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In this paper, we focus on exploring solution uniqueness, sharpness, androbust recovery to sparse regularization with a gauge J. Our approach is basedon the uniqueness of Lagrange multipliers for its dual problem. We observe thatthe Lagrange bidual problem of a gauge regularization problem is itself and itsdual problem satisfies Slater's condition and hence the strong duality holds.Therefore, the solution set to the original gauge regularization problem is theLagrange multiplier set for its dual problem and is the same for any solutionto the dual problem. Based on criteria for the uniqueness of the Lagrangemultipliers for the dual problem, we obtain several equivalent sufficientconditions for solution uniqueness to the gauge regularization problem whichare also necessary conditions under some closedness condition. We also provethat these sufficient conditions are sufficient and necessary conditions for asharp solution, which ensures the robust recovery with a linear rate. Someapplications to weighted sorted $\ell_1$-norm sparse regularization, sparseanalysis regularization, nuclear norm optimization, conic gauge optimization,and semidefinite conic gauge optimization are given.

On Solution Uniqueness and Robust Recovery for Sparse Regularization with a Gauge: from Dual Point of View