Abstract: The injective norm is a natural generalization to tensors of the operator norm of a matrix. It is also of interest in quantum information, where it measures the genuine multipartite entanglement of quantum states, and in spin glasses, where it corresponds to the ground state of a spherical spin glass. Random-matrix operator norms are frequently studied with the moment method, which is based on deterministic relationships between operator norms and traces which do not obviously generalize to tensors. We present a new kind of "moment method for tensors" that circumvents this difficulty and gives upper bounds on the expected injective norm of real and complex random tensors. Relative to prior approaches, the moment method has the benefit of being nonasymptotic, relatively elementary, and applicable to non-Gaussian models, while giving bounds that are sometimes tight. Joint work with Stephane Dartois.
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