|
Abstract:
|
The lasso is an important tool in high-dimensional statistics, but in many situations, we would like to go further and impose structural constraints on the lasso coefficients; specific examples include requiring that the lasso fit be piecewise constant (known as the fused lasso) or piecewise polynomial (known as trend filtering). The generalized lasso problem, where we augment the usual lasso criterion with a sparsity-inducing penalty on a linear transformation of the lasso coefficients, can capture a wide variety of these scenarios. However, in a high-dimensional setup, the generalized lasso problem may not have a unique solution. We provide necessary and sufficient conditions for uniqueness, and show that when the entries of the data matrix are drawn from a continuous probability distribution, then the generalized lasso solution is unique with probability one, even in a high-dimensional setup and even for loss functions other than the squared loss. We also show how to handle the situation when the generalized lasso solution is not unique, and extend the LARS algorithm for computing the generalized lasso solution path to accommodate non-uniqueness.
|
