##Data Science Seminar##\\ Hosted by the Department of Mathematics and Statistics * Date: Tuesday, October 15, 2024 * Time: 12:00pm -- 1:00pm * Room: Whitney Hall 100E * Speaker: Dr. Yang Ning (Cornell University) * Title: Estimation and Inference in Multivariate Response Regression with Hidden Variables. **//Abstract//** \\ This paper studies the estimation of the coefficient matrix $\Theta $ in multivariate regression with hidden variables, $Y = (\Theta)^TX + (B^*)^TZ + E$, where Y is a m-dimensional response vector, X is a p-dimensional vector of observable features, Z represents a K-dimensional vector of unobserved hidden variables, possibly correlated with X, and E is an independent error. The number of hidden variables K is unknown and both m and p are allowed but not required to grow with the sample size n. Since only Y and X are observable, we provide necessary conditions for the identifiability of $\Theta $. The same set of conditions are shown to be sufficient when the error E is homoscedastic. Our identifiability proof is constructive and leads to a novel and computationally efficient estimation algorithm, called HIVE. The first step of the algorithm is to estimate the best linear prediction of Y given X in which the unknown coefficient matrix exhibits an additive decomposition of $\Theta$ and a dense matrix originated from the correlation between X and the hidden variable Z. Under the row sparsity assumption on $\Theta $, we propose to minimize a penalized least squares loss by regularizing $\Theta $ via a group-lasso penalty and regularizing the dense matrix via a multivariate ridge penalty. Non-asymptotic deviation bounds of the in-sample prediction error are established. Our second step is to estimate the row space of B∗ by leveraging the covariance structure of the residual vector from the first step. In the last step, we remove the effect of hidden variable by projecting Y onto the complement of the estimated row space of B∗. Non-asymptotic error bounds of our final estimator are established. The model identifiability, parameter estimation and statistical guarantees are further extended to the setting with heteroscedastic errors. \\ Biography of the speaker: Dr. Ning is an associate professor in the Department of Statistics and Data Science at Cornell University. He received his Ph.D in Biostatistics from the Johns Hopkins University and was a post-doc at Princeton University and University of Waterloo. His research interests focus on high-dimensional statistics, and statistical inference in machine learning problems.