### 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) |
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Title of host publication | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

Pages | 162-169 |

Number of pages | 8 |

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 |

### Other

Other | Proceedings of the 34th Annual ACM Symposium on Theory of Computing |
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Country | Canada |

City | Montreal, Que. |

Period | 5/19/02 → 5/21/02 |

### Fingerprint

### ASJC Scopus subject areas

- Software

### Cite this

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing*(pp. 162-169)

**Fitting algebraic curves to noisy data.** / Arora, Sanjeev; Khot, Subhash.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing.*pp. 162-169, Proceedings of the 34th Annual ACM Symposium on Theory of Computing, Montreal, Que., Canada, 5/19/02.

}

TY - GEN

T1 - Fitting algebraic curves to noisy data

AU - Arora, Sanjeev

AU - Khot, Subhash

PY - 2002

Y1 - 2002

N2 - Motivated by applications in vision and pattern detection, we introduce the following problem. We are given pairs of datapoints (x1, y1), (x2, y2),...,(xm, ym), a noise parameter δ > 0, a degree bound d, and a threshold ρ > 0. We desire "every" degree d polynomial h satisfying h(xi) ∈ [yi - δ, yi + δ] for at least ρ fraction of i's. We assume by rescaling the data that each xi, yi ∈ [-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(dρ.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.

AB - Motivated by applications in vision and pattern detection, we introduce the following problem. We are given pairs of datapoints (x1, y1), (x2, y2),...,(xm, ym), a noise parameter δ > 0, a degree bound d, and a threshold ρ > 0. We desire "every" degree d polynomial h satisfying h(xi) ∈ [yi - δ, yi + δ] for at least ρ fraction of i's. We assume by rescaling the data that each xi, yi ∈ [-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(dρ.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.

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M3 - Conference contribution

SP - 162

EP - 169

BT - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

ER -