Study recovers Riemannian quantities from noisy data densities.
problem Recovering geometric structure from noisy data on submanifolds.
method Derive uniform small-noise expansions of noisy density and its derivatives; construct estimators for tangent spaces, intrinsic dimension, and second fundamental form.
result Fundamental Riemannian quantities identifiable from density derivatives.
Frequency bias affects neural network training on non-uniform data.
problem Understanding how frequency bias impacts neural networks trained on non-uniformly distributed data.
method Used the Neural Tangent Kernel (NTK) model to explore the effect of variable density on training dynamics.
result Convergence time for learning a pure harmonic function depends on the local density at a point.
New findings on PAC learning and marginal distribution estimation.
problem Understanding how PAC learning relates to marginal distribution estimation under distributional constraints.
method Revisited the connection between PAC learning, uniform convergence, and density estimation, considering a known family of marginal distributions.
result PAC learning is sandwiched between two refined models of density estimation, differing only in whether the learner knows the set of well-estimated events in H.
Paper proves integrability and entropy compactness for Kähler potentials with uniform log-log threshold.
problem Integrability and entropy compactness for Kähler potentials with specific density.
method Skoda-Zeriahi type integrability theorem and log-log threshold detection.
result Positivity of integrability threshold and entropy compactness for uniform log-log threshold.
Continuous-time PCD for MLE with explicit error bounds.
problem Maximum likelihood estimation of unnormalised densities.
method Continuous-time formulation as coupled SDEs, deriving UiT bounds.
result Explicit error bounds between PCD iterates and MLE solution.
New method combines strengths of two PCL approaches without density ratio estimation.
problem Estimating causal functions in Proxy Causal Learning with unobserved confounders and proxies.
method Kernel-based doubly robust estimators combining treatment and outcome bridges, density ratio-free.
result Outperforms existing methods on PCL benchmarks, including a prior doubly robust method.
We approximate the heat kernel h(x,y,t) on a compact connected Riemannian manifold M without boundary uniformly in (x,y,t)∈M×M×[a,b], a>0, by n-fold integrals over Mn of the densities of Brownian bridges. Moreover, we provide an estimate for the uniform convergence rate. As an immediate coro…
Estimates financial market impacts of COVID-19 using time-varying kernel density.
problem Estimating the impact of COVID-19 on financial markets over time.
method Time-varying kernel density estimation with Kolmogorov-Smirnov statistic.
result Determines the chronology and regional disparities of financial market impacts.
Unique floating and buoyancy surfaces identify convex polytopes.
problem Identifying convex polytopes from their flotation and buoyancy surfaces.
method Proving uniqueness of surfaces for polytopes with uniform or prescribed density.
result Floating and buoyancy surfaces uniquely determine convex polytopes.
Neural networks are dense among Lipschitz functions with fixed Lipschitz constant.
problem Characterizing neural network approximations to Lipschitz functions.
method Analyzing L-Lipschitz neural networks and their density in L-Lipschitz functions. result One layer neural networks are dense in the set of all L-Lipschitz functions. The study of the geometry of n-uniform measures in Rd has been an important question in many fields of analysis since Preiss' seminal proof of the rectifiability of measures with positive and finite density. The classification of uniform measures remains an open question to this day. In fact there is on…
We address the following problem: given two smooth densities on a manifold, find an optimal diffeomorphism that transforms one density into the other. Our framework builds on connections between the Fisher-Rao information metric on the space of probability densities and right-invariant metrics on the infinite-dimension…
The paper provides consistency results for KDE on manifolds with irregular kernels.
problem Analyzing density estimation on manifolds with complex kernels.
method Strong uniform consistency with rates for KDE on Riemannian manifolds with Riemann integrable kernels.
result Strong uniform consistency with rates for KDE on manifolds.
Use of an autoencoder (AE) as a normal model is a state-of-the-art technique for unsupervised-anomaly detection in sounds (ADS). The AE is trained to minimize the sample mean of the anomaly score of normal sounds in a mini-batch. One problem with this approach is that the anomaly score of rare-normal sounds becomes hig…
A new method for sampling on manifolds reduces density estimation errors.
problem Sampling on implicitly defined manifolds in various applications.
method Polynomial-Maximization Moment (PMM) estimator replacing local k-nearest-neighbour density estimate.
result Reduces density estimation errors by 22--36% on asymmetric gamma and boundary-spacing regimes.
