2020/04/06 by Razvan Gurău, Gurau, Razvan · 2 citations
Computer Science · Mathematics · Medicine · #60B99 #Advanced Neuroimaging Techniques and Applications #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2004.02660
openalex publication_date 2020/04/06 · openalex created_date 2020/04/10 · openalex updated_date 2026/07/28
We propose a simple generalization of the matrix resolvent to a resolvent for real symmetric tensors T∈ ⊗p ℝN of order p≥ 3. The tensor resolvent yields an integral representation for a class of tensor invariants and its singular locus can be understood in terms of the real eigenvalues of tensors. We then consider a random Gaussian (real symmetric) tensor. We show that in the large N limit the expected resolvent has a finite cut in the complex plane and that the associated "spectral density", that is the discontinuity at the cut, obeys a universal law which generalizes the Wigner semicircle law to arbitrary order. Finally, we consider a spiked tensor for p≥ 3, that is the sum of a fixed tensor b v⊗ p with v∈ ℝN (the signal) and a random Gaussian tensor T (the noise). We show that in the large N limit the expected resolvent undergoes a sharp transition at some threshold value of the signal to noise ratio b which we compute analytically.