There are several versions of universal approximation theorems known, including the very well-known ones from [1, 2, 3]. Each of them states that some collection of neural networks is dense in some space of continuous functions with respect to the uniform norm. In this short note, we present what we believe to be a new and atypically elementary proof of one such theorem. If σ is a 0-1 squashing function (a.k.a. a sigmoidal function), we show that the collection of neural networks with three hidden layers and activation function σ (except at the output) is dense in the space C(K) of real-valued continuous functions on a compact set K R

The ever-growing complexity of mathematical proofs makes their manual verification by mathematicians very cognitively demanding. Autoformalization seeks to address this by translating proofs written in natural language into a formal representation that is computer-verifiable via interactive theorem provers. In this paper, we introduce a semantic parsing approach, based on the Universal Transformer architecture, that translates elementary mathematical proofs into an equivalent formalization in the language of the Coq interactive theorem prover. The same architecture is also trained to translate simple imperative code decorated with Hoare triples into formally verifiable proofs of correctness in Coq. Experiments on a limited domain of artificial and human-written proofs show that the models generalize well to intermediate lengths not seen during training and variations in natural language.

The phase retrieval problem has garnered significant attention since the development of the PhaseLift algorithm, which is a convex program that operates in a lifted space of matrices. Because of the substantial computational cost due to lifting, many approaches to phase retrieval have been developed, including non-convex optimization algorithms which operate in the natural parameter space, such as Wirtinger Flow. Very recently, a convex formulation called PhaseMax has been discovered, and it has been proven to achieve phase retrieval via linear programming in the natural parameter space under optimal sample complexity. The current proofs of PhaseMax rely on statistical learning theory or geometric probability theory. Here, we present a short and elementary proof that PhaseMax exactly recovers real-valued vectors from random measurements under optimal sample complexity.