David I. Inouye

Ruqi Bai, Saurabh Bagchi, David I. Inouye

While prior domain generalization (DG) benchmarks consider train-test dataset heterogeneity, we evaluate Federated DG which introduces federated learning (FL) specific challenges. Additionally, we explore domain-based heterogeneity in clients' local datasets - a realistic Federated DG scenario. Prior Federated DG evaluations are limited in terms of the number or heterogeneity of clients and dataset diversity. To address this gap, we propose an Federated DG benchmark methodology that enables control of the number and heterogeneity of clients and provides metrics for dataset difficulty. We then apply our methodology to evaluate 14 Federated DG methods, which include centralized DG methods adapted to the FL context, FL methods that handle client heterogeneity, and methods designed specifically for Federated DG. Our results suggest that despite some progress, there remain significant performance gaps in Federated DG particularly when evaluating with a large number of clients, high client heterogeneity, or more realistic datasets. Please check our extendable benchmark code here: https://github.com/inouye-lab/FedDG_Benchmark.

Wonwoong Cho, Ziyu Gong, David I. Inouye

Unsupervised distribution alignment estimates a transformation that maps two or more source distributions to a shared aligned distribution given only samples from each distribution. This task has many applications including generative modeling, unsupervised domain adaptation, and socially aware learning. Most prior works use adversarial learning (i.e., min-max optimization), which can be challenging to optimize and evaluate. A few recent works explore non-adversarial flow-based (i.e., invertible) approaches, but they lack a unified perspective and are limited in efficiently aligning multiple distributions. Therefore, we propose to unify and generalize previous flow-based approaches under a single non-adversarial framework, which we prove is equivalent to minimizing an upper bound on the Jensen-Shannon Divergence (JSD). Importantly, our problem reduces to a min-min, i.e., cooperative, problem and can provide a natural evaluation metric for unsupervised distribution alignment. We show empirical results on both simulated and real-world datasets to demonstrate the benefits of our approach. Code is available at https://github.com/inouye-lab/alignment-upper-bound.

Wonwoong Cho, Hareesh Ravi, Midhun Harikumar et al.

As Diffusion Models have shown promising performance, a lot of efforts have been made to improve the controllability of Diffusion Models. However, how to train Diffusion Models to have the disentangled latent spaces and how to naturally incorporate the disentangled conditions during the sampling process have been underexplored. In this paper, we present a training framework for feature disentanglement of Diffusion Models (FDiff). We further propose two sampling methods that can boost the realism of our Diffusion Models and also enhance the controllability. Concisely, we train Diffusion Models conditioned on two latent features, a spatial content mask, and a flattened style embedding. We rely on the inductive bias of the denoising process of Diffusion Models to encode pose/layout information in the content feature and semantic/style information in the style feature. Regarding the sampling methods, we first generalize Composable Diffusion Models (GCDM) by breaking the conditional independence assumption to allow for some dependence between conditional inputs, which is shown to be effective in realistic generation in our experiments. Second, we propose timestep-dependent weight scheduling for content and style features to further improve the performance. We also observe better controllability of our proposed methods compared to existing methods in image manipulation and image translation.

Zeyu Zhou, Tianci Liu, Ruqi Bai et al.

In high-stake domains such as healthcare and hiring, the role of machine learning (ML) in decision-making raises significant fairness concerns. This work focuses on Counterfactual Fairness (CF), which posits that an ML model's outcome on any individual should remain unchanged if they had belonged to a different demographic group. Previous works have proposed methods that guarantee CF. Notwithstanding, their effects on the model's predictive performance remains largely unclear. To fill in this gap, we provide a theoretical study on the inherent trade-off between CF and predictive performance in a model-agnostic manner. We first propose a simple but effective method to cast an optimal but potentially unfair predictor into a fair one without losing the optimality. By analyzing its excess risk in order to achieve CF, we quantify this inherent trade-off. Further analysis on our method's performance with access to only incomplete causal knowledge is also conducted. Built upon it, we propose a performant algorithm that can be applied in such scenarios. Experiments on both synthetic and semi-synthetic datasets demonstrate the validity of our analysis and methods.

