Related papers: Smallest and largest generalized eigenvalues of la…

Let $(X,d)$ be a compact metric space. We consider the behavior of probability measures $\mu$ with the property that $$ \int_{X} d(x, y) d\mu(y) \qquad \mbox{is independent of}~x \in X.$$ It appears that such measures, when they exist,…

Kernel methods are successful approaches for different machine learning problems. This success is mainly rooted in using feature maps and kernel matrices. Some methods rely on the eigenvalues/eigenvectors of the kernel matrix, while for…

This paper pursues a twofold goal. First, we introduce and study in detail a new notion of variational analysis called generalized metric subregularity, which is a far-going extension of the conventional metric subregularity conditions. Our…

In this paper we bring to light an unprecedented property of the eigenvalues of a matrix A with the eigenvalues and eigenvectors of a submatrix of A. This property can be used, through the technique developed here, to determine some of…

A universal energy eigenvalue equation is proposed in this paper. It is proven that the unique set of eigenfunctions or preferred basis exists for any non-isolated sub-system. Applying the new eigenvalue equation to the relative motion of a…

This short note studies the fluctuations of the largest eigenvalue of symmetric random matrices with correlated Gaussian entries having positive mean. Under the assumption that the covariance kernel is absolutely summable, it is proved that…

We present three methods to construct majorizing measures in various settings. These methods are based on direct constructions of increasing sequences of partitions through a simple exhaustion procedure rather than on the construction of…

We generalize the measurement using an expanded concept of cover, in order to provide a new approach to size of set other than cardinality. The generalized measurement has application backgrounds such as a generalized problem in dimension…

We propose a definition of magnitude for a length space with a Borel measure, which involves integrals over the set of geodesics. This quantity agrees with the magnitude of finite metric spaces, up to re-scaling the metric to ensure the…

We consider the distance from a (square or rectangular) matrix pencil to the nearest matrix pencil in 2-norm that has a set of specified eigenvalues. We derive a singular value optimization characterization for this problem and illustrate…

We study various generalizations of concentration of measure on the unit sphere, in particular by means of log-Sobolev inequalities. First, we show Sudakov-type concentration results and local semicircular laws for weighted random matrices.…

Let X be a strictly pseudoconcave domain in a closed polarized complex manifold (Y,L) where L is a (semi-)positive line bundle over Y. Any given Hermitian metric on L, together with a volume form, induces by restriction to X a Hilbert space…

Let m be a unidimensional measure with dimension d. A natural question is to ask if the measure m is comparable with the Hausdorff measure (or the packing measure) in dimension d. We give an answer (which is in general negative) to this…

Let $L$ be a linear operator on univariate polynomials of bounded degree taking values in real symmetric matrices, whose moment matrix is positive semidefinite. Assume that $L$ admits a positive matrix-valued representing measure $\mu$. Any…

We develop tools to study arithmetically induced singular continuous spectrum in the neighborhood of the arithmetic transition in the hyperbolic regime. This leads to first transition-capturing upper bounds on packing and multifractal…

The approximation of probability measures on compact metric spaces and in particular on Riemannian manifoldsby atomic or empirical ones is a classical task in approximation and complexity theory with a wide range of applications. Instead of…

We obtain the explicit rate of convergence $N^{-1/2 + \epsilon}$ for the gaps of generalized Wigner matrices in the bulk of the spectrum, for distributions of matrix entries possibly atomic and supported on enough points. The proof proceeds…

Eigenmaps are important in analysis, geometry, and machine learning, especially in nonlinear dimension reduction. Approximation of the eigenmaps of a Laplace operator depends crucially on the scaling parameter $\epsilon$. If $\epsilon$ is…

Pickrell has fully characterized the unitarily invariant probability measures on infinite Hermitian matrices, and an alternative proof of this classification has been found by Olshanski and Vershik. Borodin and Olshanski deduced from this…

Using standard tools of harmonic analysis, we state and solve the problem of moments for non-negative measures supported on the unit ball of a Sobolev space of multivariate periodic trigonometric functions. We describe outer and inner…