Matrix　factorization　has　always　been　an　encouraging　field,　which　attempts　to　extract　discriminative　features　from　high-dimensional　data.　However,　it　suffers　from　negative　generalization　ability　and　high　computational　complexity　when　handling　large-scale　data.　In　this　paper,　we　propose　a　learnable　deep　matrix　factorization　via　the　projected　gradient　descent　method,　which　learns　multi-layer　low-rank　factors　from　scalable　metric　distances　and　flexible　regularizers.　Accordingly,　solving　a　constrained　matrix　factorization　problem　is　equivalently　transformed　into　training　a　neural　network　with　an　appropriate　activation　function　induced　from　the　projection　onto　a　feasible　set.　Distinct　from　other　neural　networks,　the　proposed　method　activates　the　connected　weights　not　just　the　hidden　layers.　As　a　result,　it　is　proved　that　the　proposed　method　can　learn　several　existing　well-known　matrix　factorizations,　including　singular　value　decomposition,　convex,　nonnegative　and　semi-nonnegative　matrix　factorizations.　Finally,　comprehensive　experiments　demonstrate　the　superiority　of　the　proposed　method　against　other　state-of-the-arts.　©　2023　Elsevier　Ltd