Genevera I. Allen

Kai R. Wycik, Tiffany M. Tang, Tarek M. Zikry et al.

Clustering is widely used across the sciences as the foundation for downstream data-driven scientific discoveries. However, clustering results are highly sensitive to the choice of algorithm, preprocessing, and the number of clusters $k$, producing scientific claims that are often not reproducible. The current state of the art for validating clustering solutions consists of clustering validation indices (CVIs) such as Silhouette, Davies-Bouldin, and Calinski-Harabasz, which rely on geometric assumptions that break down on the heavy-tailed, high-dimensional, and nonlinearly structured data encountered in biomedical research. Resampling-based alternatives - grounded in the ideas of clustering stability and generalizability - have been proposed but remain scattered across specialized tools with no unified, accessible software. We fill this gap with CARVE (Cluster Analysis with Resampling for Validation and Exploration), an open-source Python and R package that jointly evaluates multiple clustering algorithms and hyperparameters, returning stability and generalizability diagnostics at the global, cluster, and sample level together with principled selection rules and consensus-based cluster labels. Across six synthetic benchmarks CARVE consistently recovers near-optimal clusterings where classical indices degrade substantially. On experimental genomics and proteomics data sets, CARVE recovers finer biological structure when classical CVIs collapse entirely. CARVE is available with a scikit-learn-compatible Python API and an analogous R interface compatible with Seurat workflows.

Luqin Gan, Lili Zheng, Genevera I. Allen

To promote new scientific discoveries from complex data sets, feature importance inference has been a long-standing statistical problem. Instead of testing for parameters that are only interpretable for specific models, there has been increasing interest in model-agnostic methods, often in the form of feature occlusion or leave-one-covariate-out (LOCO) inference. Existing approaches often make distributional assumptions, which can be difficult to verify in practice, or require model refitting and data splitting, which are computationally intensive and lead to losses in power. In this work, we develop a novel, mostly model-agnostic and distribution-free inference framework for feature importance that is computationally efficient and statistically powerful. Our approach is fast as we avoid model refitting by leveraging a form of random observation and feature subsampling called minipatch ensembles; this approach also improves statistical power by avoiding data splitting. Our framework can be applied on tabular data and with any machine learning algorithm, together with minipatch ensembles, for regression and classification tasks. Despite the dependencies induced by using minipatch ensembles, we show that our approach provides asymptotic coverage for the feature importance score of any model under mild assumptions. Finally, our same procedure can also be leveraged to provide valid confidence intervals for predictions, hence providing fast, simultaneous quantification of the uncertainty of both predictions and feature importance. We validate our intervals on a series of synthetic and real data examples, including non-linear settings, showing that our approach detects the correct important features and exhibits many computational and statistical advantages over existing methods.

Madeline Navarro, Camille Little, Genevera I. Allen et al.

In this work, we propose data augmentation via pairwise mixup across subgroups to improve group fairness. Many real-world applications of machine learning systems exhibit biases across certain groups due to under-representation or training data that reflects societal biases. Inspired by the successes of mixup for improving classification performance, we develop a pairwise mixup scheme to augment training data and encourage fair and accurate decision boundaries for all subgroups. Data augmentation for group fairness allows us to add new samples of underrepresented groups to balance subpopulations. Furthermore, our method allows us to use the generalization ability of mixup to improve both fairness and accuracy. We compare our proposed mixup to existing data augmentation and bias mitigation approaches on both synthetic simulations and real-world benchmark fair classification data, demonstrating that we are able to achieve fair outcomes with robust if not improved accuracy.

