arXiv research

Surveying random sections on Kähler manifolds, leading to metrics.

Study shows mass distribution of random holomorphic sections follows a central limit theorem.

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measu…

The paper equidistributes zeros of random polynomials and sections on manifolds.

Study shows normal distribution in divisor counts of random sections on complex manifolds.

The paper estimates variance of random sections on complex manifolds.

In this work we prove an universality result regarding the equidistribution of zeros of random holomorphic sections associated to a sequence of singular Hermitian holomorphic line bundles on a compact Kähler complex space $X$. Namely, under mild moment assumptions, we show that the asymptotic distribution of zeros of r…

This thesis predicts the distribution of smoothed zeros of random sections on line bundles.

The paper connects bundle curvature to random zero currents.

Tian's theorem connects Chern classes of bundles to random section zeros and degeneracy sets.

Study Bergman kernels and zero distributions of random sections on Kähler manifolds.

Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.

The paper studies random systems of holomorphic sections on compact Kähler manifolds and proves equidistribution results.

This is a technical report which explores the estimation methodologies on hyper-parameters in Markov Random Field and Gaussian Hidden Markov Random Field. In first section, we briefly investigate a theoretical framework on Metropolis-Hastings algorithm. Next, by using MH algorithm, we simulate the data from Ising model…

Study of random sections on complex spaces converging to equilibrium metrics.

We prove that the Euler form of a metric connection on real oriented vector bundle $E$ over a compact oriented manifold $M$ can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilisticall…

The sectional curvature of a compact Riemannian manifold M can be seen as a random variable on the Grassmann bundle of 2-planes in TM endowed with the Fubini-Study volume density. In this article we calculate the moments of this random variable by integrating suitable local Riemannian invariants and discuss the distrib…

The study proves a central limit theorem for Gaussian holomorphic sections on Kähler manifolds.

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle over compact manifold canonically defines a metric on the bundle together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet-Chern theorem stating that if the bundle and the manifold are oriented, then t…

We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.

We extend Kac-Rice formula to compute expected intersections of random submanifolds.

Develops calculus for random submanifolds using zonoids.

Geometric quantization results for Riemann surfaces with semi-positive line bundles.

GMMNs model cross-sectional dependence for better option pricing and simulation.

Linear statistics of random zero sets are integrals of smooth differential forms over the zero set and as such are smooth analogues of the volume of the random zero set inside a fixed domain. We derive an asymptotic expansion for the variance of linear statistics of the zero divisors of random holomorphic sections of p…

We study the asymptotics of Fubini-Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kaehler complex space.

We give, as $L$ grows to infinity, an explicit lower bound of order $L^{n/m}$ for the expected Betti numbers of the vanishing locus of a random linear combination of eigenvectors of $P$ with eigenvalues below $L$. Here, $P$ denotes an elliptic self-adjoint pseudo-differential operator of order $m\textgreater{}0$, bound…

We study a tower of normal coverings over a compact Kähler manifold with holomorphic line bundles. When the line bundle is sufficiently positive, we obtain an effective estimate, which implies the Bergman stability. As a consequence, we deduce the equidistribution for zero currents of random holomorphic sections. Furth…

Study compares Islamic banks' accounting and market performance.

Fold maps associated to geodesic random walks on curved spaces.

We prove geometric and cohomological stabilization results for the universal smooth degree $d$ hypersurface section of a fixed smooth projective variety as $d$ goes to infinity. We show that relative configuration spaces of the universal smooth hypersurface section stabilize in the completed Grothendieck ring of variet…

The paper introduces tests for missing data models based on graph assumptions.

We show that integration over a $G$-manifold $M$ can be reduced to integration over a minimal section $Σ$ with respect to an induced weighted measure and integration over a homogeneous space $G/N$. We relate our formula to integration formulae for polar actions and calculate some weight functions. In case of a compact …

We build a simple diagnostic criterion for approximate factor structure in large cross-sectional equity datasets. Given a model for asset returns with observable factors, the criterion checks whether the error terms are weakly cross-sectionally correlated or share at least one unobservable common factor. It only requir…

The study examines spaces of holomorphic sections vanishing along subvarieties in complex spaces.

We show that normalized currents of integration along the common zeros of random $m$-tuples of sections of powers of $m$ singular Hermitian big line bundles on a compact Kähler manifold distribute asymptotically to the wedge product of the curvature currents of the metrics. If the Hermitian metrics are Hölder with sing…

Spatial econometric research typically relies on the assumption that the spatial dependence structure is known in advance and is represented by a deterministic spatial weights matrix. Contrary to classical approaches, we investigate the estimation of sparse spatial dependence structures for regular lattice data. In par…

A novel hybrid data-driven approach is developed for forecasting power system parameters with the goal of increasing the efficiency of short-term forecasting studies for non-stationary time-series. The proposed approach is based on mode decomposition and a feature analysis of initial retrospective data using the Hilber…

Cross-sectional "Information Coefficient" (IC) is a widely and deeply accepted measure in portfolio management. The paper gives an insight into IC in view of high-dimensional directional statistics: IC is a linear operator on the components of a centralizing-unitizing standardized random vector of next-period cross-sec…

Let $X$ be a compact normal complex space of dimension $n$, and $L$ be a holomorphic line bundle on $X$. Suppose $Σ=(Σ_1,\ldots,Σ_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $τ=(τ_1,\ldots,τ_\ell)$ is an $\ell$-tuple of positive real numbers, and consider the space $H^0_0 (X, L^p)$…

The paper characterizes the geometry and topology of spin random fields.

This paper proposes asal, a new GAN based active learning method that generates high entropy samples. Instead of directly annotating the synthetic samples, ASAL searches similar samples from the pool and includes them for training. Hence, the quality of new samples is high and annotations are reliable. To the best of o…

The paper studies the distribution of random degeneracy sets on complex manifolds.

We discuss positive closed currents and Fubini-Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini-Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curva…

High-performing equity factor with Sharpe ratio above 13 out-of-sample.

We prove a motivic stabilization result for the cohomology of the local systems on configuration spaces of varieties over $\mathbb{C}$ attached to character polynomials. Our approach interprets the stabilization as a probabilistic phenomenon based on the asymptotic independence of certain *motivic random variables*, an…

Geometrically proves majorizing measure theorem on Hadamard manifolds.

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 structure on the space of global holomorphic sections on Y with values in the k…