Error Bounds for Reconstruction of a Function f from a Finite Sequence $\{ \operatorname{sgn} ( f( t_i ) + x_i ) \}$

Consider reconstructing a function $f( t ),0\leqq t\leqq 1$, from knowledge only of $\{ ( t_i ,s_i ),1\leqq i\leqq n \}$, where $s_i = \operatorname{sgn} ( f( t_i ) + x_i )$, $1\leqq i\leqq n$, and the $x_i $ are additive “corruptions.” Without the components $x_i $, f could not be reconstructed. However, for f continuous and for random uniform noise x, Masry and Cambanis (IEEE Trans. Inform. Theory, IT-26 (1980), pp. 50–58; IT-27 (1981), pp. 84–96) show that f can be consistently estimated almost surely and in mean square as $n \to \infty $ and establish rates of pointwise convergence. Through a somewhat different treatment, in which the approximation of $f( t_i )$ is identified as a numerical integration problem rather than a statistical problem, we obtain simple exact bounds on the error of estimation, allow the noise x to be arbitrary (random or deterministic) and deal with the case of f having discontinuities. The bounds yield substantially improved rates of convergence when the noise values $x_i $, ...