Reconstructing algebraic functions from mixed data

The authors consider the task of reconstructing algebraic functions given by black boxes. Unlike traditional settings, they are interested in black boxes which represent several algebraic functions-f₁,..., f_k, where at each input x, the box arbitarrily chooses a subset of f₁(x),..., f_k(x) to output. They show how to reconstruct the functions f₁,..., f_k from the black box. This allows them to group the same points into sets, such that for each set, all outputs to points in the set are from the same algebraic function. The methods are robust in the presence of errors in the black box. The model and techniques can be applied in the areas of computer vision, machine learning, curve fitting and polynomial approximation, self-correcting programs and bivariate polynomial factorization.