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Applications of singularity theory on deep neural networks

Abstract

Singularity Theory is a branch of mathematics that studies sets, mappings, and the relationship among them. Roughly speaking, singularists are concerned with studying those points in a set where, in a certain sense, a mathematical model does not behave as ``expected". Due to its fundamental importance in characterizing the behavior of sets, basic notions of singularities are already taught in the first year calculus course where students use basics tools from real analysis to solve variational problems in biology, chemistry and physics, detecting the maximals and minimals points (extrema) of functions using simple math operations such as derivatives. The tools of singularity theory can also be applied in other branches of science and technology including robotics, autonomous vehicles, computer graphics, vision and image processing, animation and multimedia, human computer interface, network structures, speech processing, bioinformatics, etc. These areas have enriched singularity theory with new interesting problems and have shown its diverse intersections with differential geometry, differential equations, bifurcation theory, dynamical system, etc.In this project we intend to combine tools and techniques from pure and applied math to define, detect, and mitigate singularities on deep neural networks, as described in the project. It will be developed under scientific cooperation between the ICMC/USP/São Carlos(Brazil) and MSU/iProBE-Lab(USA) researchers teams. (AU)

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