Projects

Robust Estimation of Tree Structured Gaussian Graphical Models

Consider jointly Gaussian random variables whose conditional independence structure is specified by a graphical model. If we observe realizations of the variables, we can compute the covariance matrix, and it is well known that the support of the inverse covariance matrix corresponds to the edges of the graphical model. Instead, suppose we only have noisy observations. If the noise is independent, this allows us to compute the sum of the covariance matrix and an unknown diagonal. The inverse of this sum is (in general) dense. We ask: can the original independence structure be recovered? We address this question for tree structured graphical model. We prove the unidentifiability of this problem and provide the additional constraints to make this problem identifiable. We also provide an $\mathcal{O}(n^3)$ algorithm to find the underlying tree structure.

A Bayesian approach to determine the composition of heterogeneous cancer tissue

We tackle the challenge of determining the composition of a heterogeneous cancer tissue. There exist high cost methods providing cell level resolution to observe important feature values. This method can be used to get accurate estimates of hetergeneous cell composition but the cost is prohibitive. We propose an algorithm that can get accurate estimates of the heterogeneous cancer tissue composition in a cost effective manner. It relies on one-time high cost cell level resolution measurements which can then be utilized in low cost low resolution aggregate measurements to estimate the composition of the heterogeneous cancer tissue.