Mathematical Principles for Deconstructing Deep Learning: Theory and Application to Electromagnetic Signals

Witz, Evan

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Mathematical Principles for Deconstructing Deep Learning: Theory and Application to Electromagnetic Signals

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In the field of deep learning, there can often exist a divide between the theoretical and that of the applied. In this work, we attempt to bridge that gap by developing a theoretical framework for studying neural networks of a certain architecture, which can then be applied in the construction of real networks, which we apply to data coming from electromagnetic signal processing. These networks combine the best performance properties of more standard architectures with a beautifully rich algebraic structure. The theoretical portion of this work begins by studying the properties of networks with polynomial activation. We prove here a theorem demonstrating an algebraic decomposition of what we call Kronecker networks, as well as a concrete path toward a Universal Approximation Theorem for polynomial networks. In the applied portion, our focus is on the study of range localization of over-water electromagnetic signals. We attempt to use a received signal and deep learning to determine the distance from which the signal was sent. We explore the various atmospheric phenomena that can influence the signal and give a comparison to a statistical model, the maximum likelihood estimator. Finally, we bridge the two portions of this work by demonstrating that a network architecture inspired by our polynomial activation can achieve similar performance to a more standard architecture while enjoying a rich algebraic structure.

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