### Abstract

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}), ..., (x_{m}, y_{m}) ∈ [-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 |h(x_{i}) - y_{i}|≤δ for at least ρ fraction of the indices i. If δ = 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 δ>0, the problem as stated becomes ill-posed and one needs a careful reformulation (see the Introduction). We prove a few basic results about this (reformulated) problem. We show that the problem has no polynomial-time algorithm (our counterexample works for ρ = 0.5). This is shown by exhibiting an instance of the problem where the number of solutions is as large as exp(d^{0.5-ε}) 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.".

Original language | English (US) |
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Pages (from-to) | 325-340 |

Number of pages | 16 |

Journal | Journal of Computer and System Sciences |

Volume | 67 |

Issue number | 2 |

DOIs | |

State | Published - Sep 2003 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Networks and Communications
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Journal of Computer and System Sciences*,

*67*(2), 325-340. https://doi.org/10.1016/S0022-0000(03)00012-6