Abstract
It has been recently shown that a deep neural network with i.i.d. random parameters is equivalent to a Gaussian process in the limit of infinite network width. The Gaussian process associated to the neural network is fully described by a recursive covariance kernel determined by the architecture of the network, and which is expressed in terms of expectation. We give a numerically workable analytic expression of the neural network recursive covariance based on Hermite polynomials. We give explicit forms of this recursive covariance for the cases of neural networks with activation function the Heaviside, ReLU and sigmoid.
A. Arratia—Supported by grant TIN2017-89244-R from MINECO (Ministerio de Economía, Industria y Competitividad) and the recognition 2017SGR-856 (MACDA) from AGAUR (Generalitat de Catalunya).
A. Cabaña—Supported by grant RTI2018-096072-B-I00D Ministerio de Ciencia, Innovación y Universidades.
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Arratia, A., Cabaña, A., León, J.R. (2020). Deep and Wide Neural Networks Covariance Estimation. In: Farkaš, I., Masulli, P., Wermter, S. (eds) Artificial Neural Networks and Machine Learning – ICANN 2020. ICANN 2020. Lecture Notes in Computer Science(), vol 12396. Springer, Cham. https://doi.org/10.1007/978-3-030-61609-0_16
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DOI: https://doi.org/10.1007/978-3-030-61609-0_16
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