Dimension　reduction　for　high-order　tensors　is　a　challenging　problem.　In　conventional　approaches,　dimension　reduction　for　higher　order　tensors　is　implemented　via　Tucker　decomposition　to　obtain　lower　dimensional　tensors.　This　paper　introduces　a　probabilistic　vectorial　dimension　reduction　model　for　tensorial　data.　The　model　represents　a　tensor　by　using　a　linear　combination　of　the　same　order　basis　tensors,　thus　it　offers　a　learning　approach　to　directly　reduce　a　tensor　to　a　vector.　Under　this　expression,　the　projection　base　of　the　model　is　based　on　the　tensor　CandeComp/PARAFAC　(CP)　decomposition　and　the　number　of　free　parameters　in　the　model　only　grows　linearly　with　the　number　of　modes　rather　than　exponentially.　A　Bayesian　inference　has　been　established　via　the　variational　Expectation　Maximization　(EM)　approach.　A　criterion　to　set　the　parameters　(a　factor　number　of　CP　decomposition　and　the　number　of　extracted　features)　is　empirically　given.　The　model　outperforms　several　existing　principal　component　analysis-based　methods　and　CP　decomposition　on　several　publicly　available　databases　in　terms　of　classification　and　clustering　accuracy.