Study shows normal distribution in divisor counts of random sections on complex manifolds.
problem Distribution of divisors on complex manifolds.
method Central limit theorem for smooth linear statistics of Gaussian sections.
result Asymptotic normality of divisor counts.
Study on zeros of Gaussian sections on semipositive line bundles on punctured Riemann surfaces.
problem Distribution of zeros of Gaussian sections on semipositive line bundles.
method Analysis of Bergman kernels and random zeros in high tensor powers.
result Equidistribution, large deviation estimates, central limit theorem, and number variances for zeros in the semi-classical limit.
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…
The study proves a central limit theorem for Gaussian holomorphic sections on Kähler manifolds.
problem Understanding statistical properties of zeros of random holomorphic sections.
method Proves a central limit theorem for smooth linear statistics of zero divisors of Gaussian sections in line bundles over Kähler manifolds.
result Derives first-order asymptotics and upper decay estimates for Bergman kernels.
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…
The paper studies random systems of holomorphic sections on compact Kähler manifolds and proves equidistribution results.
problem Estimating the distribution of zeros of random holomorphic sections on compact Kähler manifolds.
method Asymptotic variance estimate for smooth linear statistics, equidistribution result derivation.
result Smooth positive closed form ω^k can be approximated by currents of integration along analytic subsets of X.
The paper estimates variance of random sections on complex manifolds.
problem Estimating variance of random holomorphic sections on compact Kahler manifolds.
method Analyzes a sequence of smooth Hermitian holomorphic line bundles on a compact Kahler manifold X, considering specific probability measures.
result Provides variance estimates for various measures including Gaussian and Fubini-Study measures.
We extend Kac-Rice formula to compute expected intersections of random submanifolds.
problem Computing expected intersections of random submanifolds.
method Generalized Kac-Rice formula using measure theory and integration.
result Formula computes expected cardinality of preimages of submanifolds via random maps.
Surveying random sections on Kähler manifolds, leading to metrics.
problem Understanding statistics of random sections on Kähler manifolds.
method Analyzing tensor powers of line bundles.
result Induced metrics from random sections.
Study shows mass distribution of random holomorphic sections follows a central limit theorem.
problem Understanding mass distribution of random holomorphic sections.
method Proved a central limit theorem for mass distribution of random holomorphic sections associated with positive line bundles.
result Almost every sequence of random holomorphic sections exhibits quantum ergodicity.
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.
problem Equidistribution of zeros of random polynomials and sections on manifolds.
method Weighted pluripotential theory, asymptotic Bernstein-Markov measures, variance estimation.
result Equidistribution holds for non-i.i.d. random coefficients and non-homogeneous 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 X 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.
problem Predicting the distribution of smoothed zeros of random sections on line bundles.
method Developing smoothing operators on discrete surfaces and computing the expected sum of indices on each face.
result Predictions on the distribution of smoothed section's signed zeros with multiplicity.
The paper connects bundle curvature to random zero currents.
problem Understanding the relationship between bundle curvature and random zero currents.
method Heat flow on Hermitian line bundles over Riemannian manifolds.
result Random zero currents connect bundle curvature to ground state zero current.
Tian's theorem connects Chern classes of bundles to random section zeros and degeneracy sets.
problem Understanding the distribution of zeros and degeneracy sets of random holomorphic sections.
method Analyzing the pullback of Chern classes and computing currents of integration.
result The limit distribution of zeros of random sections is determined by the Chern form.
Study Bergman kernels and zero distributions of random sections on Kähler manifolds.
problem Asymptotic distribution of common zeros of random sections on Kähler manifolds.
method Analysis of Bergman kernels and equidistribution for sequences of line bundles.
result Established asymptotic expansion of Bergman kernels and equidistribution of zeros.
Geometrically proves majorizing measure theorem on Hadamard manifolds.
problem Volume size relation between random process index space and its convex hull.
method Assumed Hadamard manifold, derived upper bound for volume ratio, applied to prove majorizing measure theorem.
result Upper bound for volume ratio between index space and convex hull.
