Eigenvalues of block structured asymmetric random matrices

Abstract

We study the spectrum of an asymmetric random matrix with block structured variances. The rows and columns of the random square matrix are divided into $D$ partitions with arbitrary size (linear in $N$). The parameters of the model are the variances of elements in each block, summarized in $g\in\mathbb{R}^{D\times D}_+$. Using the Hermitization approach and by studying the matrix-valued Stieltjes transform we show that these matrices have a circularly symmetric spectrum, we give an explicit formula for their spectral radius and a set of implicit equations for the full density function. We discuss applications of this model to neural networks

Low-dimensional dynamics of structured random networks

“…Adapting the arguments of [32] in Sec. 5, as done in [11], it can be shown the resolvent of random matrices with independent and non identically distributed entries is also close to $-\frac{1}{z}I$ outside the support of the bulk spectrum. Hence, outlying eigenvalues will fluctuate around the deterministic low-rank eigenvalues.…”

“…The two limitations of our previous studies [11, 13] for interpreting multi-unit recordings are (1) that the cell-type identity of each neuron in the network has to be known and (2) that predictions are averaged over all neurons of a specific type.…”

“…In [11] we used random matrix theory techniques to derive, for the case of block-structured J , an implicit equation that the full spectral density of J satisfies. The circular symmetry of the spectrum for that case is obvious because the equations [see Eq.…”

“…The circular symmetry of the spectrum for that case is obvious because the equations [see Eq. (3.6) in [11]] depend on the complex variable z only through | z | 2 . Similar implicit equations, with integrals instead of sums, can be written for the continuous case.…”

Location of the spectrum of Kronecker random matrices

“…Using the quadratic large deviation bounds (C.4) and (C.3) in [21], we obtain |d (4) i | + |d (5) i | + |d…”