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A Geometric Perspective on Matrix and Tensor Rank

Gabriel Brown, University of Texas at Austin

Abstract

Given the power of singular value decomposition, computational scientists often take the tractability of matrix low-rank approximation for granted, giving little thought to the underlying complexity and geometry of the low-rank set itself.

While this may be sufficient for the average practitioner, examining these geometries can lead to a deeper and more nuanced understanding of the problem and its solutions.

We examine known results about rank for both matrices and tensors in a modern and simplified form. This naturally leads to a theory governing the existence of solutions to the respective low-rank approximation problems. New visualizations help strengthen the intuitive understanding we aim to convey.

Biography

Gabriel was born and raised in Florida before attending the University of Notre Dame and University of Illinois Urbana-Champaign where he studied mechanical engineering with a focus on computation. His research interests are broadly in the design and analysis of numerical algorithms, including for (multi)linear algebra and optimization, and their high performance and parallel implementations.