The Geometry of Sparse Analysis Regularization

Analysis sparsity is a common prior in inverse problem or machine learning including special cases such as Total Variation regularization, Edge Lasso and Fused Lasso. We study the geometry of solution set (a polyhedron) analysis l1-regularization (with l2 data fidelity term) when it not reduced to singleton without any assumption dictionary nor degradation operator. In contrast with most theoretical work, we do focus on giving uniqueness and/or stability results, but rather describe worst-case scenario where can be big terms dimension. Leveraging fine sub-level regularizer itself, draw connection between support minimal face containing it, particular prove that extremal points recovered thanks an algebraic test. Moreover, sign pattern ambient dimension smallest it. Finally, show arbitrary sub-polyhedra level seen sparse regularization explicit parameters.