Sean Kulinski, David I. Inouye

A distribution shift can have fundamental consequences such as signaling a change in the operating environment or significantly reducing the accuracy of downstream models. Thus, understanding distribution shifts is critical for examining and hopefully mitigating the effect of such a shift. Most prior work focuses on merely detecting if a shift has occurred and assumes any detected shift can be understood and handled appropriately by a human operator. We hope to aid in these manual mitigation tasks by explaining the distribution shift using interpretable transportation maps from the original distribution to the shifted one. We derive our interpretable mappings from a relaxation of optimal transport, where the candidate mappings are restricted to a set of interpretable mappings. We then inspect multiple quintessential use-cases of distribution shift in real-world tabular, text, and image datasets to showcase how our explanatory mappings provide a better balance between detail and interpretability than baseline explanations by both visual inspection and our PercentExplained metric.

Zeyu Zhou, Ruqi Bai, Sean Kulinski et al.

Answering counterfactual queries has important applications such as explainability, robustness, and fairness but is challenging when the causal variables are unobserved and the observations are non-linear mixtures of these latent variables, such as pixels in images. One approach is to recover the latent Structural Causal Model (SCM), which may be infeasible in practice due to requiring strong assumptions, e.g., linearity of the causal mechanisms or perfect atomic interventions. Meanwhile, more practical ML-based approaches using naive domain translation models to generate counterfactual samples lack theoretical grounding and may construct invalid counterfactuals. In this work, we strive to strike a balance between practicality and theoretical guarantees by analyzing a specific type of causal query called domain counterfactuals, which hypothesizes what a sample would have looked like if it had been generated in a different domain (or environment). We show that recovering the latent SCM is unnecessary for estimating domain counterfactuals, thereby sidestepping some of the theoretic challenges. By assuming invertibility and sparsity of intervention, we prove domain counterfactual estimation error can be bounded by a data fit term and intervention sparsity term. Building upon our theoretical results, we develop a theoretically grounded practical algorithm that simplifies the modeling process to generative model estimation under autoregressive and shared parameter constraints that enforce intervention sparsity. Finally, we show an improvement in counterfactual estimation over baseline methods through extensive simulated and image-based experiments.

Ziyu Gong, Ben Usman, Han Zhao et al.

Distribution matching can be used to learn invariant representations with applications in fairness and robustness. Most prior works resort to adversarial matching methods but the resulting minimax problems are unstable and challenging to optimize. Non-adversarial likelihood-based approaches either require model invertibility, impose constraints on the latent prior, or lack a generic framework for distribution matching. To overcome these limitations, we propose a non-adversarial VAE-based matching method that can be applied to any model pipeline. We develop a set of alignment upper bounds for distribution matching (including a noisy bound) that have VAE-like objectives but with a different perspective. We carefully compare our method to prior VAE-based matching approaches both theoretically and empirically. Finally, we demonstrate that our novel matching losses can replace adversarial losses in standard invariant representation learning pipelines without modifying the original architectures -- thereby significantly broadening the applicability of non-adversarial matching methods.

Mai Elkady, Jim Lim, David I. Inouye

While normalizing flows for continuous data have been extensively researched, flows for discrete data have only recently been explored. These prior models, however, suffer from limitations that are distinct from those of continuous flows. Most notably, discrete flow-based models cannot be straightforwardly optimized with conventional deep learning methods because gradients of discrete functions are undefined or zero. Previous works approximate pseudo-gradients of the discrete functions but do not solve the problem on a fundamental level. In addition to that, backpropagation can be computationally burdensome compared to alternative discrete algorithms such as decision tree algorithms. Our approach seeks to reduce computational burden and remove the need for pseudo-gradients by developing a discrete flow based on decision trees -- building upon the success of efficient tree-based methods for classification and regression for discrete data. We first define a tree-structured permutation (TSP) that compactly encodes a permutation of discrete data where the inverse is easy to compute; thus, we can efficiently compute the density value and sample new data. We then propose a decision tree algorithm to build TSPs that learns the tree structure and permutations at each node via novel criteria. We empirically demonstrate the feasibility of our method on multiple datasets.

