Schedule for Spring 2025

Speaker: Dr. Zhiliang Ying (Columbia University)

Title: Survival analysis: mathematical foundation, historical and recent developments, and applications

Speaker: This talk provides an overview of statistical models and methods associated with the analysis of survival and event history data.  It intends to be introductory with coverage on some of the basics such as the Kaplan-Meier curve, the log-rank test, and the semi-parametric regression models such as the Cox model, the accelerated failure time (AFT) model and the linear transformation models.  Applications to health sciences, with particular emphasis on epidemiological follow-up studies and clinical trials, social sciences, and to business, such as marketing and finance, are also discussed.

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Speaker: Hane Lee

Title:  Measuring polarization in the public and Congress

Abstract: Political polarization is a pressing issue in the US and worldwide, where both governments and the public are sharply divided by different views into opposing camps whose members distrust and dislike those outside their cluster. This talk discusses two methods for measuring polarization in the public and the US Congress. First, I will introduce a Wasserstein index of political bipolarization in public opinion and apply it to analyze attitudes toward governmental COVID-19 vaccine mandates across 11 countries. Second, I will propose a fused latent and graphical approach for measuring social ties from congressional roll call votes. An application to Senate roll calls between 1990 and 2022 reveals changes in social connections that corroborate accounts of congressional polarization.

NO SEMINAR

Speaker: Professor Yuqi Gu (Columbia University)

Title:  Uncover Interpretable Latent Structures in High-dimensional Data and Deep Generative Models

Abstract:  In the era of data science and generative AI, latent structures are ubiquitously employed in various scientific disciplines and machine learning architectures. 

The first part of the talk focuses on the mixed membership model for multivariate categorical data widely used for analyzing survey responses and population genetics data. This so-called grade of membership (GoM) model offers rich modeling power but presents significant estimation challenges for high-dimensional polytomous data. Such data take the form of a three-way (quasi-)tensor, with many subjects responding to many items with varying numbers of categories. We introduce a novel approach to flatten the three-way quasi-tensor into a “fat” matrix and then perform SVD to estimate parameters by exploiting the singular subspace geometry. Our fast spectral method can accommodate a broad range of data distributions with arbitrarily locally dependent noise. We establish finite-sample entrywise error bounds for the model parameters. We also develop a new sharp two-to-infinity singular subspace perturbation theory for arbitrary locally dependent and flexibly distributed noise, a contribution of independent interest. Simulations and applications to data in political voting, population genetics, and single-cell genomics demonstrate our method's superior performance. 

The second part of this talk focuses on deep generative models (DGMs) with latent representations. Despite DGMs’ impressive empirical performance, the statistical properties for these models remain underexplored. DGMs are often overparametrized, non-identifiable, and uninterpretable black boxes, raising serious concerns when deploying them in high-stakes applications. Motivated by this, we propose an interpretable deep generative modeling framework for rich data types with discrete latent layers, called Deep Discrete Encoders (DDEs). Theoretically, we propose transparent identifiability conditions for DDEs, which imply progressively smaller sizes of the latent layers as they go deeper. Identifiability ensures consistent parameter estimation and inspires an interpretable design of the deep architecture. Computationally, we propose a scalable estimation pipeline of a layerwise nonlinear spectral initialization followed by a penalized stochastic approximation EM algorithm.  This procedure can efficiently estimate models with exponentially many latent components. Extensive simulation studies validate our theoretical claims. We apply DDEs to diverse real datasets for hierarchical topic modeling, image representation learning, and response time modeling in educational testing.

Speaker: Professor Sumit Mukherjee (Columbia University)

Title:  Mean Field Approximation in Bayesian LinearRegression

Abstract: In this talk, we study the problem of Bayesian linear regression, where the coefficients have an iid prior. We show that the log normalizing constant of the posterior is “well approximated” by the mean field prediction formula, fora wide class of design matrices. Our techniques allow the design matrix to be deterministic/random with dependent entries. If the data is generated from a“true” linear model, we compute asymptotics of the log normalizing constant, interms of an optimizing problem over the space of measures. If this optimization has a unique solution, we derive a Law of Large Numbers under the posterior.

This talk is based on joint work with Subhabrata Sen from Harvard University.

