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Breakdown Point of Model Selection When the Number of Variables Exceeds the Number of Observations

Donoho, David L.; Stodden, Victoria C.

The classical multivariate linear regression problem assumes p variables X1, X2, ... , Xp and a response vector y, each with n observations, and a linear relationship between the two: y = X beta + z, where z ~ N(0, sigma2). We point out that when p > n, there is a breakdown point for standard model selection schemes, such that model selection only works well below a certain critical complexity level depending on n/p. We apply this notion to some standard model selection algorithms (Forward Stepwise, LASSO, LARS) in the case where pGtn. We find that 1) the breakdown point is well-de ned for random X-models and low noise, 2) increasing noise shifts the breakdown point to lower levels of sparsity, and reduces the model recovery ability of the algorithm in a systematic way, and 3) below breakdown, the size of coefficient errors follows the theoretical error distribution for the classical linear model.

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Title
2006 International Joint Conference on Neural Networks, Sheraton Vancouver Wall Centre Hotel, Vancouver, BC, Canada, July 16-21, 2006
Publisher
IEEE
DOI
https://doi.org/10.1109/IJCNN.2006.246934

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October 11, 2011