Deep neural network approximation theory

Authors

Dennis Elbrächter, Dmytro Perekrestenko, Philipp Grohs, and Helmut Bölcskei

Reference

IEEE Transactions on Information Theory, invited feature paper, Vol. 67, No.5, pp. 2581-2623, May 2021.

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Abstract

This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal approximation through deep neural networks with the guiding theme being a relation between the complexity of the function (class) to be approximated and the complexity of the approximating network in terms of connectivity and memory requirements for storing the network topology and the associated quantized weights. The theory we develop establishes that deep networks are Kolmogorov-optimal approximants for markedly different function classes, such as unit balls in Besov spaces and modulation spaces. In addition, deep networks provide exponential approximation accuracy—i.e., the approximation error decays exponentially in the number of nonzero weights in the network—of the multiplication operation, polynomials, sinusoidal functions, and certain smooth functions. Moreover, this holds true even for one-dimensional oscillatory textures and the Weierstrass function—a fractal function, neither of which has previously known methods achieving exponential approximation accuracy. We also show that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks.

Keywords

Deep learning, neural networks, approximation theory, Kolmogorov-Donoho rate-distortion theory

Comments

In the third paragraph on p. 32, the formulation ``D is an orthonormal basis for the bounded set C'' is meant to say that D is an orthonormal basis for a space that contains C.


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