To better understand complexity in neural networks, we theoretically investigate the idealised phenomenon of lossless network compressibility, whereby an identical function can be implemented with a smaller network. We give an efficient formal algorithm for optimal lossless compression in the setting of single-hidden-layer hyperbolic tangent networks. To measure lossless compressibility, we define the rank of a parameter as the minimum number of hidden units required to implement the same function. Losslessly compressible parameters are atypical, but their existence has implications for nearby parameters. We define the proximate rank of a parameter as the rank of the most compressible parameter within a small $L^\infty$ neighbourhood. Unfortunately, detecting nearby losslessly compressible parameters is not so easy: we show that bounding the proximate rank is an NP-complete problem, using a reduction from Boolean satisfiability via a geometric problem involving covering points in the plane with small squares. These results underscore the computational complexity of measuring neural network complexity, laying a foundation for future theoretical and empirical work in this direction.

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