Classical and quantum algorithms for scaling problems

This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

Quantum algorithms for matrix scaling and matrix balancing

Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications, such as approximating the permanent, and pre-conditioning linear systems to make them more numerically stable. We study the power and limitations of quantum algorithms for these problems. We provide quantum implementations of two classical (in both senses of the word) methods: Sinkhorn's algorithm for matrix scaling and Osborne's algorithm for matrix balancing. Using amplitude estimation as our main tool, our quantum implementations both run in time Õ(√mn/∈4) for scaling or balancing an n×n matrix (given by an oracle) with m non-zero entries to within ℓ1-error ∈. Their classical analogs use time Õ(m/∈2), and every classical algorithm for scaling or balancing with small constant ∈ requires Ω(m) queries to the entries of the input matrix. We thus achieve a polynomial speed-up in terms of n, at the expense of a worse polynomial dependence on the obtained ℓ1-error ∈. Even for constant ∈ these problems are already non-trivial (and relevant in applications). Along the way, we extend the classical analysis of Sinkhorn's and Osborne's algorithm to allow for errors in the computation of marginals. We also adapt an improved analysis of Sinkhorn's algorithm for entrywise-positive matrices to the ℓ1-setting, obtaining an Õ(n1.5/∈3)-time quantum algorithm for ∈-ℓ1-scaling. We also prove a lower bound, showing our quantum algorithm for matrix scaling is essentially optimal for constant ∈: every quantum algorithm for matrix scaling that achieves a constant ℓ1-error w.r.t. uniform marginals needs Ω(√ mn) queries

GWIS: Genome-Wide Inferred Statistics for Functions of Multiple Phenotypes

Here we present a method of genome-wide inferred study (GWIS) that provides an approximation of genome-wide association study (GWAS) summary statistics for a variable that is a function of phenotypes for which GWAS summary statistics, phenotypic means, and covariances are available. A GWIS can be performed regardless of sample overlap between the GWAS of the phenotypes on which the function depends. Because a GWIS provides association estimates and their standard errors for each SNP, a GWIS can form the basis for polygenic risk scoring, LD score regression, Mendelian randomization studies, biological annotation, and other analyses. GWISs can also be used to boost power of a GWAS meta-analysis where cohorts have not measured all constituent phenotypes in the function. We demonstrate the accuracy of a BMI GWIS by performing power simulations and type I error simulations under varying circumstances, and we apply a GWIS by reconstructing a body mass index (BMI) GWAS based on a weight GWAS and a height GWAS. Furthermore, we apply a GWIS to further our understanding of the underlying genetic structure of bipolar disorder and schizophrenia and their relation to educational attainment. Our analyses suggest that the previously reported genetic correlation between schizophrenia and educational attainment is probably induced by the observed genetic correlation between schizophrenia and bipolar disorder and the previously reported genetic correlation between bipolar disorder and educational attainment

The Computerized Neurocognitive Battery: Validation, aging effects, and heritability across cognitive domains

Objective: The Computerized Neurocognitive Battery (CNB) enables efficient neurocognitive assessment. The authors aimed to (a) estimate validity and reliability of the battery's Dutch translation, (b) investigate effects of age across cognitive domains, and (c) estimate heritability of the CNB tests. Method: A populationrepresentative sample of 1,140 participants (aged 10-86), mainly twin-families, was tested on the CNB, providing measures of speed and accuracy in 14 cognitive domains. In a subsample (246 subjects aged 14-22), IQ data (Wechsler Intelligence Scale for Adults; WAIS) were available. Validity and reliability were assessed by Cronbach's alpha, comparisons of scores between Dutch and U.S. samples, and investigation of how a CNB-based common factor compared to a WAIS-based general factor of intelligence (g). Linear and nonlinear age dependencies covering the life span were modeled through regression. Heritability was estimated from twin data and from entire pedigree data. Results: Internal consistency of all tests was moderate to high (median = 0.86). Effects of gender, age, and education on cognitive performance closely resembled U.S. samples. The CNB-based common factor was completely captured by the WAIS-based g. Some domains, like nonverbal reasoning accuracy, peaked in young adulthood and showed steady decline. Other domains, like language reasoning accuracy, peaked in middle adulthood and were spared decline. CNB-test heritabilities were moderate (median h2 = 31%). Heritability of the CNB common factor was 70%, similar to the WAIS-based g-factor. Conclusion: The CNB can be used to assess specific neurocognitive performance, as well as to obtain a reliable proxy of general intelligence. Effects of aging and heritability differed across cognitive domains

An extended twin-pedigree study of neuroticism in the Netherlands Twin Register

For the participants in the Netherlands Twin Register (NTR) we constructed the extended pedigrees which specify all relations among nuclear and larger twin families in the register. A total of 253,015 subjects from 58,645 families were linked to each other, to the degree that we had information on the relations among participants. We describe the algorithm that was applied to construct the pedigrees. For > 30,000 adolescent and adult NTR participants data were available on harmonized neuroticism scores. We analyzed these data in the Mendel software package (Lange et al., Bioinformatics 29(12):1568–1570, 2013) to estimate the contributions of additive and non-additive genetic factors. In contrast to much of the earlier work based on twin data rather than on extended pedigrees, we could also estimate the contribution of shared household effects in the presence of non-additive genetic factors. The estimated broad-sense heritability of neuroticism was 47%, with almost equal contributions of additive and non-additive (dominance) genetic factors. A shared household effect explained 13% and unique environmental factors explained the remaining 40% of the variance in neuroticism