Genevera I. Allen, Luqin Gan, Lili Zheng

New technologies have led to vast troves of large and complex datasets across many scientific domains and industries. People routinely use machine learning techniques to not only process, visualize, and make predictions from this big data, but also to make data-driven discoveries. These discoveries are often made using Interpretable Machine Learning, or machine learning models and techniques that yield human understandable insights. In this paper, we discuss and review the field of interpretable machine learning, focusing especially on the techniques as they are often employed to generate new knowledge or make discoveries from large data sets. We outline the types of discoveries that can be made using Interpretable Machine Learning in both supervised and unsupervised settings. Additionally, we focus on the grand challenge of how to validate these discoveries in a data-driven manner, which promotes trust in machine learning systems and reproducibility in science. We discuss validation from both a practical perspective, reviewing approaches based on data-splitting and stability, as well as from a theoretical perspective, reviewing statistical results on model selection consistency and uncertainty quantification via statistical inference. Finally, we conclude by highlighting open challenges in using interpretable machine learning techniques to make discoveries, including gaps between theory and practice for validating data-driven-discoveries.

Irene Chang, Tarek M. Zikry, Genevera I. Allen

Low-dimensional embeddings are widely used as visual summaries of high-dimensional data and to enable downstream scientific discoveries. Yet, popular nonlinear dimension reduction methods, such as t-SNE and UMAP, are often selected based on visual appeal alone and without rigorous quantitative validation. A major reason is that manifold embeddings typically do not provide an out-of-sample map nor an inverse back to the original feature space; this makes held-out validation, the gold standard in supervised learning, all but impossible. To address these challenges, we develop a novel framework, MEDAL (Manifold Embedding Distillation via Autoencoder Learning), which distills a fitted manifold embedding into a reusable encoder--decoder model. MEDAL trains a constrained autoencoder whose bottleneck exactly matches any teacher embedding while the decoder reconstructs the original input; this yields an explicit map for new samples, an approximate inverse, and a pointwise reconstruction-based measure of distortion in the manifold space. This converts static manifold embeddings into models that can be evaluated on held-out data, enabling quantitative validation including comparing different dimension reduction methods as well as hyperparameter tuning. Across multiple benchmark and scientific case studies, we show that MEDAL enables held-out validation to determine optimal manifold embeddings and hyperparameters, reveals biologically coherent regions that are difficult to preserve in two dimensional embeddings, and detects distribution shift when new samples are mapped into a fixed reference manifold. MEDAL provides a general validation wrapper to any existing dimension reduction technique that will improve the rigor and

Hamza Golubovic, Matthew Shen, Genevera I. Allen et al.

Modern matrix completion problems often involve heterogeneous data whose rows simultaneously belong to many meta-categories, such as demographic and age groups in recommendation systems, or region and recording session labels in neural electrophysiological experiments. Standard low-rank estimators impose a single global latent geometry, which can recover average structure but may smooth away subgroup-specific variation, especially when observations are unevenly distributed across groups. We introduce Group-Aware Matrix Estimation (GAME), a convex estimator for overlapping subgroup-wise low-rank matrix estimation. GAME regularizes category-specific submatrices through overlapping nuclear-norm penalties, allowing related groups to borrow information while preserving local latent structure in a shared coordinate system. We provide finite-sample guarantees for both reconstruction error and subgroup-specific subspace recovery, showing how performance depends on sampling density, subgroup rank, and overlap structure. Experiments on synthetic, recommendation, ecological, and neuroscience datasets show that GAME is most beneficial in structured missingness regimes, where subgroup-aware regularization improves both reconstruction accuracy and latent subspace fidelity. Across these benchmarks, GAME is competitive or best among global low-rank, side-information, and modern imputation baselines, with the largest gains when subgroups exhibit distinct low-rank structure.

Luqin Gan, Tarek M. Zikry, Genevera I. Allen

As machine learning systems are increasingly used in high-stakes domains, there is a growing emphasis placed on making them interpretable to improve trust in these systems. In response, a range of interpretable machine learning (IML) methods have been developed to generate human-understandable insights into otherwise black box models. With these methods, a fundamental question arises: Are these interpretations reliable? Unlike with prediction accuracy or other evaluation metrics for supervised models, the proximity to the true interpretation is difficult to define. Instead, we ask a closely related question that we argue is a prerequisite for reliability: Are these interpretations stable? We define stability as findings that are consistent or reliable under small random perturbations to the data or algorithms. In this study, we conduct the first systematic, large-scale empirical stability study on popular machine learning global interpretations for both supervised and unsupervised tasks on tabular data. Our findings reveal that popular interpretation methods are frequently unstable, notably less stable than the predictions themselves, and that there is no association between the accuracy of machine learning predictions and the stability of their associated interpretations. Moreover, we show that no single method consistently provides the most stable interpretations across a range of benchmark datasets. Overall, these results suggest that interpretability alone does not warrant trust, and underscores the need for rigorous evaluation of interpretation stability in future work. To support these principles, we have developed and released an open source IML dashboard and Python package to enable researchers to assess the stability and reliability of their own data-driven interpretations and discoveries.

