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The composite L0 function serves as a sparse regularizer in manyapplications. The algorithmic difficulty caused by the composite L0regularization (the L0 norm composed with a linear mapping) is usually bypassedthrough approximating the L0 norm. We consider in this paper capped Lpapproximations with $p>0$ for the composite L0 regularization problem. For each$p>0$, the capped Lp function converges to the L0 norm pointwisely as theapproximation parameter tends to infinity. We point out that the capped Lpapproximation problem is essentially a penalty method with an Lp penaltyfunction for the composite L0 problem from the viewpoint of numericaloptimization. Our theoretical results stated below may shed a new light on thepenalty methods for solving the composite L0 problem and help the design ofinnovative numerical algorithms. We first establish the existence of optimalsolutions to the composite L0 regularization problem and its capped Lpapproximation problem under conditions that the data fitting function isasymptotically level stable and bounded below. Asymptotically level stablefunctions cover a rich class of data fitting functions encountered in practice.We then prove that the capped Lp problem asymptotically approximates thecomposite L0 problem if the data fitting function is a level bounded functioncomposed with a linear mapping. We further show that if the data fittingfunction is the indicator function on an asymptotically linear set or the L0norm composed with an affine mapping, then the composite L0 problem and itscapped Lp approximation problem share the same optimal solution set providedthat the approximation parameter is large enough.

Capped Lp approximations for the composite L0 regularization problem