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Karthik Natarajan

Publications and source records attributed to Karthik Natarajan.

At least 19 recordsLinked to original sources
arXiv2026

On the Semidefinite Representability of Continuous Quadratic Submodular Minimization With Applications to Pricing and Moment Problems

We study continuous quadratic submodular minimization with bounds and provide a polynomially sized semidefinite relaxation, which is provably tight for dimension n \leq 3. Via an explicit counterexample for n = 4, we show that the relaxation is not tight in general, although it remains empirically tight on randomly generated instances. We apply the relaxation to multi-product pricing and two moment problems arising in distributionally robust optimization and the computation of covariance bounds. Accordingly, this research advances the ongoing study of continuous submodular minimization and opens new application areas therein.

math.OC↗
arXiv2026

FoMoH: A clinically meaningful foundation model evaluation for structured electronic health records

Foundation models (FMs) promise to address core limitations of traditional supervised machine learning: (i) reliance on large amounts of labeled data, (ii) task specificity, and (iii) poor transportability. Despite methodological advances in structured electronic health record (EHR) foundation models, no systematic benchmark has validated whether these models meaningfully deliver on these promises. We introduce a benchmark of 14 clinically meaningful prediction tasks spanning patient prognosis and early diagnosis of acute and chronic conditions. We benchmark 6 state-of-the-art EHR FMs beyond population-level discrimination, emphasizing the need for evaluating their calibration and fairness, with rigorous controls for data contamination and reproducibility across more than 6 million patients from Columbia University Irving Medical Center and MIMIC-IV. Our benchmark identifies that FMs deliver on some of their promises. In particular, top-performing FMs outperform traditional baselines on discriminative performance, especially under limited labeled data, and exhibit more equitable performance across socio-medical groups. However, these models may underperform in low-prevalence settings, as pretraining losses may discard discriminative information about such conditions, and present lower calibration under limited labeled data. Further, cross-institutional transportability remains a challenge for structured EHR FMs. Together, these findings advance our understanding of EHR FMs' potential for clinical utility, highlight critical gaps that remain to be addressed, and provide a reproducible framework to track progress.

cs.LG↗
arXiv2026

Going from a Representative Agent to Counterfactuals in Combinatorial Choice

We study decision-making problems where data comprises points from a collection of binary polytopes, capturing aggregate information stemming from various combinatorial selection environments. We propose a nonparametric approach for counterfactual inference in this setting based on a representative agent model, where the available data is viewed as arising from maximizing separable concave utility functions over the respective binary polytopes. Our first contribution is to precisely characterize the selection probabilities representable under this model and show that verifying the consistency of any given aggregated selection dataset reduces to solving a polynomial-sized linear program. Building on this characterization, we develop a nonparametric method for counterfactual prediction. When data is inconsistent with the model, finding a best-fitting approximation for prediction reduces to solving a compact mixed-integer convex program. Numerical experiments based on synthetic data demonstrate the method's flexibility, predictive accuracy, and strong representational power even under model misspecification.

math.OC↗
arXiv2025

Beyond the Clinic: A Large-Scale Evaluation of Augmenting EHR with Wearable Data for Diverse Health Prediction

Electronic health records (EHRs) provide a powerful basis for predicting the onset of health outcomes. Yet EHRs primarily capture in-clinic events and miss aspects of daily behavior and lifestyle containing rich health information. Consumer wearables, by contrast, continuously measure activity, heart rate, and sleep, and more, offering complementary signals that can fill this gap. Despite this potential, there has been little systematic evaluation of the benefit that wearable data can bring to health outcome prediction on top of EHRs. In this study, we present an extensible framework for multimodal health outcome prediction that integrates EHR and wearable data streams. Using data from the All of Us Program, we systematically compared the combination of different encoding methods on EHR and wearable data, including the traditional feature engineering approach, as well as foundation model embeddings. Across ten clinical outcomes, wearable integration consistently improved model performance relative to EHR-only baselines, e.g., average delta AUROC +6.8% for major depressive disorder, +9.7% for hypertension, and +12.6% for diabetes. On average across all ten outcomes, fusing EHRs with wearable features shows 8.5% improvement in AUROC. To our knowledge, this is the first large-scale evaluation of wearable-EHR fusion, underscoring the utility of wearable-derived signals in complementing EHRs and enabling more holistic, personalized health outcome predictions. Meanwhile, our analysis elucidates future directions for optimizing foundation models for wearable data and its integration with EHR data.

