Mathematical Sciences Department Financial Math Seminar
Hubeyb Gurdogan, UCLA
Monday, April 14th
2:00pm - 3:00 pm
Zoom: Zoom meeting has ended.
Title: Impossibility of Eigenvector Alignment Estimation in High Dimensions
Abstract: Large covariance estimation plays a central role in high-dimensional statistics and underpins much of modern multivariate data analysis. A common modeling strategy introduces pairwise correlations among variables through a small number of latent factors, yielding a spiked covariance structure in which a few large eigenvalues separate from a bulk spectrum. The associated eigenvectors form a population quantity of interest, denoted B, while their empirical counterparts, H, are typically estimated via the leading eigenvectors of the sample covariance matrix. The matrix BTH, encoding the alignment between sample and population eigenvectors, offers a fine-grained measure of estimation accuracy.
Our analysis considers the high-dimensional, low-sample-size (HDLSS) regime, where the sample size is fixed and the dimension grows. Drawing on the HDLSS theory developed in recent PCA literature, the focus here is not on asymptotic limits of BTH, but rather on the feasibility of estimating it from the observed data. An impossibility theorem is established, showing that under mild conditions, no consistent estimator of BTH exists when the number of spiked eigenvalues exceeds one. This result highlights fundamental limitations of spectral methods in extreme dimensional settings and contributes to the broader understanding of PCA behavior in the HDLSS regime, which is increasingly relevant due to practical constraints in experimental design and the nonstationarity of real-world time series data.
Bio: Hubeyb Gurdogan is a Hedrick Assistant Adjunct Professor in the Department of Mathematics at UCLA. He earned his Ph.D. in Financial Engineering from the Department of Mathematics at Florida State University in December 2021, under the supervision of Alec Kercheval. Prior to his doctoral studies, he received M.S. degrees in Mathematics from Syracuse University and Bilkent University. Following his Ph.D., Dr. Gurdogan held a research appointment at UC Berkeley’s Center for Risk Management Research (CDAR), where he led a collaborative project with the Swiss Re Institute. In this role, he served as research lead on a project focused on modeling supply chain risk propagation, with particular emphasis on pricing Non-Damage Business Interruption (NDBI) insurance products. Dr. Gurdogan’s research lies at the intersection of high-dimensional statistics and random matrix theory, with applications to portfolio optimization, structural break detection, and lead-lag analysis in high-dimensional time series.