Study of Laplacians on smooth distributions in compact manifolds.
problem Understanding spectral properties of Laplacians on smooth distributions.
method Proving Laplacian as an unbounded regular self-adjoint operator in a Hilbert module over the foliation C*-algebra.
result Laplacians on smooth distributions define unbounded regular self-adjoint operators.
Smooth distributions on subcartesian spaces can be globally finitely generated.
problem Understanding smooth distributions on subcartesian spaces.
method Embedding in Euclidean space, Whitney Embedding Theorem, and distribution theory.
result Smooth generalized distributions and subbundles on connected subcartesian spaces are globally finitely generated.
Riemannian metrics and Laplacians defined for complex distributions on manifolds.
problem Defining metrics and Laplacians for distributions on manifolds of varying rank.
method Introduced a Riemannian metric and Laplace operator for generalised smooth distributions on manifolds.
result Essentially self-adjoint Laplacian on compact manifolds, hypoellipticity proven.
The paper studies Laplacians on smooth distributions and proves they are multipliers in C∗-algebras.
problem Understanding spectral properties of Laplacians on generalized smooth distributions.
method Survey of generalized smooth distributions, proof of Laplacian as a multiplier in foliation C∗-algebra. result Laplacians on smooth distributions define unbounded multipliers in foliation C∗-algebras. A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show …
Theorem proves integrability for piecewise-smooth distributions.
problem Integrability of piecewise-smooth distributions.
method Generalizations of Frobenius integrability theorem.
result Sufficient criteria for complete integrability with bi-Lipschitz coordinates.
This work investigates the properties of Gaussian-smoothed sliced divergences for comparing distributions.
problem Comparing probability distributions while preserving privacy.
method Investigates the theoretical properties of Gaussian-smoothed sliced Wasserstein distance and generalized versions.
result Gaussian smoothed sliced Wasserstein distance converges with a rate of \(O(n^{-1/2})\).
Defines Laplacian on smooth distributions and proves operator properties.
problem Defining and analyzing Laplacian on smooth distributions.
method Pseudodifferential calculus on singular foliations and subelliptic estimates.
result Operator properties of Laplacian on smooth distributions.
Extends curve theory to non-smooth data with finite curvature and torsion.
problem Applying classical curve theory to non-smooth data.
method Using distributional derivative measures of functions of bounded variation.
result Essentially unique non-smooth curve solution with finite total curvature and torsion.
The paper extends geometric surface properties to currents tangent to smooth distributions.
problem Understanding the geometric structure of currents tangent to smooth distributions.
method Analyzing integral and normal currents, focusing on their geometric properties and boundary.
result Integral currents behave like smooth surfaces, while normal currents have a more complex behavior.
SIXO improves inference by learning smoothing distributions from all observations.
problem Inference limitations due to ignoring future observations in filtering distributions.
method Density ratio estimation to warp filtering distributions into smoothing distributions, then use SMC with learned targets.
result Proves tighter log marginal lower bounds and more accurate inferences and estimates.
Efficient EP algorithm improves smoothing distribution inference in financial models.
problem Computational intractability of smoothing distribution in high dimensions.
method Adapted expectation propagation (EP) algorithms for the unified skew-normal family.
result Accuracy gains in financial illustrations over existing approximate algorithms.
Study on energy of smooth and singular distributions on manifolds.
problem Energy calculation for smooth and singular distributions on manifolds.
method Derive lower bounds and find minimizers for energy functionals.
result Lower bounds and minimizers for energy of distributions found.
Pairwise Label Smoothing improves deep model generalization by reducing overconfidence.
problem Improving deep model generalization through regularization.
method PLS smooths labels for pairs of samples, learning distribution mass during training.
result PLS significantly outperforms LS and baseline models, reducing up to 30% classification error.
New algorithm for differentially private distributed optimization of smooth, non-convex problems.
problem No differentially private distributed method for smooth, non-convex optimization problems.
method Smoothed normalization integrated with an error-feedback mechanism.
result Achieves superior convergence rate and first differentially private distributed optimization algorithm with provable convergence guarantees.
This paper extends exponential smoothing to distributional time series using Wasserstein distance.
problem Forecasting distributional time series with exponential smoothing.
method Generalized exponential smoothing in Wasserstein space, with consistent parameter estimation.
result Wasserstein exponential smoothing outperforms traditional methods in high-frequency financial and electricity demand data.
Improved OOD detection using label smoothing and k-NN density estimates.
problem Detecting out-of-distribution examples in classification models.
method Label smoothing and k-NN density estimate on intermediate activations.
result Label smoothing improves OOD detection performance, both theoretically and empirically.
