### 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) |
---|---|

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 |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics

### Cite this

*Journal of Computer and System Sciences*,

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

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

Research output: Contribution to journal › Article

*Journal of Computer and System Sciences*, vol. 67, no. 2, pp. 325-340. https://doi.org/10.1016/S0022-0000(03)00012-6

}

TY - JOUR

T1 - Fitting algebraic curves to noisy data

AU - Arora, Sanjeev

AU - Khot, Subhash

PY - 2003/9

Y1 - 2003/9

N2 - 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 |h(xi) - yi|≤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(d0.5-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.".

AB - 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 |h(xi) - yi|≤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(d0.5-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.".

UR - http://www.scopus.com/inward/record.url?scp=0142091455&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0142091455&partnerID=8YFLogxK

U2 - 10.1016/S0022-0000(03)00012-6

DO - 10.1016/S0022-0000(03)00012-6

M3 - Article

AN - SCOPUS:0142091455

VL - 67

SP - 325

EP - 340

JO - Journal of Computer and System Sciences

JF - Journal of Computer and System Sciences

SN - 0022-0000

IS - 2

ER -