Michael Weylandt, John Nagorski, Genevera I. Allen

Convex clustering is a promising new approach to the classical problem of clustering, combining strong performance in empirical studies with rigorous theoretical foundations. Despite these advantages, convex clustering has not been widely adopted, due to its computationally intensive nature and its lack of compelling visualizations. To address these impediments, we introduce Algorithmic Regularization, an innovative technique for obtaining high-quality estimates of regularization paths using an iterative one-step approximation scheme. We justify our approach with a novel theoretical result, guaranteeing global convergence of the approximate path to the exact solution under easily-checked non-data-dependent assumptions. The application of algorithmic regularization to convex clustering yields the Convex Clustering via Algorithmic Regularization Paths (CARP) algorithm for computing the clustering solution path. On example data sets from genomics and text analysis, CARP delivers over a 100-fold speed-up over existing methods, while attaining a finer approximation grid than standard methods. Furthermore, CARP enables improved visualization of clustering solutions: the fine solution grid returned by CARP can be used to construct a convex clustering-based dendrogram, as well as forming the basis of a dynamic path-wise visualization based on modern web technologies. Our methods are implemented in the open-source R package clustRviz, available at https://github.com/DataSlingers/clustRviz.

Camille Olivia Little, Debolina Halder Lina, Genevera I. Allen

Across various sectors such as healthcare, criminal justice, national security, finance, and technology, large-scale machine learning (ML) and artificial intelligence (AI) systems are being deployed to make critical data-driven decisions. Many have asked if we can and should trust these ML systems to be making these decisions. Two critical components are prerequisites for trust in ML systems: interpretability, or the ability to understand why the ML system makes the decisions it does, and fairness, which ensures that ML systems do not exhibit bias against certain individuals or groups. Both interpretability and fairness are important and have separately received abundant attention in the ML literature, but so far, there have been very few methods developed to directly interpret models with regard to their fairness. In this paper, we focus on arguably the most popular type of ML interpretation: feature importance scores. Inspired by the use of decision trees in knowledge distillation, we propose to leverage trees as interpretable surrogates for complex black-box ML models. Specifically, we develop a novel fair feature importance score for trees that can be used to interpret how each feature contributes to fairness or bias in trees, tree-based ensembles, or tree-based surrogates of any complex ML system. Like the popular mean decrease in impurity for trees, our Fair Feature Importance Score is defined based on the mean decrease (or increase) in group bias. Through simulations as well as real examples on benchmark fairness datasets, we demonstrate that our Fair Feature Importance Score offers valid interpretations for both tree-based ensembles and tree-based surrogates of other ML systems.

Andersen Chang, Tiffany M. Tang, Tarek M. Zikry et al.

Unsupervised machine learning is widely used to mine large, unlabeled datasets to make data-driven discoveries in critical domains such as climate science, biomedicine, astronomy, chemistry, and more. However, despite its widespread utilization, there is a lack of standardization in unsupervised learning workflows for making reliable and reproducible scientific discoveries. In this paper, we present a structured workflow for using unsupervised learning techniques in science. We highlight and discuss best practices starting with formulating validatable scientific questions, conducting robust data preparation and exploration, using a range of modeling techniques, performing rigorous validation by evaluating the stability and generalizability of unsupervised learning conclusions, and promoting effective communication and documentation of results to ensure reproducible scientific discoveries. To illustrate our proposed workflow, we present a case study from astronomy, seeking to refine globular clusters of Milky Way stars based upon their chemical composition. Our case study highlights the importance of validation and illustrates how the benefits of a carefully-designed workflow for unsupervised learning can advance scientific discovery.