Study of random sections on complex spaces converging to equilibrium metrics.
problem Understanding the behavior of random holomorphic sections on complex spaces.
method Analyzing the convergence of normalized Fubini-Study currents and integration currents to the equilibrium metric's curvature.
result The normalized currents of integration along zero divisors converge almost surely to the curvature current of the equilibrium metric.
We prove that the Euler form of a metric connection on real oriented vector bundle E E E over a compact oriented manifold M M 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…
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.
Orthogonal random features approximate a Bessel kernel, offering sharper bounds than random Fourier features.
problem Approximating Gaussian kernel efficiently for large datasets.
method Use of Haar orthogonal matrices to construct orthogonal random features and analyze their bias and variance.
result Orthogonal random features approximate a Bessel kernel, not the Gaussian kernel, with sharper bounds.
Given a Gaussian Markov random field, we consider the problem of selecting a subset of variables to observe which minimizes the total expected squared prediction error of the unobserved variables. We first show that finding an exact solution is NP-hard even for a restricted class of Gaussian Markov random fields, calle…
Generalizes randomized SVD for better matrix approximations using Gaussian vectors.
problem Computing accurate rank-k approximations of matrices with limited data.
method Extends randomized SVD to multivariate Gaussian vectors, incorporating prior knowledge and using Gaussian processes.
result Demonstrates improved accuracy in approximating matrices and Hilbert-Schmidt operators.
Bayesian approach approximates probability functions of Gaussian mixtures.
problem Approximating probability functions of non-spherical Gaussian mixtures.
method Bayesian decomposition, spherical radial decomposition, random sampling.
result Established differentiability and integral representation of gradient for probability functions.
Bounds on Gaussian approximation for neural networks with novel smoothing techniques.
problem Approximating the distribution of wide random neural networks.
method Stein's method, Gaussian smoothing, Laplacian operators, Cameron-Martin space.
result First bounds on Gaussian approximation of wide random neural networks.
Bounds neural network output distribution to Gaussian for random initialization.
problem Quantifying the distribution of randomly initialized deep neural networks.
method Quantitative Gaussian approximation using quadratic Wasserstein distance.
result Explicit inequalities show how network sizes affect Gaussian behavior.
Study on Gaussian random fields' singularities on manifolds.
problem Understanding singularities of Gaussian random fields on manifolds.
method Computed expected values of singularities under various conditions.
result Explicit formulae for singularities under different constraints.
The paper shows Gaussian fluctuations in eigenvalue statistics of random hyperbolic surfaces.
problem Understanding fluctuations in Laplace eigenvalues of random hyperbolic surfaces.
method Analyzing fluctuations of linear statistics of Laplace eigenvalues over moduli space of surfaces of large genus.
result The distribution of linear statistics tends to a Gaussian as the genus of surfaces increases.
New methods reduce computational cost for Gaussian Markov Random Fields with sparse constraints.
problem Inference and simulation of GMRFs are computationally prohibitive with many constraints.
method Proposes a basis transformation into blocks of constrained and non-constrained subspaces.
result Significantly outperforms existing alternatives in computational cost.
Develops calculus for random submanifolds using zonoids.
problem Calculating properties of random submanifolds defined by zero sets of vector fields.
method Defines zonoid sections and uses them to compute expected volumes and currents.
result Establishes new inequalities and formulas for random submanifolds.
We derive Gaussian approximations for random forest predictions using region-based stabilization.
problem Improving the accuracy of random forest predictions for Poisson process data.
method Region-based stabilization and Malliavin-Stein method for multivariate Gaussian approximation.
result Established Gaussian approximation bounds for random forest predictions under Poisson process.
Extends spectral number variance convergence to random matrix ensembles for twisted Laplacians.
problem Spectral number variance convergence for twisted Laplacians and Dirac operators.
method Extends Rudnick's approach to Gaussian ensembles for twisted Laplacians and Dirac operators.
result Convergence to Gaussian ensembles for twisted Laplacians and Dirac operators.
