Three structural results on the lasso problem

The lasso problem, least squares with a ℓ₁ regularization penalty, has been very successful as a tool for obtaining sparse representations of data in terms of given dictionary. It is known, but not widely appreciated, that the lasso problem need not have a unique solution. Sufficient conditions which ensure uniqueness of the solution are known but necessary and sufficient conditions have been elusive. We present three structural results on the lasso problem. First, we show that when the dictionary has more columns than rows, it is always possible to ensure that the dictionary has full row rank. Next we show that the feasible set for the dual lasso problem is bounded if and only if the dictionary has full row rank. Lastly, we give necessary and sufficient conditions for the uniqueness of a lasso solution.