We describe the first known mean-field study of landing probabilities for random walks on hypergraphs. In particular, we examine clique-expansion and tensor methods and evaluate their mean-field characteristics over a class of random hypergraph models for the purpose of seed-set community expansion. We describe paramet…
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48 results for “random expansions”
Multivariate splines linked to infinitely-wide neural networks with improved numerical performance.
problem Understanding the relationship between multivariate splines and neural networks.
method Showed multivariate splines can be represented as random features in infinitely-wide neural networks with a homogeneous activation function.
result The function space of multivariate splines is a Sobolev space on a Euclidean ball with explicit norm bounds on derivatives.
Random walks on free groups reveal asymmetric expansion factors.
problem Understanding expansion factors in free groups.
method Random walks and BGIP on metric spaces.
result Generic outer automorphisms have different forward and backward expansion factors.
Recently, the binary expansion testing framework was introduced to test the independence of two continuous random variables by utilizing symmetry statistics that are complete sufficient statistics for dependence. We develop a new test based on an ensemble approach that uses the sum of squared symmetry statistics and di…
Sparse random features improve accuracy in data-scarce settings.
problem Limited accuracy of random feature methods in data-scarce applications.
method Sparse random feature expansion using compressive sensing.
result Improved generalization bounds for sparse random features.
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…
Study on random representations of surface groups into SU(n), focusing on asymptotic expansions.
problem Understanding random representations of surface groups into special unitary groups.
method Use of a symplectic form on moduli space, establishing asymptotic expansions for trace values.
result Existence of large n asymptotic expansions for expected values of trace of elements under random representations.
Ensembles dynamic models using random feature approximations.
problem Online scalable Bayesian learning with dynamic models and ensembling.
method Random feature approximations and dynamic models using random walks.
result Better performance with alternative basis expansions like Hilbert space Gaussian processes.
The study improves volatility model pricing accuracy with new statistical expansions.
problem Improving option pricing accuracy in volatility models.
method Developed Edgeworth expansions for various volatility models.
result Enhanced statistical expansions for volatility models.
Probabilistic model for exhaustion in infinite-genus curve complexes.
problem Action rigidity in infinite-genus curve complexes.
method Costa and Farber's model for random simplicial complexes.
result Probabilistic evidence for exhaustion via rigid expansions.
We provide a theoretical foundation for non-parametric estimation of functions of random variables using kernel mean embeddings. We show that for any continuous function , consistent estimators of the mean embedding of a random variable lead to consistent estimators of the mean embedding of . For Matérn ke…
We investigate the analogy between the large N expansion in normal matrix models and the asymptotic expansion of the determinant of the Hilb map, appearing in the study of critical metrics on complex manifolds via projective embeddings. This analogy helps to understand the geometric meaning of the expansion of matrix m…
Neural networks solve SPDEs using Wiener chaos expansion.
problem Solving stochastic partial differential equations (SPDEs) numerically.
method Using neural networks in the truncated Wiener chaos expansion.
result Approximation rates for learning SPDE solutions with noise.
New sampling method for Heston model reduces complexity.
problem Efficient sampling for Heston model's time integrated variance.
method Series expansion, change of measure, Chebyshev polynomial approximations.
result Strong, efficient sampling scheme established for Heston model.
HARFE approximates sparse additive functions using random features and ridge regression.
problem Approximating high-dimensional sparse additive functions.
method Hard-ridge random feature expansion with sparse ridge regression and hard-thresholding pursuit.
result HARFE method converges with a given error bound and achieves lower error than other algorithms.
RSHT algorithm simplifies complex shapes to points.
problem Simplifying complex shapes to points in higher dimensions.
method Combines simplicial collapses and expansions.
result Reduces triangulated d-manifolds to points using RSHT.
We introduce closed-form transition density expansions for multivariate affine jump-diffusion processes. The expansions rely on a general approximation theory which we develop in weighted Hilbert spaces for random variables which possess all polynomial moments. We establish parametric conditions which guarantee existen…
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.
