We study the mixing time of the Metropolis-adjusted Langevin algorithm (MALA) for sampling from a log-smooth and strongly log-concave distribution. We establish its optimal minimax mixing time under a warm start. Our main contribution is two-fold. First, for a $d$-dimensional log-concave density with condition number $\kappa$, we show that MALA with a warm start mixes in $\tilde O(\kappa \sqrt{d})$ iterations up to logarithmic factors. This improves upon the previous work on the dependency of either the condition number $\kappa$ or the dimension $d$. Our proof relies on comparing the leapfrog integrator with the continuous Hamiltonian dynamics, where we establish a new concentration bound for the acceptance rate. Second, we prove a spectral gap based mixing time lower bound for reversible MCMC algorithms on general state spaces. We apply this lower bound result to construct a hard distribution for which MALA requires at least $\tilde\Omega(\kappa \sqrt{d})$ steps to mix. The lower bound for MALA matches our upper bound in terms of condition number and dimension. Finally, numerical experiments are included to validate our theoretical results.

Substantive changes in gene expression, metabolism, and the proteome are manifested in overall changes in microbial population growth. Quantifying how microbes grow is therefore fundamental to areas such as genetics, bioengineering, and food safety. Traditional parametric growth curve models capture the population growth behavior through a set of summarizing parameters. However, estimation of these parameters from data is confounded by random effects such as experimental variability, batch effects or differences in experimental material. A systematic statistical method to identify and correct for such confounding effects in population growth data is not currently available. Further, our previous work has demonstrated that parametric models are insufficient to explain and predict microbial response under non-standard growth conditions. Here we develop a hierarchical Bayesian non-parametric model of population growth that identifies the latent growth behavior and response to perturbation, while simultaneously correcting for random effects in the data. This model enables more accurate estimates of the biological effect of interest, while better accounting for the uncertainty due to technical variation. Additionally, modeling hierarchical variation provides estimates of the relative impact of various confounding effects on measured population growth.

Dengue is a vector-borne viral disease of humans that endemically circulates in many tropical and subtropical regions worldwide. Infection with dengue can result in a range of disease outcomes. A considerable amount of research has sought to improve our understanding of this variation in disease outcomes and to identify predictors of severe disease. Contributing to this research, patterns of viral load in dengue infected patients have been quantified, with analyses indicating that peak viral load levels, rates of viral load decline, and time to peak viremia are useful predictors of severe disease. Here, we take a complementary approach to understanding patterns of clinical manifestation and inter-individual variation in viral load dynamics. Specifically, we statistically fit mathematical within-host models of dengue to individual-level viral load data to test virological and immunological hypotheses explaining inter-individual variation in dengue viral load. We choose between alternative models using model selection criteria to determine which hypotheses are best supported by the data. We first show that the cellular immune response plays an important role in regulating viral load in secondary dengue infections. We then provide statistical support for the process of antibody-dependent enhancement (but not original antigenic sin) in the development of severe disease in secondary dengue infections. Finally, we show statistical support for serotype-specific differences in viral infectivity rates, with infectivity rates of dengue serotypes 2 and 3 exceeding those of serotype 1. These results contribute to our understanding of dengue viral load patterns and their relationship to the development of severe dengue disease. They further have implications for understanding how dengue transmissibility may depend on the immune status of infected individuals and the identity of the infecting serotype.

Tree-like structures are fundamental in nature, and it is often useful to reconstruct the topology of a tree - what connects to what - from a two-dimensional image of it. However, the projected branches often cross in the image: the tree projects to a planar graph, and the inverse problem of reconstructing the topology of the tree from that of the graph is ill-posed. We regularize this problem with a generative, parametric tree-growth model. Under this model, reconstruction is possible in linear time if one knows the direction of each edge in the graph - which edge endpoint is closer to the root of the tree - but becomes NP-hard if the directions are not known. For the latter case, we present a heuristic search algorithm to estimate the most likely topology of a rooted, three-dimensional tree from a single two-dimensional image. Experimental results on retinal vessel, plant root, and synthetic tree data sets show that our methodology is both accurate and efficient.

A general framework is presented for analyzing multiple protein structures. A family of related protein structures may be analyzed using statistical regression methods. The analysis requires alternating steps of finding correspondences among the protein structures and superimposing the corresponding landmarks. The superposition step may be performed using either affine or orthogonal transformations, thereby allowing protein structures to undergo either pure rotations or rotation plus shear operations. Regression analysis permits a separate weight for each position, allowing one to emphasize particular segments of a protein structure or to compensate for variances that differ at various positions in a structure. In addition, a novel method is introduced for finding an initial correspondence, based on matching discrete curvatures along the protein backbone. Another novel method is introduced for obtaining gap functions that adapt to the given data, thereby making dynamic programming meth...