Unified framework for robust, stable, and efficient density ratio estimation.
problem Density-chasm and support-chasm problems in density ratio estimation.
method Dequantified diffusion-Schrödinger bridge (D3RE) framework with DDBI and DSBI.
result Offers uniform approximation and bounded time scores in theory and empirical performance.
Study uniform consistency in nonparametric mixture models and mixed regression.
problem Uniform consistency in nonparametric mixture models and mixed regression models.
method Construct uniformly consistent estimators under general conditions, develop novel technical tools.
result Prove uniform consistency results for nonparametric mixtures and mixed regression models.
A new UU-test decides unimodality of datasets.
problem Deciding on the unimodality of a dataset for better data analysis.
method UU-test operates on the empirical cumulative density function (ecdf) to build a piecewise linear approximation that models the data as a Uniform Mixture Model.
result The UU-test provides a statistical model of the data in the form of a Uniform Mixture Model.
Conformal Test Martingales can be 'blind' to significant changes in data distribution.
problem The converse of exchangeability does not hold, leading to potential blindness of CTMs.
method Explicit construction of A-cryptic change-point using bivariate Gaussian distributions. result CTMs can be perfectly cryptic to a significant change in marginal means.
Paper characterizes optimal graph clustering limits under a new model.
problem Graph clustering under varying edge density signals.
method Introduced Popularity-Adjusted Block Model (PABM) to address SBM and DCBM limitations.
result Cluster recovery possible even when edge density signals vanish, highlighting local connectivity differences.
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.
New neural network improves MRI reconstruction for non-Cartesian data.
problem Improving MRI reconstruction for non-Cartesian data acquisitions.
method Density-compensated unrolled neural networks.
result Density-compensated unrolled neural networks outperform baselines.
Kolesnikov-Milman [9] established a local Lp-Brunn-Minkowski inequality for p∈(1−c/n23,1). Based on their local uniqueness results for the Lp-Minkowski problem, we prove in this paper the (global) Lp-Brunn-Minkowski inequality. Two uniqueness results are also obtained: the first one is for the …
Reflective Hamiltonian Monte Carlo struggles with high-dimensional sampling.
problem Slow mixing in reflective Hamiltonian Monte Carlo with inexact reflections.
method Quantifying instantaneous non-uniformity with Sinkhorn divergence; analyzing particle motion in spheres and cubes; constructing low-dimensional toy models.
result Particles spontaneously unmix, leading to resonances in particle density.
Paper estimates diameter for Minkowski problem solutions.
problem Estimating diameter of solutions to Minkowski problem.
method Uniform diameter estimate for Lp dual Minkowski problem. result Uniform diameter estimate for planar Lp dual Minkowski problem. UT module refines VAE latent space, improving disentanglement and interpretability.
problem Irregular latent distributions cause posterior collapse and misalignment in VAEs.
method UT module uses G-KDE clustering, GM modeling, and PIT to transform latent space into uniform distribution.
result UT module enhances disentanglement and interpretability of latent representations.
We explain a characterization of Einstein-Fano manifolds in terms of the lower bound of the density of the volume of the Kähler-Ricci Flow. This is a direct consequence of Perelman's uniform estimate for the Kähler-Ricci Flow and a C0 estimate of Tian and Zhu.
Proposes a method to partition univariate data into unimodal subsets.
problem Partitioning univariate multimodal data into unimodal subsets.
method Recursive splitting around valley points of the data density using properties of critical points on the convex hull of the ecdf plot.
result Obtains a hierarchical statistical model of the initial dataset as a mixture of UMMs.
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorit…
We present explicit geometric decompositions of the hyperbolic complements of alternating k-uniform tiling links, which are alternating links whose projection graphs are k-uniform tilings of S2, E2, or H2. A consequence of this decomposition is that the volumes of spherical alternating $k…
Unified framework for PDF estimation using MDL-based binning and tensor factorization.
problem Challenges in estimating PDFs for non-uniform, multimodal data.
method MDL-based binning with quantile cuts, tensor factorization (CPD).
result Effective PDF estimation on synthetic and real data.