Ziyu Gong, Jim Lim, David I. Inouye

Distribution matching (DM) is a versatile domain-invariant representation learning technique that has been applied to tasks such as fair classification, domain adaptation, and domain translation. Non-parametric DM methods struggle with scalability and adversarial DM approaches suffer from instability and mode collapse. While likelihood-based methods are a promising alternative, they often impose unnecessary biases through fixed priors or require explicit density models (e.g., flows) that can be challenging to train. We address this limitation by introducing a novel approach to training likelihood-based DM using expressive score-based prior distributions. Our key insight is that gradient-based DM training only requires the prior's score function -- not its density -- allowing us to train the prior via denoising score matching. This approach eliminates biases from fixed priors (e.g., in VAEs), enabling more effective use of geometry-preserving regularization, while avoiding the challenge of learning an explicit prior density model (e.g., a flow-based prior). Our method also demonstrates better stability and computational efficiency compared to other diffusion-based priors (e.g., LSGM). Furthermore, experiments demonstrate superior performance across multiple tasks, establishing our score-based method as a stable and effective approach to distribution matching. Source code available at https://github.com/inouye-lab/SAUB.

Rui Wang, Xiaoqian Wang, David I. Inouye

Shapley values have become one of the most popular feature attribution explanation methods. However, most prior work has focused on post-hoc Shapley explanations, which can be computationally demanding due to its exponential time complexity and preclude model regularization based on Shapley explanations during training. Thus, we propose to incorporate Shapley values themselves as latent representations in deep models thereby making Shapley explanations first-class citizens in the modeling paradigm. This intrinsic explanation approach enables layer-wise explanations, explanation regularization of the model during training, and fast explanation computation at test time. We define the Shapley transform that transforms the input into a Shapley representation given a specific function. We operationalize the Shapley transform as a neural network module and construct both shallow and deep networks, called ShapNets, by composing Shapley modules. We prove that our Shallow ShapNets compute the exact Shapley values and our Deep ShapNets maintain the missingness and accuracy properties of Shapley values. We demonstrate on synthetic and real-world datasets that our ShapNets enable layer-wise Shapley explanations, novel Shapley regularizations during training, and fast computation while maintaining reasonable performance. Code is available at https://github.com/inouye-lab/ShapleyExplanationNetworks.

Sean Kulinski, Nicholas R. Waytowich, James Z. Hare et al.

Spatial reasoning tasks in multi-agent environments such as event prediction, agent type identification, or missing data imputation are important for multiple applications (e.g., autonomous surveillance over sensor networks and subtasks for reinforcement learning (RL)). StarCraft II game replays encode intelligent (and adversarial) multi-agent behavior and could provide a testbed for these tasks; however, extracting simple and standardized representations for prototyping these tasks is laborious and hinders reproducibility. In contrast, MNIST and CIFAR10, despite their extreme simplicity, have enabled rapid prototyping and reproducibility of ML methods. Following the simplicity of these datasets, we construct a benchmark spatial reasoning dataset based on StarCraft II replays that exhibit complex multi-agent behaviors, while still being as easy to use as MNIST and CIFAR10. Specifically, we carefully summarize a window of 255 consecutive game states to create 3.6 million summary images from 60,000 replays, including all relevant metadata such as game outcome and player races. We develop three formats of decreasing complexity: Hyperspectral images that include one channel for every unit type (similar to multispectral geospatial images), RGB images that mimic CIFAR10, and grayscale images that mimic MNIST. We show how this dataset can be used for prototyping spatial reasoning methods. All datasets, code for extraction, and code for dataset loading can be found at https://starcraftdata.davidinouye.com

Surojit Ganguli, Zeyu Zhou, Christopher G. Brinton et al.

When each edge device of a network only perceives a local part of the environment, collaborative inference across multiple devices is often needed to predict global properties of the environment. In safety-critical applications, collaborative inference must be robust to significant network failures caused by environmental disruptions or extreme weather. Existing collaborative learning approaches, such as privacy-focused Vertical Federated Learning (VFL), typically assume a centralized setup or that one device never fails. However, these assumptions make prior approaches susceptible to significant network failures. To address this problem, we first formalize the problem of robust collaborative inference over a dynamic network of devices that could experience significant network faults. Then, we develop a minimalistic yet impactful method called Multiple Aggregation with Gossip Rounds and Simulated Faults (MAGS) that synthesizes simulated faults via dropout, replication, and gossiping to significantly improve robustness over baselines. We also theoretically analyze our proposed approach to explain why each component enhances robustness. Extensive empirical results validate that MAGS is robust across a range of fault rates-including extreme fault rates.

