Symmetry of Bound and Antibound States in the Semiclassical Limit for a General Class of Potentials

We consider the semiclassical Schr\"odinger operator $-h^2\partial_x^2+V(x)$ on a half-line, where $V$ is a compactly supported potential which is positive near the endpoint of its support. We prove that the eigenvalues and the purely imaginary resonances are symmetric up to an error $Ce^{-\delta/h}$.Comment: 10 pages, 1 figur

Learning from DPPs via Sampling: Beyond HKPV and symmetry

Determinantal point processes (DPPs) have become a significant tool for recommendation systems, feature selection, or summary extraction, harnessing the intrinsic ability of these probabilistic models to facilitate sample diversity. The ability to sample from DPPs is paramount to the empirical investigation of these models. Most exact samplers are variants of a spectral meta-algorithm due to Hough, Krishnapur, Peres and Vir\'ag (henceforth HKPV), which is in general time and resource intensive. For DPPs with symmetric kernels, scalable HKPV samplers have been proposed that either first downsample the ground set of items, or force the kernel to be low-rank, using e.g. Nystr\"om-type decompositions. In the present work, we contribute a radically different approach than HKPV. Exploiting the fact that many statistical and learning objectives can be effectively accomplished by only sampling certain key observables of a DPP (so-called linear statistics), we invoke an expression for the Laplace transform of such an observable as a single determinant, which holds in complete generality. Combining traditional low-rank approximation techniques with Laplace inversion algorithms from numerical analysis, we show how to directly approximate the distribution function of a linear statistic of a DPP. This distribution function can then be used in hypothesis testing or to actually sample the linear statistic, as per requirement. Our approach is scalable and applies to very general DPPs, beyond traditional symmetric kernels

Dictionary Learning under Symmetries via Group Representations

The dictionary learning problem can be viewed as a data-driven process to learn a suitable transformation so that data is sparsely represented directly from example data. In this paper, we examine the problem of learning a dictionary that is invariant under a pre-specified group of transformations. Natural settings include Cryo-EM, multi-object tracking, synchronization, pose estimation, etc. We specifically study this problem under the lens of mathematical representation theory. Leveraging the power of non-abelian Fourier analysis for functions over compact groups, we prescribe an algorithmic recipe for learning dictionaries that obey such invariances. We relate the dictionary learning problem in the physical domain, which is naturally modelled as being infinite dimensional, with the associated computational problem, which is necessarily finite dimensional. We establish that the dictionary learning problem can be effectively understood as an optimization instance over certain matrix orbitopes having a particular block-diagonal structure governed by the irreducible representations of the group of symmetries. This perspective enables us to introduce a band-limiting procedure which obtains dimensionality reduction in applications. We provide guarantees for our computational ansatz to provide a desirable dictionary learning outcome. We apply our paradigm to investigate the dictionary learning problem for the groups SO(2) and SO(3). While the SO(2)-orbitope admits an exact spectrahedral description, substantially less is understood about the SO(3)-orbitope. We describe a tractable spectrahedral outer approximation of the SO(3)-orbitope, and contribute an alternating minimization paradigm to perform optimization in this setting. We provide numerical experiments to highlight the efficacy of our approach in learning SO(3)-invariant dictionaries, both on synthetic and on real world data.Comment: 29 pages, 2 figure

Rigidity Phenomena in random point sets

Let X be a translation invariant point process on the Euclidean space E and let D, a subset of E, be a bounded open set whose boundary has zero Lebesgue measure. We ask what does the point configuration O obtained by taking the points of X outside D tell us about the point configuration I obtained from the points of X inside D? We focus mainly on translation invariant point processes on the plane. We show that for the Ginibre ensemble, O determines the number of points in I. We refer to this kind of behaviour as ``rigidity''. For the translation-invariant zero process of a planar Gaussian Analytic Function, we show that O determines the number as well as the centre of mass of the points in I. Further, in both models we prove that the outside says ``nothing more'' about the inside, in the sense that the conditional distribution of the inside points, given the outside, is mutually absolutely continuous with respect to the Lebesgue measure on its supporting submanifold. We further show that the conditional density (of the inside points given the outside) is, roughly speaking, comparable to a squared Vandermonde density. In particular, this shows that even under spatial conditioning, the points exhibit repulsion which is quadratic in their mutual separation. We apply these results to the study of continuum percolation on these point processes, and establish the existence of a non-trivial critical radius and the uniqueness of infinite cluster in the supercritical regime. En route, we obtain new estimates on hole probabilities for zeroes of the planar Gaussian Analytic Function. Finally, we apply these ideas to prove completeness properties of random exponential functions originating from "rigid" determinantal point processes. We conclude by establishing miscellaneous other results on determinantal point processes. These include answers to two questions of Lyons and Steif on certain models of stationary determinantal processes on the integers, one involving insertion and deletion tolerance, and the other regarding the recovery of the driving function from the process