Relationship between the optimal solutions of least squares regularized with L0-norm and constrained by k-sparsity
Résumé
Given an M × N real-valued matrix A with M < N and a data-vector d, consider that d must be expressed as a linear combination of a small number of basis vectors taken from A. Two popular options to find the sought-after sparse solution are (1) to minimize the least-squares regularized with the counting function ∥ * ∥0 (called usually the ℓ0-norm) via a trade-off parameter β > 0, and (2) to solve the least-squares constrained by k-sparsity, i.e. ∥ * ∥0 6 k. This work provides an exhaustive description of the relationship between the optimal solutions of these two nonconvex (combinatorial) optimization problems. Small-size exact numerical tests give a flavor of the meaning of the obtained theoretical results.
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