Related papers: Random Matrix Theory by the Supersymmetry Method B…
Finding constrained saddle points on Riemannian manifolds is significant for analyzing energy landscapes arising in physics and chemistry. Existing works have been limited to special manifolds that admit global regular level-set…
We suggest a novel nonlinear $\sigma$-model for the description of disordered superconductors. The main distinction from existing models lies in the fact that the saddle point equation is solved non-perturbatively in the superconducting…
By calculating all terms of the high-density expansion of the euclidean random matrix theory (up to second-order in the inverse density) for the vibrational spectrum of a topologically disordered system we show that the low-frequency…
A substantial body of work in machine learning (ML) and randomized numerical linear algebra (RandNLA) has exploited various sorts of random sketching methodologies, including random sampling and random projection, with much of the analysis…
Excited electronic states of molecules and solids play a fundamental role in fields such as catalysis and electronics. In electronic structure calculations, excited states typically correspond to saddle points on the surface described by…
The proliferation of saddle points, rather than poor local minima, is increasingly understood to be a primary obstacle in large-scale non-convex optimization for machine learning. Variable elimination algorithms, like Variable Projection…
We study a model of spinless fermions with infinite nearest-neighbor repulsion on the square ladder which has microscopic supersymmetry. It has been conjectured that in the continuum the model is described by the superconformal minimal…
A deep connection has been recently established between soft theorems and symmetries at null infinity in gravity and gauge theories, recasting the former as Ward identities of the latter. In particular, different orders (in the frequency of…
Random contractions (sub-unitary random matrices) appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with discrete time. We analyze statistical properties of complex…
We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results…
We review the ideas of how random matrix theory has to be properly applied to quantum physics; particularly we focus on how the spectrum has to be properly prepared and the random matrix correctly identified before the random matrix and the…
Random matrix theory (RMT) is based on two assumptions: (1) matrix-element independence, and (2) base invariance. Most of the proposed generalizations keep the first assumption and violate the second. Recently, several authors presented…
Greedy first-order methods, such as coordinate descent with Gauss-Southwell rule or matching pursuit, have become popular in optimization due to their natural tendency to propose sparse solutions and their refined convergence guarantees. In…
This work is on constrained large-scale non-convex optimization where the constraint set implies a manifold structure. Solving such problems is important in a multitude of fundamental machine learning tasks. Recent advances on Riemannian…
We consider minimizing a nonconvex, smooth function $f$ on a Riemannian manifold $\mathcal{M}$. We show that a perturbed version of Riemannian gradient descent algorithm converges to a second-order stationary point (and hence is able to…
Investigating the dynamics of learning in machine learning algorithms is of paramount importance for understanding how and why an approach may be successful. The tools of physics and statistics provide a robust setting for such…
Training Neural Networks (NNs) without overfitting is difficult; detecting that overfitting is difficult as well. We present a novel Random Matrix Theory method that detects the onset of overfitting in deep learning models without access to…
We establish the existence of fixed points for certain gauge theories candidate to be magnetic duals of QCD with one adjoint Weyl fermion. In the perturbative regime of the magnetic theory the existence of a very large number of fixed…
Nonconvex optimization problems such as the ones in training deep neural networks suffer from a phenomenon called saddle point proliferation. This means that there are a vast number of high error saddle points present in the loss function.…
We extend recent remarkable progress in the comparison of the dynamical energy spectrum of rotating closed strings in AdS_5xS^5 and the scaling weights of the corresponding non-near-BPS operators in planar N=4 supersymmetric gauge theory.…