Fitting algebraic curves to noisy data

Motivated by applications in vision and pattern detection, we introduce the following problem. We are given pairs of datapoints (x₁, y₁), (x₂, y₂),...,(x_m, y_m), a noise parameter δ > 0, a degree bound d, and a threshold ρ > 0. We desire "every" degree d polynomial h satisfying h(x_i) ∈ [y_i - δ, y_i + δ] for at least ρ fraction of i's. We assume by rescaling the data that each x_i, y_i ∈ [-1, 1]. If δ = 0, this is just the list decoding problem that has been popular in complexity theory and for which Sudan gave a poly(d, 1/ρ) time algorithm. We show a few basic results about the problem. We show that there is no polynomial time algorithm for this problem as defined; the number of solutions can be as large as exp(d^ρ.5-e) even if the data is generated using a 50-50 mixture of two polynomials. We give a rigorous analysis of a brute force algorithm for the version of this problem where the data is generated from a mixture of polynomials. Finally, in surprising contrast to our "lower bound", we describe a polynomial-time algorithm for reconstructing mixtures of O(1) polynomials when the mixing weights are "nondegeneration." The tools used include classical theory of approximations.