Deep belief networks can approximate any multivariate density with binary hidden units.
problem Approximating multivariate probability densities with binary hidden units.
method Sharp quantitative bounds on approximation error in terms of hidden units.
result Deep belief networks can approximate any multivariate density with binary hidden units under mild integrability requirements.
Upper bounds on shortest filling geodesics on hyperbolic surfaces.
problem Finding the shortest geodesic that covers a surface.
method Quantitative density of closed geodesics on hyperbolic surfaces.
result Upper bounds on the length of the shortest closed geodesic.
Quantifies how geodesic planes isolate in hyperbolic 3-manifolds.
problem Understanding isolation properties of geodesic planes in hyperbolic 3-manifolds.
method Quantitative estimates of geodesic planes in frame bundles, using tight areas and densities.
result Polynomial estimates of isolation properties with degree given by modified critical exponents.
The paper proves density and positive mass theorems for incomplete manifolds.
problem Proving density and positive mass theorems for manifolds with incomplete ends.
method Using harmonic asymptotics and quantitative positive mass theorem improvements.
result Improved quantitative positive mass theorem in dimensions 3 to 7.
Proves a limit on hyperplanes in complex manifolds.
problem Limiting the number of hyperplanes in complex manifolds.
method Effective density theorem for periodic orbits, Margulis functions, restricted projection theorem, equidistribution result.
result Proves a quantitative finiteness theorem for hyperplanes.
Theorem generalizes Reifenberg's for measures with bounds on β-numbers.
problem Bounding measures away from k-rectifiable sets with β-numbers.
method Assumptions on Jones' β-numbers to measure closeness to subspaces.
result Effective measure bounds on μ away from a closed k-rectifiable set.
This paper studies neural network operators and their convergence properties.
problem Understanding the approximation and convergence of neural network operators.
method Proves density results, convergence estimates, and Voronovskaya-type theorems.
result Establishes quantitative convergence estimates and derives Voronovskaya-type theorems.
The study provides optimal estimates for surfaces close to constant mean curvature.
problem Optimizing estimates for surfaces near constant mean curvature.
method Bi-Lipschitz and W2,2 parametrization for surfaces with density close to one and small Willmore energy. result Quantitative rigidity for L2-almost CMC surfaces. Graph classification improved with motif counts and graphon theory.
problem Classifying large graphs with high accuracy.
method Using motif homomorphisms and graphon theory to provide bounds and a classifier.
result Explicit quantitative bounds for graph classification under noise.
The paper proves surfaces close to spheres under specific conditions.
problem Proving rigidity of almost constant mean curvature spheres.
method Linearized analysis around the sphere, Willmore bound, and small defect.
result Almost-CMC surfaces are close to the round sphere with linear control.
Proves inequality linking function deviation to gradient norm on compact manifolds.
problem Analyzing coupled elliptic systems on compact manifolds.
method Develops a new Poincaré-Sobolev inequality with a density-free reference average.
result Poincaré constant depends on the density's gradient norm.
New method connects leverage scores and kernel density, revealing a decreasing relationship.
problem Understanding the relationship between leverage scores and kernel density.
method Introducing regularized Christoffel functions to study leverage scores for kernel methods.
result Quantitatively describes a decreasing relation between leverage score and population density for a broad class of kernels.
Method calculates financial distributions using recursive relationships.
problem Analyzing the distribution of financial functions at discrete points.
method Recursive method to calculate probability distributions.
result High accuracy demonstrated in numerical experiments.
Improved GAN estimator learns densities faster with insights from nonparametric statistics.
problem How well GAN learns densities under different smoothness properties.
method Improved GAN estimator that leverages the level of smoothness and evaluation metric.
result Achieves a faster rate of convergence and near optimal minimax lower bound in high dimensions.
The Clifford torus minimizes Willmore energy closely for small perturbations.
problem Finding the closest shape to the Clifford torus under small perturbations of Willmore energy.
method Analyzing integral 2-varifolds with specific properties and showing quantitative closeness to the Clifford torus.
result The support of the varifold is quantitatively close to the Clifford torus after a conformal transformation.
We prove a quantitative statement of the quantum ergodicity for Hecke--Maass cusp forms on the modular surface. As an application of our result, along a density 1 subsequence of even Hecke--Maass cusp forms, we obtain a sharp lower bound for the L2-norm of the restriction to a fixed compact geodesic segment of $η=…
Study linearizes 2-Wasserstein space using optimal transport maps.
problem Stability and linearization of the 2-Wasserstein space.
method Explicit embedding of probability measures into a Hilbert space using optimal transport maps.
result The embedding is (bi-)Hölder continuous, with stability results for optimal transport maps.
