  This paper introduces a decomposition-based method to investigate the
Lipschitz stability of solution mappings for general LASSO-type problems with
convex data fidelity and $\ell_1$-regularization terms. The solution mappings
are considered as set-valued mappings of the measurement vector and the
regularization parameter. Based on the proposed method, we provide two
regularity conditions for Lipschitz stability: the weak and strong conditions.
The weak condition implies the Lipschitz continuity of solution mapping at the
point in question, regardless of solution uniqueness. The strong condition
yields the local single-valued and Lipschitz continuity of solution mapping.
When applied to the LASSO and Square Root LASSO (SR-LASSO), the weak condition
is new, while the strong condition is equivalent to some sufficient conditions
for Lipschitz stability found in the literature. Specifically, our results
reveal that the solution mapping of the LASSO is globally (Hausdorff) Lipschitz
continuous without any assumptions. In contrast, the solution mapping of the
SR-LASSO is not always Lipschitz continuous. A sufficient and necessary
condition is proposed for its local Lipschitz property. Furthermore, we fully
characterize the local single-valued and Lipschitz continuity of solution
mappings for both problems using the strong condition.
