Multi-way clustering using super-symmetric non-negative tensor factorization

We consider the problem of clustering data into k >= 2 clusters given complex relations - going beyond pairwise - between the data points. The complex n-wise relations are modeled by an n-way array where each entry corresponds to an affinity measure over an n-tuple of data points. We show that a probabilistic assignment of data points to clusters is equivalent, under mild conditional independence assumptions, to a super-symmetric non-negative factorization of the closest hyper-stochastic version of the input n-way affinity array. We derive an algorithm for finding a local minimum solution to the factorization problem whose computational complexity is proportional to the number of n-tuple samples drawn from the data. We apply the algorithm to a number of visual interpretation problems including 3D multi-body segmentation and illumination-based clustering of human faces.