Gaussian determinantal processes: a new model for directionality in data
(Meet-&-greet from 3:30 pm to 4 pm)
Abstract:
Determinantal point processes (DPPs) have recently become popular tools for modeling the phenomenon of negative dependence, or repulsion, in data. However, our understanding of an analogue of a classical parametric statistical theory is rather limited for this class of models. In this work, we investigate a parametric family of Gaussian DPPs with a clearly interpretable effect of parametric modulation on the observed points. We show that parameter modulation impacts the observed points by introducing directionality in their repulsion structure, and the principal directions correspond to the directions of maximal (i.e., the most long- ranged) dependency. This model readily yields a viable alternative to principal component analysis (PCA) as a dimension reduction tool that favors directions along which the data are most spread out. This methodological contribution is complemented by a statistical analysis of a spiked model similar to that employed for covariance matrices as a framework to study PCA. These theoretical investigations unveil intriguing questions for further examination in random matrix theory, stochastic geometry, and related topics.
Based on joint work with Philippe Rigollet.
Biodata:
Subhro Ghosh is an assistant professor at the National University of Singapore and a faculty affiliate at the Institute of Data Science, NUS. Before joining NUS, he was a post doc at Princeton University, and prior to that he obtained his PhD from the University of California, Berkeley under the supervision of Yuval Peres. Earlier, he received his Bachelor in Statistics and Master in Mathematics degrees from the Indian Statistical Institute
Subhro is broadly interested in stochastics, focussing on problems from statistical physics and the math of data, and their interactions. These encompass constrained stochastic systems and their applications, including problems of learning under complex structure (e.g., latent group actions), dimension reduction, sampling and optimization, statistical networks and signal processing. Key paradigms include determinantal processes (DPP), strong Rayleigh measures and negative dependence, multi reference alignment (MRA), empirical likelihood, generative priors, Gaussian analytic functions (GAF), random tensors and stochastic geometry. The investigation of these problems naturally brings together a wide array of tools and techniques, including probability, harmonic and complex analysis, persistent homology and the theory of group representations.