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Fitting Algebraic Curves to Noisy Data Sanjeev Arora # Subhash Khot +

Summary: Fitting Algebraic Curves to Noisy Data
Sanjeev Arora # Subhash Khot +
December 14, 2002
We introduce the following problem which is motivated by applications in vision and
pattern detection : We are given pairs of datapoints (x 1 , y 1 ), (x 2 , y 2 ), . . . , (xm , ym ) #
[-1, 1] × [-1, 1], a noise parameter # > 0, a degree bound d, and a threshold # > 0. We
desire an algorithm that enlists every degree d polynomial h such that
|h(x i ) -y i | # # for at least # fraction of the indices i (1)
If # = 0, this is just the list decoding problem that has been popular in complexity theory
and for which Sudan gave a poly(m,d) time algorithm. However, for # > 0, the problem
as stated becomes ill­posed and one needs a careful reformulation (see Introduction).
We prove a few basic results about this (reformulated) problem. We show that the prob­
lem has no polynomial time algorithm. This is shown by exhibiting an instance of the
problem where the number of solutions is as large as exp(d 0.5-# ) and every pair of so­
lutions is far from each other in ## norm. On the algorithmic side, we give a rigorous
analysis of a brute force algorithm that runs in exponential time. Also, in surprising con­
trast 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


Source: Arora, Sanjeev - Department of Computer Science, Princeton University


Collections: Computer Technologies and Information Sciences