q-bio.QM↗
arXiv2025

CEHR-XGPT: A Scalable Multi-Task Foundation Model for Electronic Health Records

Electronic Health Records (EHRs) provide a rich, longitudinal view of patient health and hold significant potential for advancing clinical decision support, risk prediction, and data-driven healthcare research. However, most artificial intelligence (AI) models for EHRs are designed for narrow, single-purpose tasks, limiting their generalizability and utility in real-world settings. Here, we present CEHR-XGPT, a general-purpose foundation model for EHR data that unifies three essential capabilities - feature representation, zero-shot prediction, and synthetic data generation - within a single architecture. To support temporal reasoning over clinical sequences, CEHR-XGPT incorporates a novel time-token-based learning framework that explicitly encodes patients' dynamic timelines into the model structure. CEHR-XGPT demonstrates strong performance across all three tasks and generalizes effectively to external datasets through vocabulary expansion and fine-tuning. Its versatility enables rapid model development, cohort discovery, and patient outcome forecasting without the need for task-specific retraining.

cs.LG↗
arXiv2025

Distributionally robust optimization through the lens of submodularity

Distributionally robust optimization is used to tackle decision making problems under uncertainty where the distribution of the uncertain data is ambiguous. Many ambiguity sets have been proposed for continuous uncertainty that build on convexity and for which the resulting formulations scale polynomially in the number of random variables. However fewer ambiguity sets have been proposed for discrete uncertainty where the exact formulations scale polynomially in the number of random variables. Towards this, we define a submodular ambiguity set and showcase its expressive power in modeling both discrete and continuous uncertainty. With discrete uncertainty, we show that a class of distributionally robust optimization problems is solvable in polynomial time by viewing it through the lens of submodularity. With continuous uncertainty, we show that it is solvable approximately up to an additive error in pseudo-polynomial time. We then focus on a specific class of submodular ambiguity sets where univariate marginal information and bivariate dependence information on the random vector is specified and provide an exact reformulation as a polynomial sized linear program when the uncertainty is discrete and as a polynomial sized semidefinite program when the uncertainty is continuous. We provide numerical evidence of the modeling flexibility and expressive power of the submodular ambiguity set and demonstrate its applicability in two examples: project networks and multi-newsvendor problems. The paper highlights that the submodular ambiguity set is the natural discrete counterpart of the convex ambiguity set and supplements it for continuous uncertainty, both in modeling and computation.

math.OC↗
arXiv2025

A Nonparametric Approach with Marginals for Modeling Consumer Choice

Given data on the choices made by consumers for different offer sets, a key challenge is to develop parsimonious models that describe and predict consumer choice behavior while being amenable to prescriptive tasks such as pricing and assortment optimization. The marginal distribution model (MDM) is one such model, which requires only the specification of marginal distributions of the random utilities. This paper aims to establish necessary and sufficient conditions for given choice data to be consistent with the MDM hypothesis, inspired by the usefulness of similar characterizations for the random utility model (RUM). This endeavor leads to an exact characterization of the set of choice probabilities that the MDM can represent. Verifying the consistency of choice data with this characterization is equivalent to solving a polynomial-sized linear program. Since the analogous verification task for RUM is computationally intractable and neither of these models subsumes the other, MDM is helpful in striking a balance between tractability and representational power. The characterization is then used with robust optimization for making data-driven sales and revenue predictions for new unseen assortments. When the choice data lacks consistency with the MDM hypothesis, finding the best-fitting MDM choice probabilities reduces to solving a mixed integer convex program. Numerical results using real world data and synthetic data demonstrate that MDM exhibits competitive representational power and prediction performance compared to RUM and parametric models while being significantly faster in computation than RUM.

stat.ML↗
arXiv2025

Pairwise independent correlation gap

In this paper, we introduce the notion of a ``pairwise independent correlation gap'' for set functions with random elements. The pairwise independent correlation gap is defined as the ratio of the maximum expected value of a set function with arbitrary dependence among the elements with fixed marginal probabilities to the maximum expected value with pairwise independent elements with the same marginal probabilities. We show that for any nonnegative monotone submodular set function defined on $n$ elements, this ratio is upper bounded by $4/3$ in the following two cases: (a) $n = 3$ for all marginal probabilities and (b) all $n$ for small marginal probabilities (and similarly large marginal probabilities). This differs from the bound on the ``correlation gap'' which holds with mutual independence and showcases the fundamental difference between pairwise independence and mutual independence. We discuss the implication of the results with two examples and end the paper with a conjecture.

