  We focus on the minimization of the least square loss function either under a
$k$-sparse constraint or with a sparse penalty term. Based on recent results,
we reformulate the $\ell_0$ pseudo-norm exactly as a convex minimization
problem by introducing an auxiliary variable. We then propose an exact biconvex
reformulation of the $\ell_2-\ell_0$ constrained and penalized problems. We
give correspondence results between minimizers of the initial function and the
reformulated ones. The reformulation is biconvex and the non-convexity is due
to a penalty term. These two properties are used to derive a minimization
algorithm. We apply the algorithm to the problem of single-molecule
localization microscopy and compare the results with the well-known Iterative
Hard Thresholding algorithm. Visually and numerically the biconvex
reformulations perform better.