The paper classifies translation surfaces with constant curvature in a specific connection.
problem Classifying translation surfaces with constant curvature in a semi-symmetric non-metric connection.
method Completely classified translation surfaces of constant sectional curvature in a semi-symmetric non-metric connection.
result Translation surfaces of constant curvature are generalized cylinders, similar to the Levi-Civita connection but with additional non-constant curvature cases.
This paper proposes an empirical test of financial contagion in European equity markets during the tumultuous period of 2008-2011. Our analysis shows that traditional GARCH and Gaussian stochastic-volatility models are unable to explain two key stylized features of global markets during presumptive contagion periods: s…
Random Gaussian fields on 4D Riemannian manifolds with conformal invariance.
problem Characterizing and analyzing Gaussian fields on 4D Riemannian manifolds.
method Constructing and analyzing co-biharmonic Gaussian fields with covariance kernels defined by the Paneitz operator.
result Rigorous derivation of quantum Liouville measure for ∣ γ ∣ < 8 |γ|<\sqrt8 ∣ γ ∣ < 8 . New method improves Gaussian kernel approximations for high-frequency data.
problem Limited scalability of kernel-based models to large data sets.
method Local random feature approximations using Maclaurin expansions and polynomial sketches.
result Significant improvement in kernel approximations and downstream performance for high-frequency data.
Study improves error bounds for sparse regression with heavy-tailed covariates.
problem Estimating sparse coefficients in linear regression with heavy-tailed covariates.
method Employed an ℓ 1 \ell_1 ℓ 1 -penalized Huber regression method. result Error bound identical to Gaussian case for L L L -subexponential covariates. Improved kernel ridge regression for large datasets using weighted random binning.
problem Efficiently approximating kernel matrices for large-scale datasets.
method Introduced weighted random binning features for locality sensitive hashing.
result Weighted random binning features generate Gaussian processes of any desired smoothness.
Exact selective inference with randomization for Gaussian regression models.
problem Exact selective inference in Gaussian regression models.
method Introduces a pivot for exact selective inference with randomization, reducing the problem to a bivariate truncated Gaussian distribution.
result Our pivot leads to exact inference and produces narrower confidence intervals than related methods.
The paper analyzes and mitigates biases in scalable Gaussian Process methods.
problem Modeling biases in scalable Gaussian Process methods.
method Randomized truncation estimators to eliminate bias in exchange for increased variance.
result Randomized truncation estimators meaningfully outperform biased counterparts with minimal additional computation.
Paper calculates KL divergence for isotropic Gaussian-Markov fields.
problem Measuring divergence between isotropic Gaussian-Markov fields.
method Derives closed-form KL divergence expressions.
result Develops new similarity measures in image processing.
Geometric quantization results for Riemann surfaces with semi-positive line bundles.
problem Analyzing geometric quantization for Riemann surfaces with semi-positive line bundles.
method Exploring the Bergman kernel expansion and related results for induced Fubini-Study metrics, Toeplitz operators, and holomorphic torsion.
result Asymptotic results for holomorphic torsion and random sections.
Gradient span algorithms show consistent progress in high dimensions.
problem Understanding consistent training progress in large machine learning models.
method Proving deterministic behavior of gradient span algorithms on Gaussian random functions.
result Gradient span algorithms have asymptotically deterministic behavior in high dimensions.
Random neural networks with ReLU activations are non-Gaussian processes.
problem Understanding the behavior of neural networks with random initialization and rectified linear units.
method Proving these networks are non-Gaussian processes and deriving their properties.
result These networks can converge to non-Gaussian processes under certain conditions.
The study proves Gaussian universality of deep random features learning.
problem Understanding the test error in deep random features learning.
method Proving Gaussian universality of test error in ridge regression and arbitrary convex losses.
result Sharp asymptotic formula for test error in ridge regression setting.
A new method for approximating softmax and Gaussian kernels with reduced error.
problem Approximating softmax and Gaussian kernels with low error.
method Simplex Random Features (SimRFs) and SimRFs+.
result SimRFs provide the smallest MSE among weight-independent geometrically-coupled PRF mechanisms.