Camille Olivia Little, Genevera I. Allen

Ensemble methods, particularly boosting, have established themselves as highly effective and widely embraced machine learning techniques for tabular data. In this paper, we aim to leverage the robust predictive power of traditional boosting methods while enhancing fairness and interpretability. To achieve this, we develop Fair MP-Boost, a stochastic boosting scheme that balances fairness and accuracy by adaptively learning features and observations during training. Specifically, Fair MP-Boost sequentially samples small subsets of observations and features, termed minipatches (MP), according to adaptively learned feature and observation sampling probabilities. We devise these probabilities by combining loss functions, or by combining feature importance scores to address accuracy and fairness simultaneously. Hence, Fair MP-Boost prioritizes important and fair features along with challenging instances, to select the most relevant minipatches for learning. The learned probability distributions also yield intrinsic interpretations of feature importance and important observations in Fair MP-Boost. Through empirical evaluation of simulated and benchmark datasets, we showcase the interpretability, accuracy, and fairness of Fair MP-Boost.

Madeline Navarro, Genevera I. Allen, Michael Weylandt

Network models provide a powerful and flexible framework for analyzing a wide range of structured data sources. In many situations of interest, however, multiple networks can be constructed to capture different aspects of an underlying phenomenon or to capture changing behavior over time. In such settings, it is often useful to cluster together related networks in attempt to identify patterns of common structure. In this paper, we propose a convex approach for the task of network clustering. Our approach uses a convex fusion penalty to induce a smoothly-varying tree-like cluster structure, eliminating the need to select the number of clusters a priori. We provide an efficient algorithm for convex network clustering and demonstrate its effectiveness on synthetic examples.

Tianyi Yao, Minjie Wang, Genevera I. Allen

Gaussian graphical models provide a powerful framework for uncovering conditional dependence relationships between sets of nodes; they have found applications in a wide variety of fields including sensor and communication networks, physics, finance, and computational biology. Often, one observes data on the nodes and the task is to learn the graph structure, or perform graphical model selection. While this is a well-studied problem with many popular techniques, there are typically three major practical challenges: i) many existing algorithms become computationally intractable in huge-data settings with tens of thousands of nodes; ii) the need for separate data-driven hyperparameter tuning considerably adds to the computational burden; iii) the statistical accuracy of selected edges often deteriorates as the dimension and/or the complexity of the underlying graph structures increase. We tackle these problems by developing the novel Minipatch Graph (MPGraph) estimator. Our approach breaks up the huge graph learning problem into many smaller problems by creating an ensemble of tiny random subsets of both the observations and the nodes, termed minipatches. We then leverage recent advances that use hard thresholding to solve the latent variable graphical model problem to consistently learn the graph on each minipatch. Our approach is computationally fast, embarrassingly parallelizable, memory efficient, and has integrated stability-based hyperparamter tuning. Additionally, we prove that under weaker assumptions than that of the Graphical Lasso, our MPGraph estimator achieves graph selection consistency. We compare our approach to state-of-the-art computational approaches for Gaussian graphical model selection including the BigQUIC algorithm, and empirically demonstrate that our approach is not only more statistically accurate but also extensively faster for huge graph learning problems.

Luqin Gan, Genevera I. Allen

Consensus clustering has been widely used in bioinformatics and other applications to improve the accuracy, stability and reliability of clustering results. This approach ensembles cluster co-occurrences from multiple clustering runs on subsampled observations. For application to large-scale bioinformatics data, such as to discover cell types from single-cell sequencing data, for example, consensus clustering has two significant drawbacks: (i) computational inefficiency due to repeatedly applying clustering algorithms, and (ii) lack of interpretability into the important features for differentiating clusters. In this paper, we address these two challenges by developing IMPACC: Interpretable MiniPatch Adaptive Consensus Clustering. Our approach adopts three major innovations. We ensemble cluster co-occurrences from tiny subsets of both observations and features, termed minipatches, thus dramatically reducing computation time. Additionally, we develop adaptive sampling schemes for observations, which result in both improved reliability and computational savings, as well as adaptive sampling schemes of features, which leads to interpretable solutions by quickly learning the most relevant features that differentiate clusters. We study our approach on synthetic data and a variety of real large-scale bioinformatics data sets; results show that our approach not only yields more accurate and interpretable cluster solutions, but it also substantially improves computational efficiency compared to standard consensus clustering approaches.

