My personal favourite is compressive sensing, sparse signals beat the Nyquist sampling limit, by what seems like an unbounded amount 1000x, 1000000x, etc, because all real world stuff we care about lives on a low dimensional manifold, so unlike the noise.
replySampling above the nyquist sampling limit then representing it as something else is not beating the nyquist sampling limit, it's just compression.
replyYou may enjoy randomised algorithms (https://www.cl.cam.ac.uk/teaching/2425/RandAlgthm/)
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