In　this　paper,　MMV　Subspace　Pursuit　(M-SP)　algorithm　is　proposed　for　solving　joint　sparse　multiple　measurement　vectors　(MMV)　problem.　The　pre-selection　and　backtracking　mechanisms　are　used　in　M-SP,　so　M-SP　not　only　has　higher　recovery　performance　than　some　existing　algorithms,　but　also　significantly　reduces　the　iteration　number　for　improving　the　signal　recovery　efficiency.　Simulations　results　show　that　M-SP　and　Simultaneous　Compressive　Sampling　Matching　Pursuit　(SCoSaMP)　have　almost　identical　recovery　performance　and　iteration　times,　but　M-SP　significantly　reduces　the　computation　complexity　in　per　iteration.　For　example,　when　sparsity　K　is　5,　the　computational　complexity　of　M-SP　is　24.0%　of　that　of　SCoSaMP　in　each　iteration.　©　2019　IEEE.