In the first part of this talk, we will show how any continuous function $f:[0,1] \to [-1,1]$ can be approximated arbitrarily well on $(0,1)$ by polynomials with $\pm 1$-coefficients in the Bernstein basis, providing quantitative bounds on the degree of their approximation. In the second part of the talk, we will extend our method to the multivariate setting. We will also show how these multivariate one-bit polynomial approximations can be realized by means of neural networks whose parameters are chosen from the set $\{\pm 1\}$ only.

Joint work with Weilin Li.