This　work　analyzes　the　alternating　minimization　(AM)　method　for　solving　double　sparsity　constrained　minimization　problem,　where　the　decision　variable　vector　is　split　into　two　blocks.　The　objective　function　is　a　separable　smooth　function　in　terms　of　the　two　blocks.　We　analyze　the　convergence　of　the　method　for　the　non-convex　objective　function　and　prove　a　rate　of　convergence　of　the　norms　of　the　partial　gradient　mappings.　Then,　we　establish　a　non-asymptotic　sub-linear　rate　of　convergence　under　the　assumption　of　convexity　and　the　Lipschitz　continuity　of　the　gradient　of　the　objective　function.　To　solve　the　sub-problems　of　the　AM　method,　we　adopt　the　so-called　iterative　thresholding　method　and　study　their　analytical　properties.　Finally,　some　future　works　are　discussed.