Attempt at a short summary: A matrix M has an SVD U S V', where U and V have orthonormal columns and S is a diagonal matrix. How does this generalize to multid

Attempt at a short summary: A matrix M has an SVD U S V', where U and V have orthonormal columns and S is a diagonal matrix. How does this generalize to multidimensional arrays? A third-order tensor T has a multilinear SVD of the form S x (U, V, W), where S is a core tensor and U, V, and W are again orthonormal matrices. In the matrix SVD, U transforms the columns of S, while V transforms its rows (hence it appears on the right of S). The product S x (U, V, W) is similar: U transforms the columns of the core tensor S, V its rows and W the mode-3 vectors of S. The authors show that slices of S must be orthogonal under the inner product, e.g., <S(:,i,:), S(:,j,:)> = c * delta(i,j). In the case of a matrix SVD, this reduces to S being diagonal. An example application is predicting how a user would rate a movie at a given time by approximating (the known entries of) a user x movie x time tensor with a truncated multilinear SVD.