Speaker: Professor Bianca Dumitrascu (Columbia University)

Title:  Interpretable Representation Learning for Wound Healing Dynamics

Abstract:  Single-cell RNA-seq enables the study of cell states across diverse biological conditions, such as aging, drug treatments, and tissue injury. However, disentangling shared and condition-specific transcriptomic patterns remains a significant computational challenge, particularly in settings with missing data or complex experimental designs. In this talk, I will introduce Patches, a deep generative framework designed to disentangle these transcriptomic signals, allowing for robust integration, cross-condition prediction, and biologically interpretable insights. Using real and simulated scRNA-seq datasets, we demonstrate how Patches uncovers shared wound healing patterns and distinct changes in cell behavior, including age-dependent immune responses and drug-modulated extracellular matrix remodeling. Finally, I will discuss open problems towards a pipeline for synthetic regeneration which includes identifying and designing targeted therapeutic interventions to accelerate wound healing.

Speaker:  Yuli Slavutsky

Title:  Generalization in Zero-Shot Learning

Abstract:  In this talk we will examine generalization in the challenging setting of zero-shot learning, where training data includes only a subset of the classes that may be encountered when using the model in practice.

Assuming that training and test classes are sampled from the same distribution, can we predict the model's ability to generalize to a larger, unobserved set of classes? We show that it is possible if the learned data representation remains unchanged when new classes are added. For that, we define a new measure of separation, the reversed ROC (rROC), prove that classification accuracy is a function of the rROC, and leverage this result to develop an algorithm for predicting the expected accuracy on unobserved class sets.

In many real-life applications, however, the distribution of classes might shift in deployment, due to changes in class attributes (for example, a shift in gender or race distribution in the task of person identification). Moreover, during training it is usually unknown which attribute is expected to cause the shift. In such cases, the performance on training classes is no longer indicative of the performance on test classes, presenting a new challenge: how to learn data representations that are robust to unknown attribute shifts? To address this, we present a new framework that combines hierarchical sampling with out-of-distribution generalization techniques, and demonstrate its effectiveness in achieving improved performance on diverse class distributions.

This talk is based on joint work with Yuval Benjamini.

NO SEMINAR: SPRING BREAK

Speaker: Ruchira Ray 

Title: Statistical guarantees for data-driven posterior tempering

Abstract: Posterior tempering reduces the influence of the likelihood in the Bayesian posterior by raising the likelihood to a constant fractional power. The resulting posterior — also known as a power posterior — has been shown to exhibit appealing properties, including robustness to model misspecification and asymptotic normality (Bernstein-von Mises). However, practical recommendations for selecting the power and statistical guarantees for the resulting power posterior remain open questions. We engage with these issues by connecting posterior tempering to penalized estimation. Cross-validation-based approaches to tuning the power parameter in these settings suggest a novel asymptotic regime where the distribution of the selected power is a mixture with a point mass at infinity and the remaining mass converging to 0. We formalize the limiting distribution of the power posterior in this new regime. Furthermore, in the regime where the power converges to 0, we provide sufficient conditions for (i) asymptotic normality of the power posterior, (ii) consistency of the power posterior moments, and (iii) “root(n)” consistency of the power posterior mean. Our results allow for the power to depend on the data in an arbitrary way.

Speaker: Shubhangi Ghosh

Title: Signal-to-noise-ratio aware minimax analysis of sparse linear regression

Abstract:The minimax framework has been one of the cornerstones of theoretical statistics, and has contributed to the popularity of many well-known estimators, such as the regularized M-estimators and regularized linear regression estimators for high-dimensional problems. In this paper, we demonstrate that numerous theoretical results within the classical minimax framework are inadequate in explaining empirical observations. In some instances, these minimax outcomes offer insights that contradict empirical findings. For example, although LASSO has been proven to be minimax optimal for the sparse linear regression problem, numerous empirical studies have shown its suboptimal performance across various signal-to-noise (SNR) levels. In this study, we aim to introduce an enhanced version of the minimax framework that not only elucidates these disparities but also offers more precise insights into the optimality of different estimators.

Our novel approach has two two distinctive components: (1) it integrates the signal-to-noise ratio into the construction of the parameter space. (2) It obtains accurate approximation of the minimax risk through asymptotic arguments. The theoretical findings derived from this refined framework provide new insights and practical guidance. For instance, in the context of estimating sparse signals under the linear regression model, our approach demonstrates that in the low SNR, ridge regression surpasses all other estimators, even when the regression coefficients are sparse.

