### 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 nondegeneracy conditions it reconstructs the generating graph with high probability using Θ(dε^{-2} δ^{-4} log n) samples, where e, δ depend on the local interactions. For most local interactions ε,δ are of order exp(- O(d)). Our results are optimal as a function of n up to a multiplicative constant depending on d and the strength of the local interactions. Our results seem to be the first results for general models that guarantee that the generating model is reconstructed. Furthermore, we provide explicit O(n^{d+2} ε^{-2} δ^{-4} 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. We also discuss the effect of observing noisy samples and 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 nonidentifiability even for generic noise and interactions. Finally, we briefly show that in some simple cases, models with hidden nodes can also be recovered.

Original language | English (US) |
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Pages (from-to) | 563-578 |

Number of pages | 16 |

Journal | SIAM Journal on Computing |

Volume | 42 |

Issue number | 2 |

DOIs | |

State | Published - Jul 18 2013 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Mathematics(all)

### Keywords

- Algorithms
- Correlation decay
- Markov random fields

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## Cite this

*SIAM Journal on Computing*,

*42*(2), 563-578. https://doi.org/10.1137/100796029