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A lattice-based approach to the expressivity of deep ReLU neural networks
lib:14183bd2c202519d (v1.0.0)
Authors: Vincent Corlay,Joseph J. Boutros,Philippe Ciblat,Loic Brunel
ArXiv: 1902.11294
Document: PDF DOI
Abstract URL: https://arxiv.org/abs/1902.11294v2
We present new families of continuous piecewise linear (CPWL) functions in Rn having a number of affine pieces growing exponentially in $n$. We show that these functions can be seen as the high-dimensional generalization of the triangle wave function used by Telgarsky in 2016. We prove that they can be computed by ReLU networks with quadratic depth and linear width in the space dimension. We also investigate the approximation error of one of these functions by shallower networks and prove a separation result. The main difference between our functions and other constructions is their practical interest: they arise in the scope of channel coding. Hence, computing such functions amounts to performing a decoding operation.
ArXiv: 1902.11294
Document: PDF DOI
Abstract URL: https://arxiv.org/abs/1902.11294v2
We present new families of continuous piecewise linear (CPWL) functions in Rn having a number of affine pieces growing exponentially in $n$. We show that these functions can be seen as the high-dimensional generalization of the triangle wave function used by Telgarsky in 2016. We prove that they can be computed by ReLU networks with quadratic depth and linear width in the space dimension. We also investigate the approximation error of one of these functions by shallower networks and prove a separation result. The main difference between our functions and other constructions is their practical interest: they arise in the scope of channel coding. Hence, computing such functions amounts to performing a decoding operation.
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