Speaker: Professor Cynthia Rush (Columbia University

Title: Approximate Message Passing for High-Dimensional Estimation and Inference

Abstract: Approximate Message Passing (AMP) refers to a class of efficient, iterative algorithms that have been applied in a number of statistical problems, such as linear regression, generalized linear models and low-rank matrix estimation. Moreover, this class of algorithms is popular and practical for use in many engineering applications, including imaging, communications and deep learning.  The broad goal of this talk is to provide a mathematical introduction to AMP and a recipe for how to use AMP theory to establish precise asymptotic guarantees for a variety of statistical procedures (e.g., spectral methods, LASSO, $L_1$ interpolation) in the high-dimensional regime, under suitable assumptions on the data.

Speakers:  Anirban Nath & Navid Ardeshir, PhD students

Anirban Nath

Abstract:  Network-valued time series are currently a common form of network data. However, the study of the aggregate behavior of network sequences generated from network-valued stochastic processes is relatively rare. Most of the existing research focuses on the simple setup where the networks are independent (or conditionally independent) across time, and all edges are updated synchronously at each time step. In this paper, we study the concentration properties of the aggregated adjacency matrix and the corresponding Laplacian matrix associated with network sequences generated from lazy network-valued stochastic processes, where edges update asynchronously, and each edge follows a lazy stochastic process for its updates independent of the other edges. We demonstrate the usefulness of these concentration results in proving consistency of standard estimators in community estimation and changepoint estimation problems. We also conduct a simulation study to demonstrate the effect of the laziness parameter, which controls the extent of temporal correlation, on the accuracy of community and changepoint estimation.

Navid Ardeshir

Title: Adaptivity and Functional View of Wide Two-layer Neural Networks

Abstract:  We study the statistical and structural properties of (potentially infinite width) two-layer ReLU neural networks under norm constraints on the weights. These networks are known to achieve adaptive rates in certain nonparametric settings, particularly when the target function exhibits low-dimensional structure. This has led to the hypothesis that such adaptivity arises from favoring solutions that reflect the low-dimensional structure of the data. We challenge this hypothesis by analyzing a class of ridge functions that vary along a single direction. Despite the intrinsic low-dimensionality of these target functions, we show that all interpolating solutions within this neural network class are inherently multivariate functions. Our results reveal a gap between the data’s structural simplicity and the structural properties of the learned interpolants, shedding new light on the inductive biases of wide two-layer ReLU networks.

Speaker:  Professor Marco Avella Medina (Columbia University)

Title: M-estimation, noisy optimization and user-level differential privacy

Abstract:  We propose a general optimization-based framework for computing differentially private M-estimators. We first show that robust statistics can be used in conjunction with noisy gradient descent or noisy Newton methods in order to obtain optimal private estimators with global linear or quadratic convergence, respectively. We establish local and global convergence guarantees, under both local strong convexity and self-concordance, showing that our private estimators converge with high probability to a small neighborhood of the nonprivate M-estimators. We then extend this optimization framework to the more restrictive setting of local differential privacy (LDP) where a group of users communicates with an untrusted central server. Contrary to most works that aim to protect a single data point, here we assume each user possesses multiple data points and focus on user-level privacy which aims to protect the entire set of data points belonging to a user. Our main algorithm is a noisy gradient descent algorithm, combined with a user-level LDP mean estimation procedure to privately compute the average gradient across users at each step. We will highlight the challenges associated with guaranteeing user-level LDP and present finite sample global linear convergence guarantees for the iterates of our algorithm.

Speaker: Dr. Steven Campbell (Columbia University)

Title: Optimal Execution among N Traders with Transient Price Impact

Abstract:  We study N-player optimal execution games in an Obizhaeva--Wang model of transient price impact. When the game is regularized by an instantaneous cost on the trading rate, a unique equilibrium exists, and we derive its closed form. Whereas without regularization, there is no equilibrium and this can not be remedied by introducing randomization. We prove that existence is restored if (and only if) a very particular, time-dependent cost on block trades is added to the model. In that case, the equilibrium is particularly tractable. We show that this equilibrium is the limit of the regularized equilibria as the instantaneous cost parameter ε tends to zero. Moreover, we explain the seemingly ad-hoc block cost as the limit of the equilibrium instantaneous costs. Notably, in contrast to the single-player problem, the optimal instantaneous costs do not vanish in the limit ε→0. We use this tractable equilibrium to study the cost of liquidating in the presence of predators and the cost of anarchy. Our results also give a new interpretation to the erratic behaviors previously observed in discrete-time trading games with transient price impact.

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