Hyper-gates Framework for Straight-line Programs and Their Applications in Nonlinear Algebra
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Mahon, Hannah
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Abstract
The massive computations required by artificial intelligence and scientific computing have outpaced general-purpose CPUs, necessitating hardware acceleration via GPUs. Straight-Line Programs (SLPs) are an ideal solution as by definition they are branchless and do not consist of any conditionals, avoiding thread divergence, high memory consumption, and incorrect solutions due to guessing caused by branching on hardware. SLPs can enable an algorithm to be mapped as a predictable data-flow circuit, enabling massive parallelization. Nonlinear systems are foundational to science and engineering, with their solutions studied with algebraic geometry and approximated by numerical algebraic geometry algorithms like homotopy continuation. These solvers rely on automatic (auto) differentiation, and since auto differentiation can be represented as an SLP, it is of great interest to represent the entire solver as an SLP for hardware acceleration.
This thesis introduces HGates.m2, a framework that expands on SLPs by defining hyper-level gates to model nonlinear algorithms. We detail how autodifferentiation and an adaptation of the homotopy continuation predictor-corrector method are implemented within this framework. For several geometric problems, we demonstrate how to set up polynomial systems of equations using H-gates.
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2025-12
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Thesis (Masters Degree)