Minjie Wang, Genevera I. Allen

In neuroscience, researchers seek to uncover the connectivity of neurons from large-scale neural recordings or imaging; often people employ graphical model selection and estimation techniques for this purpose. But, existing technologies can only record from a small subset of neurons leading to a challenging problem of graph selection in the presence of extensive latent variables. Chandrasekaran et al. (2012) proposed a convex program to address this problem that poses challenges from both a computational and statistical perspective. To solve this problem, we propose an incredibly simple solution: apply a hard thresholding operator to existing graph selection methods. Conceptually simple and computationally attractive, we demonstrate that thresholding the graphical Lasso, neighborhood selection, or CLIME estimators have superior theoretical properties in terms of graph selection consistency as well as stronger empirical results than existing approaches for the latent variable graphical model problem. We also demonstrate the applicability of our approach through a neuroscience case study on calcium-imaging data to estimate functional neural connections.

Michael Weylandt, T. Mitchell Roddenberry, Genevera I. Allen

Clustering is a ubiquitous problem in data science and signal processing. In many applications where we observe noisy signals, it is common practice to first denoise the data, perhaps using wavelet denoising, and then to apply a clustering algorithm. In this paper, we develop a sparse convex wavelet clustering approach that simultaneously denoises and discovers groups. Our approach utilizes convex fusion penalties to achieve agglomeration and group-sparse penalties to denoise through sparsity in the wavelet domain. In contrast to common practice which denoises then clusters, our method is a unified, convex approach that performs both simultaneously. Our method yields denoised (wavelet-sparse) cluster centroids that both improve interpretability and data compression. We demonstrate our method on synthetic examples and in an application to NMR spectroscopy.

Kelly Geyer, Frederick Campbell, Andersen Chang et al.

ElectroCOrticoGraphy (ECoG) technology measures electrical activity in the human brain via electrodes placed directly on the cortical surface during neurosurgery. Through its capability to record activity at a fast temporal resolution, ECoG experiments have allowed scientists to better understand how the human brain processes speech. By its nature, ECoG data is difficult for neuroscientists to directly interpret for two major reasons. Firstly, ECoG data tends to be large in size, as each individual experiment yields data up to several gigabytes. Secondly, ECoG data has a complex, higher-order nature. After signal processing, this type of data may be organized as a 4-way tensor with dimensions representing trials, electrodes, frequency, and time. In this paper, we develop an interpretable dimension reduction approach called Regularized Higher Order Principal Components Analysis, as well as an extension to Regularized Higher Order Partial Least Squares, that allows neuroscientists to explore and visualize ECoG data. Our approach employs a sparse and functional Candecomp-Parafac (CP) decomposition that incorporates sparsity to select relevant electrodes and frequency bands, as well as smoothness over time and frequency, yielding directly interpretable factors. We demonstrate the performance and interpretability of our method with an ECoG case study on audio and visual processing of human speech.

Mohammad Taha Toghani, Genevera I. Allen

Boosting methods are among the best general-purpose and off-the-shelf machine learning approaches, gaining widespread popularity. In this paper, we seek to develop a boosting method that yields comparable accuracy to popular AdaBoost and gradient boosting methods, yet is faster computationally and whose solution is more interpretable. We achieve this by developing MP-Boost, an algorithm loosely based on AdaBoost that learns by adaptively selecting small subsets of instances and features, or what we term minipatches (MP), at each iteration. By sequentially learning on tiny subsets of the data, our approach is computationally faster than other classic boosting algorithms. Also as it progresses, MP-Boost adaptively learns a probability distribution on the features and instances that upweight the most important features and challenging instances, hence adaptively selecting the most relevant minipatches for learning. These learned probability distributions also aid in interpretation of our method. We empirically demonstrate the interpretability, comparative accuracy, and computational time of our approach on a variety of binary classification tasks.

