Related papers: Relative concentration bounds for the spectrum of …
We study concentration operators associated with either the discrete or the continuous Fourier transform, that is, operators that incorporate a spatial cut-off and a subsequent frequency cut-off to the Fourier inversion formula. Their…
We generalize finite-sample bounds for convex clustering to the setting where affinity weights appearing in the objective correspond to a general connected graph. These bounds and their analysis lead to a better understanding of clustering…
A method of resummation of infinite series of perturbation theory diagrams is applied for studying the properties of random band matrices. The topological classification of Feynman diagrams, which was actively used in last years for matrix…
Complex networks are characterized by heterogeneous distributions of the degree of nodes, which produce a large diversification of the roles of the nodes within the network. Several centrality measures have been introduced to rank nodes…
Consider any random graph model where potential edges appear independently, with possibly different probabilities, and assume that the minimum expected degree is omega(ln n). We prove that the adjacency matrix and the Laplacian of that…
We study numerically the spectrum and eigenstate properties of the Google matrix of various examples of directed networks such as vocabulary networks of dictionaries and university World Wide Web networks. The spectra have gapless structure…
We present simple, user-friendly bounds for the expected operator norm of a random kernel matrix under general conditions on the kernel function $k(\cdot,\cdot)$. Our approach uses decoupling results for U-statistics and the non-commutative…
Understanding the spectral properties of kernels offers a principled perspective on generalization and representation quality. While deep models achieve state-of-the-art accuracy in molecular property prediction, kernel methods remain…
Statistical properties of non--symmetric real random matrices of size $M$, obtained as truncations of random orthogonal $N\times N$ matrices are investigated. We derive an exact formula for the density of eigenvalues which consists of two…
This paper focuses on large neural networks whose synaptic connectivity matrices are randomly chosen from certain random matrix ensembles. The dynamics of these networks can be characterized by the eigenvalue spectra of their connectivity…
How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite…
This paper studies the concentration properties of random codes. Specifically, we show that, for discrete memoryless channels, the error exponent of a randomly generated code with pairwise-independent codewords converges in probability to…
We study largest singular values of large random matrices, each with mean of a fixed rank $K$. Our main result is a limit theorem as the number of rows and columns approach infinity, while their ratio approaches a positive constant. It…
We derive concentration inequalities for the supremum norm of the difference between a kernel density estimator (KDE) and its point-wise expectation that hold uniformly over the selection of the bandwidth and under weaker conditions on the…
Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to…
We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as random, real, asymmetric matrices with a special…
Low-rank approximation of kernels is a fundamental mathematical problem with widespread algorithmic applications. Often the kernel is restricted to an algebraic variety, e.g., in problems involving sparse or low-rank data. We show that…
The investigation of universality questions for local eigenvalue statistics continues to be a driving force in the theory of Random Matrices. For Matrix Models [53] the method of orthogonal polynomials can be used and the asymptotics of the…
We study the properties of various over-parametrized convolutional neural architectures through their respective Gaussian process and neural tangent kernels. We prove that, with normalized multi-channel input and ReLU activation, the…
We use the steepest descents method to study the integral kernel of a family of normal random matrix ensembles with eigenvalue distribution P_{N}(z_{1},...,z_{N}) = Z_{N}^{-1} e^{-N\Sigma_{i=1}^{N}V_{\alpha}(z_{i})}…