Upcoming MAMS Seminar Series
September 12, 2023
Speaker: Michael Roysdon
Title: Framework for the Theory of Higher Order Convex Bodies
Abstract: We develop the basic theory of “higher–order” convex bodies beginning with a result of Schneider from 1970 concerning the Rogers-Shephard inequality. Building upon Schneider’s work, we develop the notion of the higher order L_p-projection body and L_p-centroid body. If time permits, we will discuss proofs of the associated affine isoperimetric inequalities: Petty projection inequality, Zhang’s projection inequality, and Busemann-Petty Centroid inequality.
In particular, we discuss two interesting consequences of our results that are surprisingly new to the literature:
1) A version of the Busemann random simplex inequality for the mean width, but where the random simplex is replaced by a random polytope and where the vertices of the random polytope need not be i.i.d.
2) A extremal volume inequality for operator norms between Banach spaces.
This talk in based on joint works, and ongoing works, with Julian Haddad, Dylan Langharst, Eli Putterman, and Deping Ye.
Abstract: Bayesian hierarchical models have been demonstrated to provide efficient algorithms for finding sparse solutions to ill-posed inverse problems. The models comprise typically a conditionally Gaussian prior model for the unknown, augmented by a hyperprior model for the variances. A widely used choice for the hyperprior is a member of the family of generalized gamma distributions. Most of the work in the literature has concentrated on numerical approximation of the maximum a posteriori (MAP) estimates, and less attention has been paid on sampling methods or other means for uncertainty quantification. Sampling from the hierarchical models is challenging mainly for two reasons: The hierarchical models are typically high-dimensional, thus suffering from the curse of dimensionality, and the strong correlation between the unknown of interest and its variance can make sampling rather inefficient. This work addresses mainly the first one of these obstacles. By using a novel reparametrization, it is shown how the posterior distribution can be transformed into one dominated by a Gaussian white noise, allowing sampling by using the preconditioned Crank-Nicholson (pCN) scheme that has been shown to be efficient for sampling from distributions dominated by a Gaussian component. Furthermore, a novel idea for speeding up the pCN in a special case is developed, and the question of how strongly the hierarchical models are concentrated on sparse solutions is addressed in light of a computed example.
The work is in collaboration with Professor Daniela Calvetti.
from a Bayesian hierarchical model of total and categorical count data. The scheme
applies to a Negative Binomial – Binomial (NBB) hierarchical regression model with
logit links and normal priors on regression coefficients. The approach is shown to be
very efficient and in most cases out-performs the Stan program. We apply the
hierarchical modeling framework and the Póya-Gamma data augmentation scheme to
human mitochondrial DNA data.