Finding the smallest or largest element of a tensor from its low-rank factors
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We consider the problem of finding the smallest or largest entry of a tensor of order N that is specified via its rank decomposition. Stated in a different way, we are given N sets of R-dimensional vectors and we wish to select one vector from each set such that the sum of the Hadamard product of the selected vectors is minimized or maximized. This is a fundamental tensor problem with numerous applications in embedding similarity search, recommender systems, graph mining, multivariate probability, and statistics. We show that this discrete optimization problem is NP-hard for any tensor rank higher than one, but also provide an equivalent continuous problem reformulation which is amenable to disciplined non-convex optimization. We propose a suite of gradient-based approximation algorithms whose performance in preliminary experiments appears to be promising.
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