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Coin Flipping Neural Networks
We show that a CFNN can approximate the indicator of a $d$-dimensional ball to arbitrary accuracy with only 2 layers and $\mathcal{O}(1)$ neurons, where a 2-layer deterministic network was shown to require $\Omega(e^d)$ neurons, an exponential improvement (arXiv:1610. 09887).
Slicing the hypercube is not easy
We prove that at least $\Omega(n^{0. 51})$ hyperplanes are needed to slice all edges of the $n$-dimensional hypercube.
It's Not What Machines Can Learn, It's What We Cannot Teach
Can deep neural networks learn to solve any task, and in particular problems of high complexity?
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