Neural networks and rational functions
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Neural networks and rational functions efficiently approximate each other. In more detail, it is shown here that for any ReLU network, there exists a rational function of degree O(polylog(1/ϵ)) which is ϵ-close, and similarly for any rational function there exists a ReLU network of size O(polylog(1/ϵ)) which is ϵ-close. By contrast, polynomials need degree Ω(poly(1/ϵ)) to approximate even a single ReLU. When converting a ReLU network to a rational function as above, the hidden constants depend exponentially on the number of layers, which is shown to be tight; in other words, a compositional representation can be beneficial even for rational functions.
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