math.OC↗
arXiv2024

Environment Scan of Generative AI Infrastructure for Clinical and Translational Science

This study reports a comprehensive environmental scan of the generative AI (GenAI) infrastructure in the national network for clinical and translational science across 36 institutions supported by the Clinical and Translational Science Award (CTSA) Program led by the National Center for Advancing Translational Sciences (NCATS) of the National Institutes of Health (NIH) at the United States. With the rapid advancement of GenAI technologies, including large language models (LLMs), healthcare institutions face unprecedented opportunities and challenges. This research explores the current status of GenAI integration, focusing on stakeholder roles, governance structures, and ethical considerations by administering a survey among leaders of health institutions (i.e., representing academic medical centers and health systems) to assess the institutional readiness and approach towards GenAI adoption. Key findings indicate a diverse range of institutional strategies, with most organizations in the experimental phase of GenAI deployment. The study highlights significant variations in governance models, with a strong preference for centralized decision-making but notable gaps in workforce training and ethical oversight. Moreover, the results underscore the need for a more coordinated approach to GenAI governance, emphasizing collaboration among senior leaders, clinicians, information technology staff, and researchers. Our analysis also reveals concerns regarding GenAI bias, data security, and stakeholder trust, which must be addressed to ensure the ethical and effective implementation of GenAI technologies. This study offers valuable insights into the challenges and opportunities of GenAI integration in healthcare, providing a roadmap for institutions aiming to leverage GenAI for improved quality of care and operational efficiency.

cs.CY↗
arXiv2024

CEHR-GPT: Generating Electronic Health Records with Chronological Patient Timelines

Synthetic Electronic Health Records (EHR) have emerged as a pivotal tool in advancing healthcare applications and machine learning models, particularly for researchers without direct access to healthcare data. Although existing methods, like rule-based approaches and generative adversarial networks (GANs), generate synthetic data that resembles real-world EHR data, these methods often use a tabular format, disregarding temporal dependencies in patient histories and limiting data replication. Recently, there has been a growing interest in leveraging Generative Pre-trained Transformers (GPT) for EHR data. This enables applications like disease progression analysis, population estimation, counterfactual reasoning, and synthetic data generation. In this work, we focus on synthetic data generation and demonstrate the capability of training a GPT model using a particular patient representation derived from CEHR-BERT, enabling us to generate patient sequences that can be seamlessly converted to the Observational Medical Outcomes Partnership (OMOP) data format.

cs.LG↗
arXiv2022

Tight Probability Bounds with Pairwise Independence

While useful probability bounds for $n$ pairwise independent Bernoulli random variables adding up to at least an integer $k$ have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide several results in this direction. Firstly, when $k = 1$, the tightest upper bound on the probability of the union of $n$ pairwise independent events is provided in closed-form for any input marginal probability vector $\mathbf{p} \in [0,1]^n$. To prove the result, we show the existence of a positively correlated Bernoulli random vector with transformed bivariate probabilities, which is of independent interest. Building on this, we show that the ratio of the Boole union bound and the tight pairwise independent bound is upper bounded by $4/3$ and that the ratio is attained. Applications of the result in correlation gap analysis and distributionally robust bottleneck optimization are discussed. The result is extended to find the tightest lower bound on the probability of the intersection of $n$ pairwise independent events. Secondly, for any $k \geq 2$ and input marginal probability vector $\mathbf{p} \in [0,1]^n$, new upper bounds are derived by exploiting ordering of probabilities. Numerical examples are provided to illustrate when the bounds provide improvement over existing bounds. Lastly, we identify specific instances when the existing and the new bounds are tight, for example, with identical marginal probabilities.

math.OC↗
arXiv2022

Probability bounds for $n$ random events under $(n-1)$-wise independence

A collection of $n$ random events is said to be $(n - 1)$-wise independent if any $n - 1$ events among them are mutually independent. We characterise all probability measures with respect to which $n$ random events are $(n - 1)$-wise independent. We provide sharp upper and lower bounds on the probability that at least $k$ out of $n$ events with given marginal probabilities occur over these probability measures. The bounds are shown to be computable in polynomial time.