Tianyi Yao, Genevera I. Allen

Feature selection often leads to increased model interpretability, faster computation, and improved model performance by discarding irrelevant or redundant features. While feature selection is a well-studied problem with many widely-used techniques, there are typically two key challenges: i) many existing approaches become computationally intractable in huge-data settings with millions of observations and features; and ii) the statistical accuracy of selected features degrades in high-noise, high-correlation settings, thus hindering reliable model interpretation. We tackle these problems by proposing Stable Minipatch Selection (STAMPS) and Adaptive STAMPS (AdaSTAMPS). These are meta-algorithms that build ensembles of selection events of base feature selectors trained on many tiny, (adaptively-chosen) random subsets of both the observations and features of the data, which we call minipatches. Our approaches are general and can be employed with a variety of existing feature selection strategies and machine learning techniques. In addition, we provide theoretical insights on STAMPS and empirically demonstrate that our approaches, especially AdaSTAMPS, dominate competing methods in terms of feature selection accuracy and computational time.

Minjie Wang, Tianyi Yao, Genevera I. Allen

Clustering has long been a popular unsupervised learning approach to identify groups of similar objects and discover patterns from unlabeled data in many applications. Yet, coming up with meaningful interpretations of the estimated clusters has often been challenging precisely due to its unsupervised nature. Meanwhile, in many real-world scenarios, there are some noisy supervising auxiliary variables, for instance, subjective diagnostic opinions, that are related to the observed heterogeneity of the unlabeled data. By leveraging information from both supervising auxiliary variables and unlabeled data, we seek to uncover more scientifically interpretable group structures that may be hidden by completely unsupervised analyses. In this work, we propose and develop a new statistical pattern discovery method named Supervised Convex Clustering (SCC) that borrows strength from both information sources and guides towards finding more interpretable patterns via a joint convex fusion penalty. We develop several extensions of SCC to integrate different types of supervising auxiliary variables, to adjust for additional covariates, and to find biclusters. We demonstrate the practical advantages of SCC through simulations and a case study on Alzheimer's Disease genomics. Specifically, we discover new candidate genes as well as new subtypes of Alzheimer's Disease that can potentially lead to better understanding of the underlying genetic mechanisms responsible for the observed heterogeneity of cognitive decline in older adults.

Minjie Wang, Genevera I. Allen

In mixed multi-view data, multiple sets of diverse features are measured on the same set of samples. By integrating all available data sources, we seek to discover common group structure among the samples that may be hidden in individualistic cluster analyses of a single data-view. While several techniques for such integrative clustering have been explored, we propose and develop a convex formalization that will inherit the strong statistical, mathematical and empirical properties of increasingly popular convex clustering methods. Specifically, our Integrative Generalized Convex Clustering Optimization (iGecco) method employs different convex distances, losses, or divergences for each of the different data views with a joint convex fusion penalty that leads to common groups. Additionally, integrating mixed multi-view data is often challenging when each data source is high-dimensional. To perform feature selection in such scenarios, we develop an adaptive shifted group-lasso penalty that selects features by shrinking them towards their loss-specific centers. Our so-called iGecco+ approach selects features from each data-view that are best for determining the groups, often leading to improved integrative clustering. To fit our model, we develop a new type of generalized multi-block ADMM algorithm using sub-problem approximations that more efficiently fits our model for big data sets. Through a series of numerical experiments and real data examples on text mining and genomics, we show that iGecco+ achieves superior empirical performance for high-dimensional mixed multi-view data.

Tianyi Yao, Genevera I. Allen

Knowledge of functional groupings of neurons can shed light on structures of neural circuits and is valuable in many types of neuroimaging studies. However, accurately determining which neurons carry out similar neurological tasks via controlled experiments is both labor-intensive and prohibitively expensive on a large scale. Thus, it is of great interest to cluster neurons that have similar connectivity profiles into functionally coherent groups in a data-driven manner. In this work, we propose the clustered Gaussian graphical model (GGM) and a novel symmetric convex clustering penalty in an unified convex optimization framework for inferring functional clusters among neurons from neural activity data. A parallelizable multi-block Alternating Direction Method of Multipliers (ADMM) algorithm is used to solve the corresponding convex optimization problem. In addition, we establish convergence guarantees for the proposed ADMM algorithm. Experimental results on both synthetic data and real-world neuroscientific data demonstrate the effectiveness of our approach.