Enhances random forest performance with exogenous randomness.
problem Improving random forest performance through exogenous randomness.
method Developed non-asymptotic MSE expansions for individual trees and forests, identified two types of randomness, and conducted simulations.
result Exogenous randomness, particularly feature subsampling, reduces both bias and variance of random forests.
In graph theory there are intimate connections between the expansion properties of a graph and the spectrum of its Laplacian. In this paper we define a notion of combinatorial expansion for simplicial complexes of general dimension, and prove that similar connections exist between the combinatorial expansion of a compl…
Paper tackles high-order inference in structured prediction tasks.
problem Maximizing a score function on the space of labels in high-order Markov random fields.
method Generative model approach with two-stage convex optimization algorithm.
result Success in general high-order inference problems driven by hyperedge expansion properties.
Bayesian inference with finitely wide neural networkscond-mat.dis-nn
Bayesian inference for wide neural networks using Edgeworth expansion.
problem Analyzing the non-Gaussian behavior of wide neural networks in Bayesian inference.
method Proposed a non-Gaussian distribution using multivariate Edgeworth expansion for finite-width neural networks.
result Derived non-Gaussian posterior distribution in Bayesian regression tasks.
Friedman-Ramanujan functions in random hyperbolic geometry and application to spectral gaps Imath.SP
New volume functions for random hyperbolic surfaces link to spectral gaps.
problem Analyzing spectral gaps in random hyperbolic surfaces.
method Introduced new volume functions , derived their asymptotic expansions, and linked them to spectral gaps.
result Coefficients in the asymptotic expansion of are Friedman-Ramanujan functions.
Maximal concentration bounds for stochastic approximation with heavy-tailed noise.
problem Analyzing the convergence of stochastic approximation algorithms under heavy-tailed Markovian noise.
method Novel Lyapunov function and black-box truncation argument.
result Tail behavior of the error can be sub-Gaussian, sub-Weibull, or lighter than any Pareto but heavier than any Weibull.
AL-SPCE improves reliability analysis for complex systems with active learning and SPCE.
problem Efficiently analyzing reliability of complex, computationally expensive models with intrinsic randomness.
method Active learning framework using stochastic polynomial chaos expansions (SPCE) to reduce computational burden.
result AL-SPCE maintains high accuracy in reliability estimates while significantly improving efficiency.
Mean Field Portfolio Gamesq-fin.MF
Study Nash equilibrium in mean field portfolio games with random market parameters.
problem Modeling wealth and relative performance in competitive financial markets.
method Martingale optimality principle approach to characterize Nash equilibrium in mean field FBSDE.
result Unique Nash equilibrium found under weak interaction assumption and market parameters independence.
Measurements of cosmic microwave background (CMB) anisotropy are ideal experiments for discovering the non-trivial global topology of the universe. To evaluate the CMB anisotropy in multiply-connected compact cosmological models, one needs to compute the eigenmodes of the Laplace-Beltrami operator. Using the direct bou…
The composition of multiple Gaussian Processes as a Deep Gaussian Process (DGP) enables a deep probabilistic nonparametric approach to flexibly tackle complex machine learning problems with sound quantification of uncertainty. Existing inference approaches for DGP models have limited scalability and are notoriously cum…
Asymptotic expansion of induced Grassmannian Chern forms and distribution of random degeneracy setsmath.CV
The paper studies the distribution of random degeneracy sets on complex manifolds.
problem Distribution of random degeneracy sets on compact Kähler manifolds.
method Asymptotic expansion of induced Grassmannian Chern forms, meromorphic transforms, and Wishart distribution.
result Normalized currents converge to curvature forms with quantitative estimates.
We consider first order expansions of convex penalized estimators in high-dimensional regression problems with random designs. Our setting includes linear regression and logistic regression as special cases. For a given penalty function and the corresponding penalized estimator , we construct a quantity ,…
The challenges for non-intrusive methods for Polynomial Chaos modeling lie in the computational efficiency and accuracy under a limited number of model simulations. These challenges can be addressed by enforcing sparsity in the series representation through retaining only the most important basis terms. In this work, w…
Improved surrogate model for field-valued QoIs using LF and HF simulations.
problem Accurate and efficient modeling of field-valued quantities under uncertain inputs.
method Bifidelity Karhunen-Loève expansion with active learning.
result Consistent improvements in predictive accuracy and sample efficiency.
