Embedding Super-Symmetric Tensors of Higher-Order Similarities of High-Dimensional Data

In this paper we propose an algorithm for non-linear embedding of affinity tensors obtained by measuring higher-order similarities between high-dimensional points. We achieve this by preserving the original triadic similarities using another triadic similarity function obtained by sum of squares of diadic similarities in a low-dimension. We show that this formulation reduces to solving for the nonlinear embedding of a graph which has a specific kind of a graph Laplacian. We provide an iterative algorithm for minimizing the loss, and also propose a simple linear-constraint that prevents non-zero solutions for embedding problems unlike the existing variants of quadratic orthonormality constraints used in the literature, that require eigen decompositions to solve for the embedding.