The paper finds bounds on shortest dense curves on surfaces.
problem Finding shortest dense curves on surfaces.
method Quantitative density of closed geodesics and orthogeodesics.
result Upper bounds on shortest dense curves.
SGD favors flat minima exponentially more than sharp minima in deep learning.
problem Understanding how SGD selects flat minima in deep learning.
method Developed a density diffusion theory (DDT) to analyze minima selection.
result SGD exponentially favors flat minima over sharp minima due to Hessian-dependent noise.
Study on surfaces containing twistor lines, proving their density and bounds.
problem Understanding the distribution and maximum number of twistor lines on algebraic surfaces.
method Analyzing ideal sheaves, proving density in Grassmannian, and calculating bounds for surface degrees.
result Twistor lines are Zariski dense in the Grassmannian and have maximum bounds on their number.
Algorithm optimizes lockdown policies balancing health and economy.
problem Balancing health and economic impacts of lockdowns during pandemics.
method Reinforcement learning to automatically compute lockdown policies.
result Algorithm learns optimal lockdown policies from disease and population data.
FourNet approximates financial transition densities using Fourier transforms.
problem Approximating transition densities in finance with high accuracy.
method FourNet is a novel FFNN with Gaussian activation, learning from characteristic functions.
result FourNet can approximate transition densities arbitrarily well with finite neurons.
This research improves demand forecasting by predicting complete probability density functions using machine learning.
problem Forecasting complete probability density functions for better operational decision making.
method Supervised machine learning method 'Cyclic Boosting' for explainable predictions.
result Predicted probability density functions are fully explainable and avoid 'black-box' models.
A new sampling method using log-concave Markov chains.
problem Sampling from unnormalized densities efficiently.
method Decomposes sampling into log-concave Markov chains with noisy measurements.
result Shows remarkable capacity to 'tunnel' between modes of a distribution.
This study investigates that a characteristic time scale on an exchange rate market (USD/JPY) is examined for the period of 1998 to 2000. Calculating power spectrum densities for the number of tick quotes per minute and averaging them over the year yield that the mean power spectrum density has a peak at high frequenci…
L-Cool improves image and language translation by cooling low-density samples.
problem Improving translation performance on fringe samples in unsupervised domain translation.
method Performing Langevin dynamics to move low-density samples towards high-density areas.
result L-Cool enhances state-of-the-art methods in image and language translation tasks.
Paper proves diffusion models work on manifolds.
problem Current diffusion models assume densities are w.r.t. Lebesgue measure, limiting their applicability.
method Introduced convergence results for diffusion models on more general target distributions.
result Quantitative bounds on Wasserstein distance for target and generated distributions.
New estimates quantify blow-up rates of spacelike singularities in gravitational collapse.
problem Quantifying the blow-up rates of spacelike singularities in gravitational collapse.
method Deriving new quantitative estimates using spherical symmetry and double-null coordinates.
result Polynomial blow-up rates O(1/rN) for various quantities, with improved estimates for r∂ur and r∂vr. Develops a fast and accurate option pricing method using Legendre polynomials.
problem Pricing European options with smoothness issues in payoff functions.
method Legendre series expansion and characteristic function relation for density function.
result Highly accurate approximations for European call options.
Rank-statistic method approximates f-divergences without density-ratio estimation.
problem Approximating f-divergences without explicit density-ratio estimation. method Mapping distribution rank histograms to discrete f-divergence and averaging over random projections. result The rank-statistic estimator is a lower bound of the true f-divergence and converges under mild conditions. Conditional diffusion models can approximate target distributions well with Gaussian-mixture reverse kernels.
problem Approximating target distributions in conditional diffusion models.
method Using finite Gaussian mixtures with ReLU-network logits as reverse kernels, reducing the problem to static conditional density approximation.
result The resulting neural reverse-kernel class is dense in conditional KL divergence under exact terminal matching.
Paper analyzes convergence rates of mean-field SVGD method.
problem Establishing quantitative rates of convergence for mean-field SVGD.
method Quantitative analysis of mean-field SVGD dynamics on torus.
result Explicit polynomial convergence rates in L2-norm for Riesz-type kernels.
