### Abstract

We study graph estimation and density estimation in high dimensions, using a family of density estimators based on forest structured undirected graphical models. For density estimation, we do not assume the true distribution corresponds to a forest; rather, we form kernel density estimates of the bivariate and univariate marginals, and apply Kruskal's algorithm to estimate the optimal forest on held out data. We prove an oracle inequality on the excess risk of the resulting estimator relative to the risk of the best forest. For graph estimation, we consider the problem of estimating forests with restricted tree sizes. We prove that finding a maximum weight spanning forest with restricted tree size is NP-hard, and develop an approximation algorithm for this problem. Viewing the tree size as a complexity parameter, we then select a forest using data splitting, and prove bounds on excess risk and structure selection consistency of the procedure. Experiments with simulated data and microarray data indicate that the methods are a practical alternative to sparse Gaussian graphical models.

Original language | English (US) |
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Title of host publication | COLT 2010 - The 23rd Conference on Learning Theory |

Pages | 394-406 |

Number of pages | 13 |

State | Published - Dec 1 2010 |

Event | 23rd Conference on Learning Theory, COLT 2010 - Haifa, Israel Duration: Jun 27 2010 → Jun 29 2010 |

### Publication series

Name | COLT 2010 - The 23rd Conference on Learning Theory |
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### Other

Other | 23rd Conference on Learning Theory, COLT 2010 |
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Country | Israel |

City | Haifa |

Period | 6/27/10 → 6/29/10 |

### All Science Journal Classification (ASJC) codes

- Education

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## Cite this

*COLT 2010 - The 23rd Conference on Learning Theory*(pp. 394-406). (COLT 2010 - The 23rd Conference on Learning Theory).