Yulia Baker, Tiffany M. Tang, Genevera I. Allen

Data integration methods that analyze multiple sources of data simultaneously can often provide more holistic insights than can separate inquiries of each data source. Motivated by the advantages of data integration in the era of "big data", we investigate feature selection for high-dimensional multi-view data with mixed data types (e.g. continuous, binary, count-valued). This heterogeneity of multi-view data poses numerous challenges for existing feature selection methods. However, after critically examining these issues through empirical and theoretically-guided lenses, we develop a practical solution, the Block Randomized Adaptive Iterative Lasso (B-RAIL), which combines the strengths of the randomized Lasso, adaptive weighting schemes, and stability selection. B-RAIL serves as a versatile data integration method for sparse regression and graph selection, and we demonstrate the effectiveness of B-RAIL through extensive simulations and a case study to infer the ovarian cancer gene regulatory network. In this case study, B-RAIL successfully identifies well-known biomarkers associated with ovarian cancer and hints at novel candidates for future ovarian cancer research.

David I. Inouye, Eunho Yang, Genevera I. Allen et al.

The Poisson distribution has been widely studied and used for modeling univariate count-valued data. Multivariate generalizations of the Poisson distribution that permit dependencies, however, have been far less popular. Yet, real-world high-dimensional count-valued data found in word counts, genomics, and crime statistics, for example, exhibit rich dependencies, and motivate the need for multivariate distributions that can appropriately model this data. We review multivariate distributions derived from the univariate Poisson, categorizing these models into three main classes: 1) where the marginal distributions are Poisson, 2) where the joint distribution is a mixture of independent multivariate Poisson distributions, and 3) where the node-conditional distributions are derived from the Poisson. We discuss the development of multiple instances of these classes and compare the models in terms of interpretability and theory. Then, we empirically compare multiple models from each class on three real-world datasets that have varying data characteristics from different domains, namely traffic accident data, biological next generation sequencing data, and text data. These empirical experiments develop intuition about the comparative advantages and disadvantages of each class of multivariate distribution that was derived from the Poisson. Finally, we suggest new research directions as explored in the subsequent discussion section.

Eunho Yang, Pradeep Ravikumar, Genevera I. Allen et al.

"Mixed Data" comprising a large number of heterogeneous variables (e.g. count, binary, continuous, skewed continuous, among other data types) are prevalent in varied areas such as genomics and proteomics, imaging genetics, national security, social networking, and Internet advertising. There have been limited efforts at statistically modeling such mixed data jointly, in part because of the lack of computationally amenable multivariate distributions that can capture direct dependencies between such mixed variables of different types. In this paper, we address this by introducing a novel class of Block Directed Markov Random Fields (BDMRFs). Using the basic building block of node-conditional univariate exponential families from Yang et al. (2012), we introduce a class of mixed conditional random field distributions, that are then chained according to a block-directed acyclic graph to form our class of Block Directed Markov Random Fields (BDMRFs). The Markov independence graph structure underlying a BDMRF thus has both directed and undirected edges. We introduce conditions under which these distributions exist and are normalizable, study several instances of our models, and propose scalable penalized conditional likelihood estimators with statistical guarantees for recovering the underlying network structure. Simulations as well as an application to learning mixed genomic networks from next generation sequencing expression data and mutation data demonstrate the versatility of our methods.

