论文标题
关于Wigner半圆定律对真实对称张量的概括
On the generalization of the Wigner semicircle law to real symmetric tensors
论文作者
论文摘要
我们提出了将矩阵分解到分解为真实对称张量的简单概括$ t \ in \ otimes^p \ mathbb {r}^n $的订单$ p \ ge 3 $。张量的分解产生了一类张量不变式的组成表示，并且可以从张量的真实特征值来理解其奇异基因座。然后，我们考虑一个随机的高斯（真实对称）张量。我们表明，在大的$ n $中，预期的分解在复杂的平面中具有有限的削减，并且相关的“光谱密度”（即削减处的不连续性）遵守一项通用法律，该法律将Wigner Semicircle Law推广到任意命令。最后，我们考虑了$ p \ ge 3 $的尖刺张量，这是固定张量$ b \，v^{\ otimes p} $，带有$ v \ in \ mathbb {r}^n $（信号）和随机的高斯张量tensor $ t $（noise）。我们表明，在大的$ n $中，预期的分解会以信号比率$ b $的一定阈值阈值$ b $进行急剧过渡。
We propose a simple generalization of the matrix resolvent to a resolvent for real symmetric tensors $T\in \otimes^p \mathbb{R}^N$ of order $p\ge 3$. 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 $p\ge 3$, that is the sum of a fixed tensor $b\,v^{\otimes p}$ with $v\in \mathbb{R}^N$ (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.
