Singular value decomposition, Dimensionality reduction, Weighted graph, Generalized eigenvalue problem, Data processing

In this contribution, we focus on extending the Laplacian processing used in data-driven dimensionality reduction based on weighted graphs by incorporating the concept of singular value decomposition. We indicate a novel point of view on generalized eigenvalue problem by pointing out geometric meaning of factorization matrices. We demonstrate that classical eigenvalue problem of normalized Laplacian, generalized eigenvalue problem of pure Laplacian and singular value decomposition of specific altered Laplacian form are mutually equivalent problems and discuss some of its theoretical implications.