Seminar for Postdoctoral Fellows, Spring 2016

Title: Topological Data Analysis: Looking at the Underlying Geometry of Data

Abstract: Several modern statistical methodologies revolve around what is commonly referred to as the 'large p small n' problem where the number of variables collected for each observations greatly exceeds the total number of observations. Many methods designed for high dimensional data either reduce the number of variables or find appropriate functional combinations. One drawback of this variable selection/combination approach is the view that that each variable is by itself a measure of interest. In complex data structures, such as fMRI data, each variable is not necessarily a measure of interest unto itself, but rather a part describing the total structure of interest. Topological Data Analysis (TDA) allows us to examine the underlying topology of the data, thus allowing us to analyze observations in terms of overall structure instead of variable by variable. I will discuss the basic ideas and approaches behind TDA, focusing on summarizing the persistent homology of a data set and potential for utilizing this information in a statistical framework. I will follow with examples of applying TDA methodology to my current projects at SAMSI: classification of the tertiary structure of proteins and feature identification and comparison with functional neurological data.

Title: Towards Reduced Order Models for Multispectral Photoacoustic Tomography

Abstract: Photoacoustic tomography (PAT) is an emerging biomedical imaging modality which combines the high spatial resolution of ultrasound tomography with the high contrast of optical tomography. The goal is to reconstruct desired optical properties on the interior of a domain of interest, using measured ultrasound data collected outside the domain. Mathematically this is an inverse problem to determine the coefficients of a diffusion equation, using measured pressure values coming from an acoustic wave equation which is coupled to the diffusion equation via the photoacoustic effect. A popular and well-performing reconstruction approach for this inverse problem is via regularized least-squares minimization. However, this approach is computationally intensive for large-scale (i.e. high-resolution and/or multispectral) images, as it requires repeatedly solving the PDEs in large dimensions. We seek to reduce the computational burden by replacing the PDE models with cheaper yet accurate surrogates which have smaller dimension. In this talk, we present some results for a reduced-order model for the wave equation and discuss next steps towards reduced-order models for the diffusion equation, and for the wavelength dependence in the multispectral case. This is current work with Arvind Saibaba in the Computational Inverse Problems working group.

Title: Selecting an Optimal Predictive Approximation for the Regression Function with Bernstein Polynomials using an OBayes Approach

Abstract: The order of smoothness chosen in nonparametric estimation problems is critical, balancing a tradeoff between model generality and data overfitting. Commonly used strategies in this context are cross-validation (CV) and Generalized cross-validation (GCV). However, these alternatives may result computationally costly and preclude quantification of the uncertainty surrounding this choice. As an alternative, we take an objective Bayes approach to select an appropriate order of smoothness while simultaneously assessing its uncertainty. Although our method can be applied to any series-based smoother, our focus is on approximations arising from Bernstein polynomial expansions, which are shape preserving, among their many other desirable features. For the single predictor case, we prove that choosing as the order of smoothness the degree of the median probability model with our approach is optimal for prediction. Extending the method to multiple predictors is not a trivial task, as now one must to choose the polynomial order for the Bernstein basis functions of each predictor and for all their higher order interactions. As such, these model spaces can become prohibitively large extremely fast. To control the growth of model space we develop a semi-greedy algorithm that builds the model space adaptively using the available data. We compare the performance of our method to that of CV and GCV through simulations.  Both our method and GCV are orders of magnitude faster than CV, and all methods have comparable predictive accuracy. Two real data examples with a single predictor are also analyzed. Finally, to illustrate the extension to multiple predictors we provide preliminary results from a small simulation experiment. [This is a joint work with Sujit Ghosh, NCSU/SAMSI].

Title:  Directional Mean Curvature for Textured Image Deconvolution and Decomposition

Title: Supervised Learning in Forensic Science

Abstract: Modern learning algorithms are typically seen as prediction-only tools, meaning the interpretability and intuition provided by a more traditional modeling approach are sacrificed in order to achieve superior predictions.  In this talk, we argue that this black-box perspective need not always be the case.  We demonstrate that predictions from ensemble learners like bagged trees and random forests, when built with subsamples in lieu of full bootstrap samples, can be viewed as incomplete, infinite-order U-statistics and as such, are asymptotically normal.  Furthermore, we show that the limiting variance depends only on the size of the ensemble relative to the size of the training set and by enforcing a structure on the subsamples used in the ensemble, we can form a consistent estimate of variance at no additional computational cost.  This allows for statistical inference to be carried out in practice and we'll conclude this talk by demonstrating how such an approach can be applied to problems in criminology and forensic science in areas such as quality metric evaluation and the identification of randomly acquired characteristics on shoe prints and tire tracks.