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

The classical multivariate linear regression problem assumes p variables X
₁
, X
₂
,... ,X
_p
and a response vector y, each with n observations, and a linear relationship between the two: y = Xbeta + z, where z ~ N(0, sigma
²
). 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 coef cient errors follows the theoretical error distribution for the classical linear model.