New method avoids curse of dimensionality in structured density estimation.
problem Estimating multivariate density with Markov graph constraints.
method Introduces 'graph resilience' to control sample complexity.
result Avoids curse of dimensionality under Markov conditions.
As one type of efficient unsupervised learning methods, clustering algorithms have been widely used in data mining and knowledge discovery with noticeable advantages. However, clustering algorithms based on density peak have limited clustering effect on data with varying density distribution (VDD), equilibrium distribu…
The study finds anisotropic minimal surfaces in 3-manifolds with smooth boundaries.
problem Finding smooth anisotropic minimal surfaces in closed 3-manifolds.
method Min-max construction with elliptic integrands, uniform upper bound for density ratios.
result Obtains a smooth anisotropic minimal surface in a closed 3-manifold.
This paper considers a new family of variational distributions motivated by Sklar's theorem. This family is based on new copula-like densities on the hypercube with non-uniform marginals which can be sampled efficiently, i.e. with a complexity linear in the dimension of state space. Then, the proposed variational densi…
Generative Adversarial Networks (GAN) training process, in most cases, apply Uniform or Gaussian sampling methods in the latent space, which probably spends most of the computation on examples that can be properly handled and easy to generate. Theoretically, importance sampling speeds up stochastic optimization in supe…
Study uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.
problem Uniform convergence of random walk Laplacians to diffusion Laplacian on smooth manifolds.
method Analysis of random walks on geometric and directed kNN graphs, using concentration tools and differential geometry.
result Uniform convergence of kNN Laplacians to diffusion Laplacian, without continuity of transition kernel. Geodesics found in spacetime satisfy curvature conditions.
problem Finding geodesics in spacetime satisfying specific curvature conditions.
method Proving existence of geodesics with entropic semiconvexity and uniform L∞ densities. result Existence of geodesics satisfying the timelike curvature-dimension condition.
Estimates for complex Monge-Ampère equations lead to insights on moduli spaces and singular metrics.
problem Uniform estimates for complex Monge-Ampère equations on Kähler manifolds.
method Refined techniques to control degenerate equations and analyze families of singular Kähler-Einstein metrics.
result Uniform integrability properties and insights into moduli spaces of stable varieties.
The study shows that certain graphs are regular at boundary points.
problem Boundary regularity of anisotropic minimal Lipschitz graphs.
method Proves regularity for graphs with bounded anisotropic mean curvature and atomic energy condition.
result Regularity at boundary points with density bounded above by 1/2 + σ.
I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume element, some information on the topology and metric is encoded in the partial or…
A new method for density estimation using mixture discrepancy and moments.
problem Generalizing histogram statistics to higher dimensions.
method Density estimation via mixture discrepancy and moments (DSP-mix and MSP).
result DSP-mix and MSP are computationally tractable and maintain accuracy with increased speed.
New method improves uncertainty calibration in deep learning.
problem Systematic overconfidence in EDL on out-of-distribution inputs.
method Density-Informed Pseudo-count EDL (DIP-EDL) separates class prediction from uncertainty.
result DIP-EDL achieves asymptotic concentration and enhances robustness and uncertainty calibration.
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. Proves ε-regularity for capillary surfaces in Riemannian manifolds.
problem Regularity of minimal surfaces with capillary boundary conditions.
method Uniform first variation control and ε-regularity theorems for varifolds.
result Capillary varifolds with bounded mean curvature and close to a capillary half-plane coincide with a C1,α properly embedded hypersurface. Generative model for tabular data density regression.
problem Estimating conditional distribution of outcomes given covariates.
method Tree-based flow model for efficient sampling and likelihood evaluation.
result Our method achieves comparable or superior performance with reduced training and sampling costs.
New study shows limits of low-degree algorithms in finding large independent sets in sparse hypergraphs.
problem Finding large independent sets in sparse random hypergraphs.
method Low-degree polynomial algorithms are analyzed to determine their limits.
result Low-degree algorithms can find independent sets of density up to \(\left(\frac{\log d}{(r-1)d}
ight)^{1/(r-1)}\), but no larger.
The Hartman-Watson distribution with density fr(t) is a probability distribution defined on t≥0 which appears in several problems of applied probability. The density of this distribution is expressed in terms of an integral θ(r,t) which is difficult to evaluate numerically for small t→0. Using saddle p…