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Open Access Publications from the University of California

Scholarly Works (2 results)

  • Thesis
  • Peer Reviewed
UC Berkeley Electronic Theses and Dissertations (2023)

High-dimensional expansion, a generalization of graph expansion to higher-order edges, has recently garnered significant attention in the theoretical computer science community for the additional boost they give in applications like error-correction and approximate sampling. In this thesis, we explore two problems related to high-dimensional expansion, using tools from the geometry of polynomials as well as high-dimensional convex geometry.

First, we study approximate sampling from discrete distributions. The framework for sampling obtained from high-dimensional expansion provides both a natural set of random walks to use in MCMC algorithms, as well as a set of tools for their analysis. We show that the geometric properties (e.g. log-concavity) of a polynomial derived from the distribution allows us to speed up the implementations of these random walks.

Next, we study a random graph model called the ``random geometric graph,'' with an eventual goal of understanding its modeling capabilities as well as its high-dimensional expansion properties. Along the way, we prove new results about distinguishing the random geometric graph model from the Erdos-Renyi model, and develop a new geometric toolkit for analyzing these graphs.

    Cover page: Geometry-Inspired Sampling Algorithms and Random Graphs
    • Article
    • Peer Reviewed
    UC Irvine Previously Published Works (2013)
    Abstract Background Age-related macular degeneration (AMD) is the leading cause of vision loss in elderly, Caucasian populations. There is strong evidence that mitochondrial dysfunction and oxidative stress play a role in the cell death found in AMD retinas. The purpose of this study was to examine the association of the Caucasian mitochondrial JTU haplogroup cluster with AMD. We also assessed for gender bias and additive risk with known high risk nuclear gene SNPs, ARMS2/LOC387715 (G > T; Ala69Ser, rs10490924) and CFH (T > C; Try402His, rs1061170). Methods Total DNA was isolated from 162 AMD subjects and 164 age-matched control subjects located in Los Angeles, California, USA. Polymerase chain reaction (PCR) and restriction enzyme digestion were used to identify the J, U, T, and H mitochondrial haplogroups and the ARMS2-rs10490924 and CFH-rs1061170 SNPs. PCR amplified products were sequenced to verify the nucleotide substitutions for the haplogroups and ARMS2 gene. Results The JTU haplogroup cluster occurred in 34% (55/162) of AMD subjects versus 15% (24/164) of normal (OR = 2.99; p = 0.0001). This association was slightly greater in males (OR = 3.98, p = 0.005) than the female population (OR = 3.02, p = 0.001). Assuming a dominant effect, the risk alleles for the ARMS2 (rs10490924; p = 0.00001) and CFH (rs1061170; p = 0.027) SNPs were significantly associated with total AMD populations. We found there was no additive risk for the ARMS2 (rs10490924) or CFH (rs1061170) SNPs on the JTU haplogroup background. Conclusions There is a strong association of the JTU haplogroup cluster with AMD. In our Southern California population, the ARMS2 (rs10490924) and CFH (rs1061170) genes were significantly but independently associated with AMD. SNPs defining the JTU mitochondrial haplogroup cluster may change the retinal bioenergetics and play a significant role in the pathogenesis of AMD.
      Cover page: Mitochondrial DNA haplogroups confer differences in risk for age-related macular degeneration: a case control study
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