Dongze Wu, David I. Inouye, Yao Xie

Predicting potential and counterfactual outcomes from observational data is central to clinical decision-making, where physicians must weigh treatments for an individual patient rather than relying solely on average effects at the population level. We propose PO-Flow, a continuous normalizing flow (CNF) framework for causal inference that jointly models potential outcomes and counterfactuals. Trained via flow matching, PO-Flow provides a unified approach to average treatment effect estimation, individualized potential outcome prediction, and counterfactual prediction. Besides, PO-Flow directly learns the densities of potential outcomes, enabling likelihood-based evaluation of predictions. Furthermore, PO-Flow explores counterfactual outcome generation conditioned on the observed factual in general observational datasets, with a supporting recovery result under certain assumptions. PO-Flow outperforms modern baselines across diverse datasets and causal tasks in the potential outcomes framework.

Avi Amalanshu, Yash Sirvi, David I. Inouye

Vertical Federated Learning (VFL) is an emergent distributed machine learning paradigm for collaborative learning between clients who have disjoint features of common entities. However, standard VFL lacks fault tolerance, with each participant and connection being a single point of failure. Prior attempts to induce fault tolerance in VFL focus on the scenario of "straggling clients", usually entailing that all messages eventually arrive or that there is an upper bound on the number of late messages. To handle the more general problem of arbitrary crashes, we propose Decoupled VFL (DVFL). To handle training with faults, DVFL decouples training between communication rounds using local unsupervised objectives. By further decoupling label supervision from aggregation, DVFL also enables redundant aggregators. As secondary benefits, DVFL can enhance data efficiency and provides immunity against gradient-based attacks. In this work, we implement DVFL for split neural networks with a self-supervised autoencoder loss. When there are faults, DVFL outperforms the best VFL-based alternative (97.58% vs 96.95% on an MNIST task). Even under perfect conditions, performance is comparable.

Yi-Chung Chen, David I. Inouye, Jing Gao

Generative classifiers, which leverage conditional generative models for classification, have recently demonstrated desirable properties such as robustness to distribution shifts. However, recent progress in this area has been largely driven by diffusion-based models, whose substantial computational cost severely limits scalability. This exclusive focus on diffusion-based methods has also constrained our understanding of generative classifiers. In this work, we propose a novel generative classifier built on recent advances in visual autoregressive (VAR) modeling, which offers a new perspective for studying generative classifiers. To further enhance its performance, we introduce the Adaptive VAR Classifier$^+$ (A-VARC$^+$), which achieves a superior trade-off between accuracy and inference speed, thereby significantly improving practical applicability. Moreover, we show that the VAR-based method exhibits fundamentally different properties from diffusion-based methods. In particular, due to its tractable likelihood, the VAR-based classifier enables visual explainability via token-wise mutual information and demonstrates inherent resistance to catastrophic forgetting in class-incremental learning tasks.

Wonwoong Cho, Yanxia Zhang, Yan-Ying Chen et al.

Blending visual and textual concepts into a new visual concept is a unique and powerful trait of human beings that can fuel creativity. However, in practice, cross-modal conceptual blending for humans is prone to cognitive biases, like design fixation, which leads to local minima in the design space. In this paper, we propose a T2I diffusion adapter "IT-Blender" that can automate the blending process to enhance human creativity. Prior works related to cross-modal conceptual blending are limited in encoding a real image without loss of details or in disentangling the image and text inputs. To address these gaps, IT-Blender leverages pretrained diffusion models (SD and FLUX) to blend the latent representations of a clean reference image with those of the noisy generated image. Combined with our novel blended attention, IT-Blender encodes the real reference image without loss of details and blends the visual concept with the object specified by the text in a disentangled way. Our experiment results show that IT-Blender outperforms the baselines by a large margin in blending visual and textual concepts, shedding light on the new application of image generative models to augment human creativity.

Ruqi Bai, Yao Ji, Zeyu Zhou et al.

Models that learn spurious correlations from training data often fail when deployed in new environments. While many methods aim to learn invariant representations to address this, they often underperform standard empirical risk minimization (ERM). We propose a data-centric alternative that shifts the focus from learning invariant representations to leveraging invariant data pairs -- pairs of samples that should have the same prediction. We prove that certain counterfactuals naturally satisfy this invariance property. Based on this, we introduce Noisy Counterfactual Matching (NCM), a simple constraint-based method that improves robustness by leveraging even a small number of \emph{noisy} counterfactual pairs -- improving upon prior works that do not explicitly consider noise. For linear causal models, we prove that NCM's test-domain error is bounded by its in-domain error plus a term dependent on the counterfactuals' quality and diversity. Experiments on synthetic data validate our theory, and we demonstrate NCM's effectiveness on real-world datasets.

