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We have an $\m\x\n$ real-valued arbitrary matrix $A$ (e.g. a dictionary) with$\m<\n$ and data $d$ describing the sought-after object with the help of $A$.This work provides an in-depth analysis of the (local and global) minimizers ofan objective function $\Fd$ combining a quadratic data-fidelity term and an$\ell_0$ penalty applied to each entry of the sought-after solution, weightedby a regularization parameter $\be>0$. For several decades, this objective hasattracted a ceaseless effort to conceive algorithms approaching a goodminimizer. Our theoretical contributions, summarized below, shed new light onthe existing algorithms and can help the conception of innovative numericalschemes. To solve the normal equation associated with any $\m$-row submatrix of$A$ is equivalent to compute a local minimizer $\hu$ of $\Fd$. (Local)minimizers $\hu$ of $\Fd$ are strict if and only if the submatrix, composed ofthose columns of $A$ whose indexes form the support of $\hu$, has full columnrank. An outcome is that strict local minimizers of $\Fd$ are easily computedwithout knowing the value of $\be$. Each strict local minimizer is linear indata. It is proved that $\Fd$ has global minimizers and that they are alwaysstrict. They are studied in more details under the (standard) assumption that$\rank(A)=\m<\n$. The global minimizers with $\m$-length support are seen to beimpractical. Given $d$, critical values $\be_\k$ for any $\k\leq\m-1$ areexhibited such that if $\be>\be_\k$, all global minimizers of $\Fd$ are$\k$-sparse. An assumption on $A$ is adopted and proved to fail only on aclosed negligible subset. Then for all data $d$ beyond a closed negligiblesubset, the objective $\Fd$ for $\be>\be_\k$, $\k\leq\m-1$, has a unique globalminimizer and this minimizer is $\k$-sparse. Instructive small-size ($5\x 10$)numerical illustrations confirm the main theoretical results.

Description of the minimizers of least squares regularized~ with~ $\bm{\ell_0}$-norm. Uniqueness of the global minimizer