Eric C. Chi, Genevera I. Allen, Richard G. Baraniuk

In the biclustering problem, we seek to simultaneously group observations and features. While biclustering has applications in a wide array of domains, ranging from text mining to collaborative filtering, the problem of identifying structure in high dimensional genomic data motivates this work. In this context, biclustering enables us to identify subsets of genes that are co-expressed only within a subset of experimental conditions. We present a convex formulation of the biclustering problem that possesses a unique global minimizer and an iterative algorithm, COBRA, that is guaranteed to identify it. Our approach generates an entire solution path of possible biclusters as a single tuning parameter is varied. We also show how to reduce the problem of selecting this tuning parameter to solving a trivial modification of the convex biclustering problem. The key contributions of our work are its simplicity, interpretability, and algorithmic guarantees - features that arguably are lacking in the current alternative algorithms. We demonstrate the advantages of our approach, which includes stably and reproducibly identifying biclusterings, on simulated and real microarray data.

Genevera I. Allen, Michael Weylandt

Regularized variants of Principal Components Analysis, especially Sparse PCA and Functional PCA, are among the most useful tools for the analysis of complex high-dimensional data. Many examples of massive data, have both sparse and functional (smooth) aspects and may benefit from a regularization scheme that can capture both forms of structure. For example, in neuro-imaging data, the brain's response to a stimulus may be restricted to a discrete region of activation (spatial sparsity), while exhibiting a smooth response within that region. We propose a unified approach to regularized PCA which can induce both sparsity and smoothness in both the row and column principal components. Our framework generalizes much of the previous literature, with sparse, functional, two-way sparse, and two-way functional PCA all being special cases of our approach. Our method permits flexible combinations of sparsity and smoothness that lead to improvements in feature selection and signal recovery, as well as more interpretable PCA factors. We demonstrate the efficacy of our method on simulated data and a neuroimaging example on EEG data.

Eunho Yang, Pradeep Ravikumar, Genevera I. Allen et al.

Undirected graphical models, or Markov networks, are a popular class of statistical models, used in a wide variety of applications. Popular instances of this class include Gaussian graphical models and Ising models. In many settings, however, it might not be clear which subclass of graphical models to use, particularly for non-Gaussian and non-categorical data. In this paper, we consider a general sub-class of graphical models where the node-wise conditional distributions arise from exponential families. This allows us to derive multivariate graphical model distributions from univariate exponential family distributions, such as the Poisson, negative binomial, and exponential distributions. Our key contributions include a class of M-estimators to fit these graphical model distributions; and rigorous statistical analysis showing that these M-estimators recover the true graphical model structure exactly, with high probability. We provide examples of genomic and proteomic networks learned via instances of our class of graphical models derived from Poisson and exponential distributions.

Genevera I. Allen, Christine Peterson, Marina Vannucci et al.

High-dimensional data common in genomics, proteomics, and chemometrics often contains complicated correlation structures. Recently, partial least squares (PLS) and Sparse PLS methods have gained attention in these areas as dimension reduction techniques in the context of supervised data analysis. We introduce a framework for Regularized PLS by solving a relaxation of the SIMPLS optimization problem with penalties on the PLS loadings vectors. Our approach enjoys many advantages including flexibility, general penalties, easy interpretation of results, and fast computation in high-dimensional settings. We also outline extensions of our methods leading to novel methods for Non-negative PLS and Generalized PLS, an adaption of PLS for structured data. We demonstrate the utility of our methods through simulations and a case study on proton Nuclear Magnetic Resonance (NMR) spectroscopy data.

Genevera I. Allen

High-dimensional tensors or multi-way data are becoming prevalent in areas such as biomedical imaging, chemometrics, networking and bibliometrics. Traditional approaches to finding lower dimensional representations of tensor data include flattening the data and applying matrix factorizations such as principal components analysis (PCA) or employing tensor decompositions such as the CANDECOMP / PARAFAC (CP) and Tucker decompositions. The former can lose important structure in the data, while the latter Higher-Order PCA (HOPCA) methods can be problematic in high-dimensions with many irrelevant features. We introduce frameworks for sparse tensor factorizations or Sparse HOPCA based on heuristic algorithmic approaches and by solving penalized optimization problems related to the CP decomposition. Extensions of these approaches lead to methods for general regularized tensor factorizations, multi-way Functional HOPCA and generalizations of HOPCA for structured data. We illustrate the utility of our methods for dimension reduction, feature selection, and signal recovery on simulated data and multi-dimensional microarrays and functional MRIs.