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doi:10.1016/S0022-0000(03)00012-6 
Copyright © 2003 Elsevier Inc. All rights reserved.
Fitting algebraic curves to noisy data
Received 10 July 2002;
revised 10 December 2002.
Available online 19 July 2003.
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Abstract
We introduce the following problem which is motivated by applications in vision and pattern detection: We are given pairs of datapoints (x1,y1),(x2,y2),…,(xm,ym)
[−1,1]×[−1,1], a noise parameter δ>0, a degree bound d, and a threshold ρ>0. We desire an algorithm that enlists every degree d polynomial h such that
) and every pair of solutions is far from each other in ℓ∞ norm. On the algorithmic side, we give a rigorous analysis of a brute force algorithm that runs in exponential time. Also, in surprising contrast to our lowerbound, we give a polynomial-time algorithm for learning the polynomials assuming the data is generated using a mixture model in which the mixing weights are “nondegenerate.” Author Keywords: Curve Fitting; Noisy polynomial reconstruction; List decoding; Learning theory; Vision
