Dynamic graph and polynomial chaos based models for contact tracing data analysis and optimal testing prescription.

In this study, we address three important challenges related to disease transmissions such as the COVID-19 pandemic, namely, (a) providing an early warning to likely exposed individuals, (b) identifying individuals who are asymptomatic, and (c) prescription of optimal testing when testing capacity is limited. First, we present a dynamic-graph based SEIR epidemiological model in order to describe the dynamics of the disease propagation. Our model considers a dynamic graph/network that accounts for the interactions between individuals over time, such as the ones obtained by manual or automated contact tracing, and uses a diffusion-reaction mechanism to describe the state dynamics. This dynamic graph model helps identify likely exposed/infected individuals to whom we can provide early warnings, even before they display any symptoms and/or are asymptomatic. Moreover, when the testing capacity is limited compared to the population size, reliable estimation of individual's health state and disease transmissibility using epidemiological models is extremely challenging. Thus, estimation of state uncertainty is paramount for both eminent risk assessment, as well as for closing the tracing-testing loop by optimal testing prescription. Therefore, we propose the use of arbitrary Polynomial Chaos Expansion, a popular technique used for uncertainty quantification, to represent the states, and quantify the uncertainties in the dynamic model. This design enables us to assign uncertainty of the state of each individual, and consequently optimize the testing as to reduce the overall uncertainty given a constrained testing budget. These tools can also be used to optimize vaccine distribution to curb the disease spread when limited vaccines are available. We present a few simulation results that illustrate the performance of the proposed framework, and estimate the impact of incomplete contact tracing data.

Fast Estimation of Approximate Matrix Ranks Using Spectral Densities.

Many machine learning and data-related applications require the knowledge of approximate ranks of large data matrices at hand. This letter presents two computationally inexpensive techniques to estimate the approximate ranks of such matrices. These techniques exploit approximate spectral densities, popular in physics, which are probability density distributions that measure the likelihood of finding eigenvalues of the matrix at a given point on the real line. Integrating the spectral density over an interval gives the eigenvalue count of the matrix in that interval. Therefore, the rank can be approximated by integrating the spectral density over a carefully selected interval. Two different approaches are discussed to estimate the approximate rank, one based on Chebyshev polynomials and the other based on the Lanczos algorithm. In order to obtain the appropriate interval, it is necessary to locate a gap between the eigenvalues that correspond to noise and the relevant eigenvalues that contribute to the matrix rank. A method for locating this gap and selecting the interval of integration is proposed based on the plot of the spectral density. Numerical experiments illustrate the performance of these techniques on matrices from typical applications.

Improving the Incoherence of a Learned Dictionary via Rank Shrinkage.

This letter considers the problem of dictionary learning for sparse signal representation whose atoms have low mutual coherence. To learn such dictionaries, at each step, we first update the dictionary using the method of optimal directions (MOD) and then apply a dictionary rank shrinkage step to decrease its mutual coherence. In the rank shrinkage step, we first compute a rank 1 decomposition of the column-normalized least squares estimate of the dictionary obtained from the MOD step. We then shrink the rank of this learned dictionary by transforming the problem of reducing the rank to a nonnegative garrotte estimation problem and solving it using a path-wise coordinate descent approach. We establish theoretical results that show that the rank shrinkage step included will reduce the coherence of the dictionary, which is further validated by experimental results. Numerical experiments illustrating the performance of the proposed algorithm in comparison to various other well-known dictionary learning algorithms are also presented.