### Abstract

Markov random fields are used to model high dimensional distributions in a number of applied areas. Much recent interest has been devoted to the reconstruction of the dependency structure from independent samples from the Markov random fields. We analyze a simple algorithm for reconstructing the underlying graph defining a Markov random field on n nodes and maximum degree d given observations. We show that under mild non-degeneracy conditions it reconstructs the generating graph with high probability using Θ(d log n) samples which is optimal up to a multiplicative constant. Our results seem to be the first results for general models that guarantee that the generating model is reconstructed. Furthermore, we provide an explicit O(dn^{d+2}log n) running time bound. In cases where the measure on the graph has correlation decay, the running time is O(n^{2}log n) for all fixed d. In the full-length version we also discuss the effect of observing noisy samples. There we show that as long as the noise level is low, our algorithm is effective. On the other hand, we construct an example where large noise implies non-identifiability even for generic noise and interactions. Finally, we briefly show that in some cases, models with hidden nodes can also be recovered.

Original language | English (US) |
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Title of host publication | Approximation, Randomization and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 11th International Workshop, APPROX 2008 and 12th International Workshop, RANDOM 2008, Proceedings |

Pages | 343-356 |

Number of pages | 14 |

DOIs | |

State | Published - Sep 22 2008 |

Externally published | Yes |

Event | 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008 - Boston, MA, United States Duration: Aug 25 2008 → Aug 27 2008 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 5171 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008 |
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Country | United States |

City | Boston, MA |

Period | 8/25/08 → 8/27/08 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques - 11th International Workshop, APPROX 2008 and 12th International Workshop, RANDOM 2008, Proceedings*(pp. 343-356). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5171 LNCS). https://doi.org/10.1007/978-3-540-85363-3_28