### Abstract

Motivated by applications in vision and pattern detection, we introduce the following problem. We are given pairs of datapoints (x_{1}, y_{1}), (x_{2}, y_{2}),...,(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.

Original language | English (US) |
---|---|

Pages (from-to) | 162-169 |

Number of pages | 8 |

Journal | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

DOIs | |

State | Published - 2002 |

Event | Proceedings of the 34th Annual ACM Symposium on Theory of Computing - Montreal, Que., Canada Duration: May 19 2002 → May 21 2002 |

### All Science Journal Classification (ASJC) codes

- Software