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Kähler Geometry of Framed Quiver Moduli and Machine Learning

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Abstract

We develop an algebro-geometric formulation for neural networks in machine learning using the moduli space of framed quiver representations. We find natural Hermitian metrics on the universal bundles over the moduli whose expressions are independent of dimension vector, and show that their Ricci curvatures give a Kähler metric on the moduli. Moreover, we use toric moment maps to construct activation functions and prove the universal approximation theorem for the softmax function (also known as Boltzmann distribution) using toric geometry of the complex projective space.

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Acknowledgements

We are grateful to Marco Antonio Armenta for informing us about the work [3] and the further useful discussions. We express our gratitude to Shing-Tung Yau for his generous encouragement. The work of S.C. Lau in this paper is partially supported by the Simons collaboration grant.

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  1. Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, MA, 02215, USA

    George Jeffreys & Siu-Cheong Lau

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  1. George Jeffreys
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Correspondence to George Jeffreys.

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Jeffreys, G., Lau, SC. Kähler Geometry of Framed Quiver Moduli and Machine Learning. Found Comput Math 23, 1899–1957 (2023). https://doi.org/10.1007/s10208-022-09587-3

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