The　paper　proposes　a　first-order　fast　optimization　algorithm　on　Riemannian　manifolds　(FOA)　to　address　the　problem　of　speeding　up　optimization　algorithms　for　a　class　of　composite　functions　on　Riemannian　manifolds.　The　theoretical　analysis　for　FOA　shows　that　the　algorithm　achieves　the　optimal　rate　of　convergence　for　function　values　sequence.　The　experiments　on　the　matrix　completion　task　show　that　FOA　has　better　performance　than　other　existing　first-order　optimization　methods　on　Riemannian　manifolds.　A　subspace　pursuit　method　(SP-RPRG(ALM))　based　on　FOA　is　also　proposed　to　solve　the　low-rank　representation　model　with　the　augmented　Lagrange　method　(ALM)　on　the　low-rank　matrix　variety.　Experimental　results　on　synthetic　data　and　public　databases　are　presented　to　demonstrate　that　both　FOA　and　SP-RPRG　(ALM)　can　achieve　superior　performance　in　terms　of　faster　convergence　and　higher　accuracy.　We　have　made　the　experimental　code　public　at　https://github.com/Haoran2014.　(c)　2018　Elsevier　B.V.　All　rights　reserved.