The　PCA　dimensionality　reduction　algorithm　for　2D　data　with　the　Laplacian　noise　model,　i.e.,　L1-2DPCA,　not　only　preserves　the　structural　relation　among　2D　data,　but　also　is　robust　for　data　outliers.　The　algorithm　relies　on　the　EM　algorithm　with　great　computational　cost.　In　order　to　learn　intrinsic　information　more　consistently,　this　paper　takes　a　view　of　manifold　optimization　for　the　model　based　on　the　L1-norm　average　reconstruction　error　and　develops　an　efficient　optimization　algorithm　(called　as　L1-2DPCAM)　on　product　manifolds.　This　way　has　avoided　from　using　a　greedy　strategy　employed　in　the　existing　state-of-the-art　2DPCA　algorithms.　The　proposed　algorithm　has　been　assessed　on　the　tasks　of　image　reconstruction　and　classification.　The　experimental　results　demonstrate　that　the　proposed　algorithm　outperforms　two　state-of-the-art　algorithms.　©　2017　IEEE.