Study on the Lp behavior of helix curves' energy density.
problem Analyzing the Lp asymptotics of helix curves' Möbius energy density. method Contour integration and Laurent expansion near poles.
result Established the precise Lp blowup rate of the Möbius energy density. Random groups prove length constraints on product of conjugates.
problem Quantify products of conjugates in random groups.
method Sharp van Kampen diagram argument and boundary block-counting.
result Prove a sharp inequality for products of conjugates in random groups.
Improved MLMC method for robust and efficient probability and density estimation.
problem Stability and poor complexity of MLMC for low-regularity functionals.
method Numerical smoothing combined with MLMC for deterministic quadrature methods.
result Significant improvement in strong convergence and robustness of MLMC method.
Improved disentangled representation learning using a non-parametric latent density model.
problem Limited disentanglement in VAE due to constraints on latent density independence and complexity.
method Utilized the Indian Buffet Process (IBP) as a non-parametric latent density model to allow richer modeling capacity.
result IBP-VAE outperformed state-of-the-art VAEs in disentangling latent factors across various datasets.
Improves GANs by sampling from an energy-based model induced by discriminator scores.
problem Improving the quality of images generated by GANs.
method DDLS (Discriminator Driven Latent Sampling) using the sum of latent prior log-density and discriminator output score.
result Significantly improves Inception Score on CIFAR-10 dataset.
Study on visibility properties of spiral sets in higher dimensions.
problem Characterizing density properties of spiral sets.
method Employing visibility concepts from discrete geometry.
result Established conditions for various density properties of spirals.
Fast, reliable, and error-bounded option pricing with neural networks
problem Fast, reliable, and error-bounded option pricing
method Mixture Density Network
result Out-of-sample CDF error of 1.4imes10−4 Analyzes multifractality caused by fat-tailed distributions in time series.
problem Quantifying multifractality induced by fat-tailed distributions in time series data.
method Examines different types of fat-tailed distributions using Tsallis statistics and nonextensive analysis.
result Developed semi-analytical formulas to distinguish true multifractality from spurious multifractality.
New methods estimate point-wise dependency from neural MI models.
problem Estimating point-wise dependency between different events.
method Developed two methods: Probabilistic Classifier and Density-Ratio Fitting.
result Demonstrated effectiveness in MI estimation, self-supervised representation learning, and cross-modal retrieval.
Study investigates singularity formation in α-Yang-Mills-Higgs fields on spheres.
problem Singularity formation in α-Yang-Mills-Higgs fields on spheres. method Established α-energy identity, no-neck property through Hodge decomposition and new conservation law. result Unified and quantitative framework for singularity formation in variational gauge theories.
Model estimates lung well-aerated volume from CT images, independent of patient and imaging parameters.
problem Lack of clear connection between quantitative metrics in lung CT images and physiology.
method Patient-independent model using Gaussian fit to lower CT histogram data points.
result Model estimates well-aerated volume (WAVE) independent of CT reconstruction parameters and respiratory cycle.
A new model characterizes undocumented and asymptomatic infections to quantify COVID-19 uncertainties.
problem Quantifying uncertainties in COVID-19 infections and contagion.
method SUDR model: characterizes undocumented and documented infections, captures probabilistic density, and incorporates Bayesian inference.
result Demonstrates deeper understanding of COVID-19 uncertainties compared to classic models.
The study tightens the sample complexity for learning nonparametric mixture components.
problem Learning nonparametric distributions in a finite mixture model.
method Assumes each component is a convolution of a Gaussian and a compactly supported density, and uses a quantitative Tauberian theorem.
result Tight bounds on sample complexity required for estimating each component, showing it lies between polynomial and exponential.
Minimal submanifolds either fill space or are confined with geometric restrictions.
problem Understanding the behavior of minimal submanifolds in space.
method A dichotomy principle and volume doubling theorem.
result Quantitative restrictions on confined minimal submanifolds, including volume growth and optimal density rates.
New trust matrix quantifies breakdowns in deep neural networks.
problem Understanding trust breakdowns in deep learning models.
method Introduces trust matrix and conditional trust densities to analyze deep neural networks.
result Trust matrices reveal areas needing improvement for deep neural networks.
The paper identifies regions where investment strategies match expected performance.
problem Inconsistent performance of Markowitz efficient portfolios.
method Density forecasting to measure ex-ante accuracy and identify the consistency region.
result Investment strategies based on consistent portfolios outperform efficient ones.