For a good choice of alpha, the Lasso can fully recover the exact set of non-zero variables using only few observations, provided certain specific conditions are met. In particular, the number of samples should be “sufficiently large”, or L1 models will perform at random, where “sufficiently large” depends on the number of non-zero coefficients, the logarithm of the number of features, the amount of noise, the smallest absolute value of non-zero coefficients, and the structure of the design matrix X. In addition, the design matrix must display certain specific properties, such as not being too correlated. On the use of Lasso for sparse signal recovery, see this example on compressive sensing: Compressive sensing: tomography reconstruction with L1 prior (Lasso).

There is no general rule to select an alpha parameter for recovery of non-zero coefficients. It can be set by cross-validation (LassoCV or LassoLarsCV), though this may lead to under-penalized models: including a small number of non-relevant variables is not detrimental to prediction score. BIC (LassoLarsIC) tends, on the opposite, to set high values of alpha.

References

Richard G. Baraniuk “Compressive Sensing”, IEEE Signal Processing Magazine [120] July 2007 http://users.isr.ist.utl.pt/~aguiar/CS_notes.pdf