Wonwoong Cho, Yan-Ying Chen, Matthew Klenk et al.

Text-to-Image (T2I) Diffusion Models have achieved remarkable performance in generating high quality images. However, enabling precise control of continuous attributes, especially multiple attributes simultaneously, in a new domain (e.g., numeric values like eye openness or car width) with text-only guidance remains a significant challenge. To address this, we introduce the Attribute (Att) Adapter, a novel plug-and-play module designed to enable fine-grained, multi-attributes control in pretrained diffusion models. Our approach learns a single control adapter from a set of sample images that can be unpaired and contain multiple visual attributes. The Att-Adapter leverages the decoupled cross attention module to naturally harmonize the multiple domain attributes with text conditioning. We further introduce Conditional Variational Autoencoder (CVAE) to the Att-Adapter to mitigate overfitting, matching the diverse nature of the visual world. Evaluations on two public datasets show that Att-Adapter outperforms all LoRA-based baselines in controlling continuous attributes. Additionally, our method enables a broader control range and also improves disentanglement across multiple attributes, surpassing StyleGAN-based techniques. Notably, Att-Adapter is flexible, requiring no paired synthetic data for training, and is easily scalable to multiple attributes within a single model.

Mai Elkady, Thu Bui, Bruno Ribeiro et al.

There has been a growing excitement that implicit graph generative models could be used to design or discover new molecules for medicine or material design. Because these molecules have not been discovered, they naturally lie in unexplored or scarcely supported regions of the distribution of known molecules. However, prior evaluation methods for implicit graph generative models have focused on validating statistics computed from the thick support (e.g., mean and variance of a graph property). Therefore, there is a mismatch between the goal of generating novel graphs and the evaluation methods. To address this evaluation gap, we design a novel evaluation method called Vertical Validation (VV) that systematically creates thin support regions during the train-test splitting procedure and then reweights generated samples so that they can be compared to the held-out test data. This procedure can be seen as a generalization of the standard train-test procedure except that the splits are dependent on sample features. We demonstrate that our method can be used to perform model selection if performance on thin support regions is the desired goal. As a side benefit, we also show that our approach can better detect overfitting as exemplified by memorization.

Sean Kulinski, Saurabh Bagchi, David I. Inouye

While previous distribution shift detection approaches can identify if a shift has occurred, these approaches cannot localize which specific features have caused a distribution shift -- a critical step in diagnosing or fixing any underlying issue. For example, in military sensor networks, users will want to detect when one or more of the sensors has been compromised, and critically, they will want to know which specific sensors might be compromised. Thus, we first define a formalization of this problem as multiple conditional distribution hypothesis tests and propose both non-parametric and parametric statistical tests. For both efficiency and flexibility, we then propose to use a test statistic based on the density model score function (i.e. gradient with respect to the input) -- which can easily compute test statistics for all dimensions in a single forward and backward pass. Any density model could be used for computing the necessary statistics including deep density models such as normalizing flows or autoregressive models. We additionally develop methods for identifying when and where a shift occurs in multivariate time-series data and show results for multiple scenarios using realistic attack models on both simulated and real world data.

Zeyu Zhou, Ziyu Gong, Pradeep Ravikumar et al.

The unsupervised task of aligning two or more distributions in a shared latent space has many applications including fair representations, batch effect mitigation, and unsupervised domain adaptation. Existing flow-based approaches estimate multiple flows independently, which is equivalent to learning multiple full generative models. Other approaches require adversarial learning, which can be computationally expensive and challenging to optimize. Thus, we aim to jointly align multiple distributions while avoiding adversarial learning. Inspired by efficient alignment algorithms from optimal transport (OT) theory for univariate distributions, we develop a simple iterative method to build deep and expressive flows. Our method decouples each iteration into two subproblems: 1) form a variational approximation of a distribution divergence and 2) minimize this variational approximation via closed-form invertible alignment maps based on known OT results. Our empirical results give evidence that this iterative algorithm achieves competitive distribution alignment at low computational cost while being able to naturally handle more than two distributions.

