Question

Consider the optimization problem

$$\min_x ||Ax||_1 + \lambda||x-b||^2,$$

where $A \in \mathbb{R}^{n \times n}$, $x,b \in \mathbb{R}^n$ and $\lambda$ is strictly greater than 0. (This problem is closely related to the "lasso" problem in basis pursuit.) Can anything be said about the value of $\lambda$ for which $Ax^*$ is sparsest? Clearly some values are bad: for instance, if $\lambda$ is huge and $b$ is dense then it is unlikely that $Ax^\star$ will be very sparse.

In other words: among all $\lambda > 0$ there is at least one value $\lambda^\star$ such that $||Ax^\star(\lambda)||_0$ is minimized. Are there, say, bounds on $\lambda^\star$ in terms of $A$ and $b$? I'd also be interested in results pertaining to basis pursuit or other similar problems.

Edit: I'm primarily interested in problems where ideal sparsity cannot be achieved, i.e., $||Ax^\star(\lambda^\star)||_0 > 0.$ (Assume that $A$ is square w/ full rank and $b \ne 0$.)

Answer 1

The ultimate sparseness occurs when $Ax^*(\lambda)=0$, which is the case when the minimizer $x^*$ is the projection of $b$ onto $\ker A$. For this to happen, $\lambda$ must be small enough so that the restriction of $A$ to the orthogonal complement of its kernel is bounded from below by a constant greater than $2\lambda \mathrm{dist}(b,\ker A)$. Here the lower bound for operator is understood in the $\ell^2\to\ell^1$ norm.