly across the sciences. The mathematical results also highlight important new findings regarding the interplay of statistical significance and scientific significance. Finally, the methodology is applied to various real-world datasets to show that it may outperform widely used existing models, including Random Forests and Deep Neural Networks with very few iterations.

We present a geometrically enhanced Markov chain Monte Carlo sampler for networks based on a discrete curvature measure defined on graphs. Specifically, we incorporate the concept of graph Forman curvature into sampling procedures on both the nodes and edges of a network explicitly, via the transition probability of the Markov chain, as well as implicitly, via the target stationary distribution, which gives a novel, curved Markov chain Monte Carlo approach to learning networks. We show that integrating curvature into the sampler results in faster convergence to a wide range of network statistics demonstrated on deterministic networks drawn from real-world data.

Operator learning techniques have recently emerged as a powerful tool for learning maps between infinite-dimensional Banach spaces. Trained under appropriate constraints, they can also be effective in learning the solution operator of partial differential equations (PDEs) in an entirely self-supervised manner. In this work we analyze the training dynamics of deep operator networks (DeepONets) through the lens of Neural Tangent Kernel (NTK) theory, and reveal a bias that favors the approximation of functions with larger magnitudes. To correct this bias we propose to adaptively re-weight the importance of each training example, and demonstrate how this procedure can effectively balance the magnitude of back-propagated gradients during training via gradient descent. We also propose a novel network architecture that is more resilient to vanishing gradient pathologies. Taken together, our developments provide new insights into the training of DeepONets and consistently improve their predictive accuracy by a factor of 10-50x, demonstrated in the challenging setting of learning PDE solution operators in the absence of paired input-output observations. All code and data accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/ImprovedDeepONets.}