The Lugannani-Rice formula is a saddlepoint approximation method for estimating the tail probability distribution function, which was originally studied for the sum of independent identically distributed random variables. Because of its tractability, the formula is now widely used in practical financial engineering as …
Neural Chaos uses neural networks instead of polynomials for stochastic modeling.
problem Challenges in constructing surrogate models with uncertainty quantification for complex or high-dimensional stochastic processes.
method Adopting spectral expansion formalism with neural network basis functions, identifying them data-drivenly without prior assumptions.
result Demonstrates effectiveness of the proposed scheme through numerical examples of varying complexity.
In this paper, we address the inverse problem, or the statistical machine learning problem, in Markov random fields with a non-parametric pair-wise energy function with continuous variables. The inverse problem is formulated by maximum likelihood estimation. The exact treatment of maximum likelihood estimation is intra…
Filters in a Convolutional Neural Network (CNN) contain model parameters learned from enormous amounts of data. In this paper, we suggest to decompose convolutional filters in CNN as a truncated expansion with pre-fixed bases, namely the Decomposed Convolutional Filters network (DCFNet), where the expansion coefficient…
The paper solves the problem of fitting an ellipsoid to random points efficiently.
problem Finding an ellipsoid that passes through random Gaussian points.
method Constructing a fitting ellipsoid using a decomposition of a random matrix and graph matrix theory.
result The ellipsoid fitting problem transitions from feasible to infeasible at a sharp threshold of .
We show that kernel-based quadrature rules for computing integrals can be seen as a special case of random feature expansions for positive definite kernels, for a particular decomposition that always exists for such kernels. We provide a theoretical analysis of the number of required samples for a given approximation e…
New method uses sparse random features for crashworthiness analysis.
problem Efficient surrogate modelling for uncertainty quantification.
method Sparse Random Features combined with self-supervised dimensionality reduction.
result Superiority over state-of-the-art techniques in crashworthiness analysis.
PCA for Point Processesmath.ST
A new PCA method for analyzing point processes.
problem Analyzing variability in replicated point processes.
method Functional Principal Component Analysis (fPCA) on cumulative mass functions.
result Established convergence and introduced principal measures.
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.
Deep ReLU networks show that 4 layers suffice for unique input recovery.
problem Injectivity capacity of deep ReLU networks.
method Developed a program connecting deep ReLU injectivity to an -extension of the spherical perceptrons, using random duality theory.
result Only 4 layers are needed for unique input recovery, showing expansion saturation effect.
We review the utility-based valuation method for pricing derivative securities in incomplete markets. In particular, we review the practical approach to the utility-based pricing by the means of computing the first order expansion of marginal utility-based prices with respect to a small number of random endowments.
This paper analyzes DONE, an online optimization algorithm that iteratively minimizes an unknown function based on costly and noisy measurements. The algorithm maintains a surrogate of the unknown function in the form of a random Fourier expansion (RFE). The surrogate is updated whenever a new measurement is available,…
BEGIN network models binary data without parametric assumptions.
problem Conditional independence in non-parametric families of binary data.
method BEGIN network models binary data using sparse linear representations and block factorizations.
result BEGIN network captures conditional independence for arbitrary binary and multinomial variables.
We provide a direct proof of Cramér's theorem for geodesic random walks in a complete Riemannian manifold . We show how to exploit the vector space structure of the tangent spaces to study large deviation properties of geodesic random walks in . Furthermore, we reveal the geometric obstructions one runs into …
A new fast method simulates stochastic volatility models.
problem Simulating stochastic volatility models efficiently.
method Karhunen-Loève expansions to express stochastic volatility as sine series, followed by analytical derivation of integrals.
result Simulation is several hundred times faster than existing methods.
This work provides a semi-analytic approximation method for decoupled forwardbackward SDEs (FBSDEs) with jumps. In particular, we construct an asymptotic expansion method for FBSDEs driven by the random Poisson measures with σ-finite compensators as well as the standard Brownian motions around the small-variance limit …