Topics in the arithmetic of polynomials over finite fields

In this thesis, we investigate various topics regarding the arithmetic of polynomials over finite fields. In particular, we explore the analogy between the integers and this polynomial ring, and exploit the additional structure of the latter in order to derive arithmetic statistics which go beyond what can currently be proved in the integer setting. First, we adapt the Selberg-Delange method to prove an asymptotic formula for counting polynomials with a given number of prime factors. We then extend this formula to cases in which these polynomials are restricted first to arithmetic progressions, and then to `short intervals'. In both cases, we obtain better ranges for the associated parameters than in the integer setting, by using Weil's Riemann Hypothesis for curves over finite fields. Then, we investigate highly composite polynomials and the divisor function for polynomials over a finite field, as inspired by Ramanujan's work on highly composite numbers. We determine a family of highly composite polynomials which is not too sparse, and use it to compute the maximum order of the divisor function up to an error which is much smaller than in the case of integers, even when the Riemann Hypothesis is assumed there. Afterwards, we take a brief aside to discuss the connection between the Generalised Divisor Problem and the Lindelöf Hypothesis in the integer setting. Next, we prove that for a certain set of multiplicative functions on the polynomial ring, the bound in Halász's Theorem can be improved. Conversely, we determine a criterion for when the general bound is actually attained, and construct an example which satisfies this criterion. Finally, in the other direction, we develop a formula for the Möbius function of a number field which is related to Pellet's Formula for the Möbius function of the polynomial ring

Are You Going to Have Heart Failure Soon?

Presented at the Georgia Tech Career, Research, and Innovation Development Conference (CRIDC), January 27-28, 2020, Georgia Tech Global Learning Center, Atlanta, GA.The Career, Research, and Innovation Development Conference (CRIDC) is designed to equip on-campus and online graduate students with tools and knowledge to thrive in an ever-changing job market.Ardavan Afshar, in the School of Computational Science and Engineering at Georgia Tech, was the winner of a College of Computing Travel Award

SWIFT: Scalable Wasserstein Factorization for Sparse Nonnegative Tensors

Existing tensor factorization methods assume that the input tensor follows some specific distribution (i.e. Poisson, Bernoulli, and Gaussian), and solve the factorization by minimizing some empirical loss functions defined based on the corresponding distribution. However, it suffers from several drawbacks: 1) In reality, the underlying distributions are complicated and unknown, making it infeasible to be approximated by a simple distribution. 2) The correlation across dimensions of the input tensor is not well utilized, leading to sub-optimal performance. Although heuristics were proposed to incorporate such correlation as side information under Gaussian distribution, they can not easily be generalized to other distributions. Thus, a more principled way of utilizing the correlation in tensor factorization models is still an open challenge. Without assuming any explicit distribution, we formulate the tensor factorization as an optimal transport problem with Wasserstein distance, which can handle non-negative inputs. We introduce SWIFT, which minimizes the Wasserstein distance that measures the distance between the input tensor and that of the reconstruction. In particular, we define the N-th order tensor Wasserstein loss for the widely used tensor CP factorization and derive the optimization algorithm that minimizes it. By leveraging sparsity structure and different equivalent formulations for optimizing computational efficiency, SWIFT is as scalable as other well-known CP algorithms. Using the factor matrices as features, SWIFT achieves up to 9.65% and 11.31% relative improvement over baselines for downstream prediction tasks. Under the noisy conditions, SWIFT achieves up to 15% and 17% relative improvements over the best competitors for the prediction tasks.Comment: Accepted by AAAI-2

Imputation of missing values for electronic health record laboratory data

Laboratory data from Electronic Health Records (EHR) are often used in prediction models where estimation bias and model performance from missingness can be mitigated using imputation methods. We demonstrate the utility of imputation in two real-world EHR-derived cohorts of ischemic stroke from Geisinger and of heart failure from Sutter Health to: (1) characterize the patterns of missingness in laboratory variables; (2) simulate two missing mechanisms, arbitrary and monotone; (3) compare cross-sectional and multi-level multivariate missing imputation algorithms applied to laboratory data; (4) assess whether incorporation of latent information, derived from comorbidity data, can improve the performance of the algorithms. The latter was based on a case study of hemoglobin A1c under a univariate missing imputation framework. Overall, the pattern of missingness in EHR laboratory variables was not at random and was highly associated with patients’ comorbidity data; and the multi-level imputation algorithm showed smaller imputation error than the cross-sectional method