Ruqi Bai, Saurabh Bagchi, David I. Inouye

Adversarial examples (AEs) are images that can mislead deep neural network (DNN) classifiers via introducing slight perturbations into original images. This security vulnerability has led to vast research in recent years because it can introduce real-world threats into systems that rely on neural networks. Yet, a deep understanding of the characteristics of adversarial examples has remained elusive. We propose a new way of achieving such understanding through a recent development, namely, invertible neural models with Lipschitz continuous mapping functions from the input to the output. With the ability to invert any latent representation back to its corresponding input image, we can investigate adversarial examples at a deeper level and disentangle the adversarial example's latent representation. Given this new perspective, we propose a fast latent space adversarial example generation method that could accelerate adversarial training. Moreover, this new perspective could contribute to new ways of adversarial example detection.

David I. Inouye, Liu Leqi, Joon Sik Kim et al.

In practical applications of machine learning, it is necessary to look beyond standard metrics such as test accuracy in order to validate various qualitative properties of a model. Partial dependence plots (PDP), including instance-specific PDPs (i.e., ICE plots), have been widely used as a visual tool to understand or validate a model. Yet, current PDPs suffer from two main drawbacks: (1) a user must manually sort or select interesting plots, and (2) PDPs are usually limited to plots along a single feature. To address these drawbacks, we formalize a method for automating the selection of interesting PDPs and extend PDPs beyond showing single features to show the model response along arbitrary directions, for example in raw feature space or a latent space arising from some generative model. We demonstrate the usefulness of our automated dependence plots (ADP) across multiple use-cases and datasets including model selection, bias detection, understanding out-of-sample behavior, and exploring the latent space of a generative model.

Chih-Kuan Yeh, Cheng-Yu Hsieh, Arun Sai Suggala et al.

We consider objective evaluation measures of saliency explanations for complex black-box machine learning models. We propose simple robust variants of two notions that have been considered in recent literature: (in)fidelity, and sensitivity. We analyze optimal explanations with respect to both these measures, and while the optimal explanation for sensitivity is a vacuous constant explanation, the optimal explanation for infidelity is a novel combination of two popular explanation methods. By varying the perturbation distribution that defines infidelity, we obtain novel explanations by optimizing infidelity, which we show to out-perform existing explanations in both quantitative and qualitative measurements. Another salient question given these measures is how to modify any given explanation to have better values with respect to these measures. We propose a simple modification based on lowering sensitivity, and moreover show that when done appropriately, we could simultaneously improve both sensitivity as well as fidelity.

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.

David I. Inouye, Pradeep Ravikumar, Inderjit S. Dhillon

We present a novel k-way high-dimensional graphical model called the Generalized Root Model (GRM) that explicitly models dependencies between variable sets of size k > 2---where k = 2 is the standard pairwise graphical model. This model is based on taking the k-th root of the original sufficient statistics of any univariate exponential family with positive sufficient statistics, including the Poisson and exponential distributions. As in the recent work with square root graphical (SQR) models [Inouye et al. 2016]---which was restricted to pairwise dependencies---we give the conditions of the parameters that are needed for normalization using the radial conditionals similar to the pairwise case [Inouye et al. 2016]. In particular, we show that the Poisson GRM has no restrictions on the parameters and the exponential GRM only has a restriction akin to negative definiteness. We develop a simple but general learning algorithm based on L1-regularized node-wise regressions. We also present a general way of numerically approximating the log partition function and associated derivatives of the GRM univariate node conditionals---in contrast to [Inouye et al. 2016], which only provided algorithm for estimating the exponential SQR. To illustrate GRM, we model word counts with a Poisson GRM and show the associated k-sized variable sets. We finish by discussing methods for reducing the parameter space in various situations.

David I. Inouye, Pradeep Ravikumar, Inderjit S. Dhillon

We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models [Yang et al. 2015] did not allow positive dependencies for the exponential and Poisson generalizations. However, in many real-world datasets, variables clearly have positive dependencies. For example, the airport delay time in New York---modeled as an exponential distribution---is positively related to the delay time in Boston. With this motivation, we give an example of our model class derived from the univariate exponential distribution that allows for almost arbitrary positive and negative dependencies with only a mild condition on the parameter matrix---a condition akin to the positive definiteness of the Gaussian covariance matrix. Our Poisson generalization allows for both positive and negative dependencies without any constraints on the parameter values. We also develop parameter estimation methods using node-wise regressions with $\ell_1$ regularization and likelihood approximation methods using sampling. Finally, we demonstrate our exponential generalization on a synthetic dataset and a real-world dataset of airport delay times.