Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach

Using large $N$ arguments, we propose a scheme for calculating the two-point eigenvector correlation function for non-normal random matrices in the large $N$ limit. The setting generalizes the quaternionic extension of free probability to two-point functions. In the particular case of biunitarily invariant random matrices, we obtain a simple, general expression for the two-point eigenvector correlation function, which can be viewed as a further generalization of the single ring theorem. This construction has some striking similarities to the freeness of the second kind known for the Hermitian ensembles in large $N$. On the basis of several solved examples, we conjecture two kinds of microscopic universality of the eigenvectors - one in the bulk, and one at the rim. The form of the conjectured bulk universality agrees with the scaling limit found by Chalker and Mehlig [JT Chalker, B Mehlig, PRL, \textbf{81}, 3367 (1998)] in the case of the complex Ginibre ensemble.Comment: 20 pages + 4 pages of references, 12 figs; v2: typos corrected, refs added; v3: more explanator

Spectra of large time-lagged correlation matrices from Random Matrix Theory

We analyze the spectral properties of large, time-lagged correlation matrices using the tools of random matrix theory. We compare predictions of the one-dimensional spectra, based on approaches already proposed in the literature. Employing the methods of free random variables and diagrammatic techniques, we solve a general random matrix problem, namely the spectrum of a matrix $\frac{1}{T}XAX^{\dagger}$, where $X$ is an $N\times T$ Gaussian random matrix and $A$ is \textit{any} $T\times T$, not necessarily symmetric (Hermitian) matrix. As a particular application, we present the spectral features of the large lagged correlation matrices as a function of the depth of the time-lag. We also analyze the properties of left and right eigenvector correlations for the time-lagged matrices. We positively verify our results by the numerical simulations.Comment: 44 pages, 11 figures; v2 typos corrected, final versio

From synaptic interactions to collective dynamics in random neuronal networks models: critical role of eigenvectors and transient behavior

The study of neuronal interactions is currently at the center of several neuroscience big collaborative projects (including the Human Connectome, the Blue Brain, the Brainome, etc.) which attempt to obtain a detailed map of the entire brain matrix. Under certain constraints, mathematical theory can advance predictions of the expected neural dynamics based solely on the statistical properties of such synaptic interaction matrix. This work explores the application of free random variables (FRV) to the study of large synaptic interaction matrices. Besides recovering in a straightforward way known results on eigenspectra of neural networks, we extend them to heavy-tailed distributions of interactions. More importantly, we derive analytically the behavior of eigenvector overlaps, which determine stability of the spectra. We observe that upon imposing the neuronal excitation/inhibition balance, although the eigenvalues remain unchanged, their stability dramatically decreases due to strong non-orthogonality of associated eigenvectors. It leads us to the conclusion that the understanding of the temporal evolution of asymmetric neural networks requires considering the entangled dynamics of both eigenvectors and eigenvalues, which might bear consequences for learning and memory processes in these models. Considering the success of FRV analysis in a wide variety of branches disciplines, we hope that the results presented here foster additional application of these ideas in the area of brain sciences.Comment: 24 pages + 4 pages of refs, 8 figure

Narain transform for spectral deformations of random matrix models

We start from applying the general idea of spectral projection (suggested by Olshanski and Borodin and advocated by Tao) to the complex Wishart model. Combining the ideas of spectral projection with the insights from quantum mechanics, we derive in an effortless way all spectral properties of the complex Wishart model: first, the Marchenko-Pastur distribution interpreted as a Bohr-Sommerfeld quantization condition for the hydrogen atom; second, hard (Bessel), soft (Airy) and bulk (sine) microscopic kernels from properly rescaled radial Schrödinger equation for the hydrogen atom. Then, generalizing the ideas based on Schrödinger equation to the case when Hamiltonian is non-Hermitian, we propose an analogous construction for spectral projections of universal kernels for bi-orthogonal ensembles. In particular, we demonstrate that the Narain transform is a natural extension of the Hankel transform for the products of Wishart matrices, yielding an explicit form of the universal kernel at the hard edge. We also show how the change of variables of the rescaled kernel allows us to make the link to the universal kernel of the Muttalib-Borodin ensemble. The proposed construction offers a simple alternative to standard methods of derivation of microscopic kernels. Finally, we speculate, that a suitable extension of the Bochner theorem for Sturm-Liouville operators may provide an additional insight into the classification of microscopic universality classes in random matrix theory

Risk of lymph node metastases in multifocal papillary thyroid cancer associated with Hashimoto's thyroiditis

AIMS: The aim of this study was to evaluate the risk factors of lymph nodes metastases (LNM) in patients with papillary thyroid cancer (PTC) and coexisting Hashimoto’s thyroiditis (HT). PATIENTS AND METHODS: This was a retrospective cohort study of patients with PTC and HT who had undergone total thyroidectomy (TT) with central neck dissection (CND) over an 11-year period (between 2002 and 2012). Pathological reports of all eligible patients were reviewed. Multivariable analysis was performed to identify risk factors of LNM. RESULTS: During the study period, PTC was diagnosed in 130 patients with HT who had undergone TT with CND (F/M ratio = 110:20; median age, 52.4 ± 12.7 years). Multifocal lesions were observed in 28 (21.5 %) patients. LNM were identified in 25 of 28 (89.3 %) patients with multifocal PTC and HT versus 69 of 102 (67.5 %) patients with a solitary focus of PTC and HT (p = 0.023). In multivariable analysis, multifocal disease was identified as an independent risk factor for LNM (odds ratio, 3.99; 95 % confidence interval, 1.12 to 14.15; p = 0.033). CONCLUSIONS: Multifocal PTC in patients with HT is associated with an increased risk of LNM. Nevertheless, the clinical importance of this finding needs to be validated in well-designed prospective studies

Unveiling the significance of eigenvectors in diffusing non-hermitian matrices by identifying the underlying Burgers dynamics

Following our recent letter, we study in detail an entry-wise diffusion of non-hermitian complex matrices. We obtain an exact partial differential equation (valid for any matrix size $N$ and arbitrary initial conditions) for evolution of the averaged extended characteristic polynomial. The logarithm of this polynomial has an interpretation of a potential which generates a Burgers dynamics in quaternionic space. The dynamics of the ensemble in the large $N$ is completely determined by the coevolution of the spectral density and a certain eigenvector correlation function. This coevolution is best visible in an electrostatic potential of a quaternionic argument built of two complex variables, the first of which governs standard spectral properties while the second unravels the hidden dynamics of eigenvector correlation function. We obtain general large $N$ formulas for both spectral density and 1-point eigenvector correlation function valid for any initial conditions. We exemplify our studies by solving three examples, and we verify the analytic form of our solutions with numerical simulations.Comment: 24 pages, 11 figure

Dynamical Isometry is Achieved in Residual Networks in a Universal Way for any Activation Function

We demonstrate that in residual neural networks (ResNets) dynamical isometry is achievable irrespectively of the activation function used. We do that by deriving, with the help of Free Probability and Random Matrix Theories, a universal formula for the spectral density of the input-output Jacobian at initialization, in the large network width and depth limit. The resulting singular value spectrum depends on a single parameter, which we calculate for a variety of popular activation functions, by analyzing the signal propagation in the artificial neural network. We corroborate our results with numerical simulations of both random matrices and ResNets applied to the CIFAR-10 classification problem. Moreover, we study the consequence of this universal behavior for the initial and late phases of the learning processes. We conclude by drawing attention to the simple fact, that initialization acts as a confounding factor between the choice of activation function and the rate of learning. We propose that in ResNets this can be resolved based on our results, by ensuring the same level of dynamical isometry at initialization