math.PR↗
arXiv2022

Discrete Optimal Transport with Independent Marginals is #P-Hard

We study the computational complexity of the optimal transport problem that evaluates the Wasserstein distance between the distributions of two K-dimensional discrete random vectors. The best known algorithms for this problem run in polynomial time in the maximum of the number of atoms of the two distributions. However, if the components of either random vector are independent, then this number can be exponential in K even though the size of the problem description scales linearly with K. We prove that the described optimal transport problem is #P-hard even if all components of the first random vector are independent uniform Bernoulli random variables, while the second random vector has merely two atoms, and even if only approximate solutions are sought. We also develop a dynamic programming-type algorithm that approximates the Wasserstein distance in pseudo-polynomial time when the components of the first random vector follow arbitrary independent discrete distributions, and we identify special problem instances that can be solved exactly in strongly polynomial time.

math.OC↗
arXiv2022

Correlation Robust Influence Maximization

We propose a distributionally robust model for the influence maximization problem. Unlike the classic independent cascade model \citep{kempe2003maximizing}, this model's diffusion process is adversarially adapted to the choice of seed set. Hence, instead of optimizing under the assumption that all influence relationships in the network are independent, we seek a seed set whose expected influence under the worst correlation, i.e. the "worst-case, expected influence", is maximized. We show that this worst-case influence can be efficiently computed, and though the optimization is NP-hard, a ($1 - 1/e$) approximation guarantee holds. We also analyze the structure to the adversary's choice of diffusion process, and contrast with established models. Beyond the key computational advantages, we also highlight the extent to which the independence assumption may cost optimality, and provide insights from numerical experiments comparing the adversarial and independent cascade model.

cs.SI↗
arXiv2021

CEHR-BERT: Incorporating temporal information from structured EHR data to improve prediction tasks

Embedding algorithms are increasingly used to represent clinical concepts in healthcare for improving machine learning tasks such as clinical phenotyping and disease prediction. Recent studies have adapted state-of-the-art bidirectional encoder representations from transformers (BERT) architecture to structured electronic health records (EHR) data for the generation of contextualized concept embeddings, yet do not fully incorporate temporal data across multiple clinical domains. Therefore we developed a new BERT adaptation, CEHR-BERT, to incorporate temporal information using a hybrid approach by augmenting the input to BERT using artificial time tokens, incorporating time, age, and concept embeddings, and introducing a new second learning objective for visit type. CEHR-BERT was trained on a subset of Columbia University Irving Medical Center-York Presbyterian Hospital's clinical data, which includes 2.4M patients, spanning over three decades, and tested using 4-fold cross-validation on the following prediction tasks: hospitalization, death, new heart failure (HF) diagnosis, and HF readmission. Our experiments show that CEHR-BERT outperformed existing state-of-the-art clinical BERT adaptations and baseline models across all 4 prediction tasks in both ROC-AUC and PR-AUC. CEHR-BERT also demonstrated strong transfer learning capability, as our model trained on only 5% of data outperformed comparison models trained on the entire data set. Ablation studies to better understand the contribution of each time component showed incremental gains with every element, suggesting that CEHR-BERT's incorporation of artificial time tokens, time and age embeddings with concept embeddings, and the addition of the second learning objective represents a promising approach for future BERT-based clinical embeddings.

cs.LG↗
arXiv2021

Extremal Probability Bounds in Combinatorial Optimization

In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the marginal distributions of the objective coefficient vector. The bounds are ``extremal'' since they are valid across all joint distributions with the given marginals. We analyze the complexity of computing the bounds assuming discrete marginals and identify instances when the bounds are computable in polynomial time. For compact 0/1 V-polytopes, we show that the tightest upper bound is weakly NP-hard to compute by providing a pseudopolynomial time algorithm. On the other hand, the tightest lower bound is shown to be strongly NP-hard to compute for compact 0/1 V-polytopes by restricting attention to Bernoulli random variables. For compact 0/1 H-polytopes, for the special case of PERT networks arising in project management, we show that the tightest upper bound is weakly NP-hard to compute by providing a pseudopolynomial time algorithm. The results in the paper complement existing results in the literature for computing the probability with independent random variables.

math.OC↗
arXiv2019

Tree Bounds for Sums of Bernoulli Random Variables: A Linear Optimization Approach

We study the problem of computing the tightest upper and lower bounds on the probability that the sum of $n$ dependent Bernoulli random variables exceeds an integer $k$. Under knowledge of all pairs of bivariate distributions denoted by a complete graph, the bounds are NP-hard to compute. When the bivariate distributions are specified on a tree graph, we show that tight bounds are computable in polynomial time using linear optimization. These bounds provide robust probability estimates when the assumption of conditional independence in a tree structured graphical model is violated. Generalization of the result to finding probability bounds of order statistic for more general random variables and instances where the bounds provide the most significant improvements over univariate bounds is also discussed in the paper.

math.OC↗