We study diffeologies on locally convex spaces and their application to smooth multiplication of distributions.
problem Constructing smooth multiplication of distributions on locally convex spaces.
method Using diffeological colimits and wavefront-set criterion.
result Proving smooth multiplication of microlocally multipliable distributions.
High-dimensional smoothing techniques struggle with robustness guarantees against various attacks.
problem Challenges in extending randomized smoothing to other attack models in high-dimensional space.
method Analysis of isotropic and generalized Gaussian smoothing distributions, proving bounds on certified robustness radii.
result Certifiable robustness radii decrease as $O(1/d^{ rac{1}{2} - rac{1}{p}})$ with dimension d for p>2. New method improves robustness of smoothed classifiers against adversarial attacks.
problem Improving robustness of smoothed classifiers against adversarial attacks.
method Proposes worst-case adversarial loss over input distributions as a robustness certificate, and uses duality and smoothness properties to provide an easy-to-compute upper bound.
result Shows superior robustness performance over state-of-the-art certified or heuristic methods.
LSAM optimizes deep learning training with improved efficiency.
problem Inefficiency in distributed large-batch training with Sharpness-Aware Minimization (SAM).
method Integrates SAM's adversarial steps with an asynchronous distributed sampling strategy.
result Higher final accuracy compared to data-parallel SAM.
Smooth contact mappings in a flat (2,3,5)-distribution are shown to be smoother.
problem Characterizing smoothness of contact mappings in a specific geometric setting.
method Study of differential identities and rigidity of stratified Lie groups.
result Smooth contact mappings are actually smoother than initially assumed.
New method improves counterfactual distribution learning for high-dimensional outcomes.
problem Counterfactual distribution learning for high-dimensional outcomes with concentrated structure.
method Geometry-adaptive diffusion-guided smoothing estimators combining causal nuisance adjustment and local outcome geometry.
result Geometry-adaptive methods show steeper error decay in semi-synthetic experiments.
Study on diffeologies on locally convex spaces and smooth multiplication of distributions.
problem Geometric characterization and smoothness of distribution multiplication.
method Investigation of canonical and c∞-diffeologies on locally convex spaces, proving geometric characterizations, and comparing diffeologies. result Established a framework for nonlinear distribution theory beyond manifolds, realizing microlocally multipliable distributions as a diffeological colimit.
Whereas subriemannian geometry usually deals with smooth horizontal distributions, partially hyperbolic dynamical systems provide many examples of subriemannian geometries defined by non-smooth (namely, Hölder continuous) distributions. These distributions are of great significance for the behavior of the parent dynami…
New insights into learning from distributional adversaries and private data.
problem Understanding minimal assumptions for learning and generalization under distributional constraints.
method Generalized smoothness as a characterization of learnability and privacy under distributional adversaries.
result Near complete characterization of families that admit learnability and privacy under distributional adversaries.
Develops a distributed strategy for Pareto optimization of aggregate costs with smoothed regularizers.
problem Optimizing aggregate costs with non-smooth regularizers in a network of agents.
method Distributed strategy using infimal convolution to smooth regularizers, seeking Pareto optimal solution via diffusion.
result Pareto solution of smoothed problem can be made arbitrarily close to original non-smooth problem.
Proves Sard conjecture for specific distributions, controlling divergence of vector fields.
problem Proving the Sard conjecture for certain types of distributions.
method Constructs a singular distribution capturing essential abnormal lifts, proving the conjecture for rank 3 distributions in dimension 4 and generic corank 1 distributions.
result Proves the Sard conjecture for generic co-rank one distributions.
Gibbs sampler mixes quickly for certain smooth distributions.
problem Drawing samples from log-smooth log-concave distributions.
method Analyzes Gibbs sampler on log-smooth and strongly log-concave distributions.
result Gibbs sampler mixes in O⋆(κ2n7.5) steps. ID3 learns juntas under smoothed product distributions.
problem Understanding the ID3 algorithm's performance on juntas.
method Smoothed analysis of ID3 for learning k-Juntas.
result ID3 learns k-Juntas in polynomial time when k = log n.
The paper improves smoothed analysis for online problems with adaptive adversaries.
problem Online prediction, discrepancy minimization, and online optimization with adaptive adversaries.
method General technique to prove smoothed guarantees against adaptive adversaries, reducing to simpler oblivious adversaries.
result Strong smoothed guarantees for three online problems, matching or improving previous results.
Paper proposes an algorithm for sampling from complex mixture distributions without requiring smoothness.
problem Sampling from a mixture of weakly smooth potentials.
method Unadjusted Langevin algorithm with Euler discretization for a mixture of weakly smooth distributions.
result Convergence in Kullback-Leibler divergence and Lβ-Wasserstein metric with polynomial dependence on dimension. A deep learning method for estimating discrete conditional distributions efficiently.
problem Estimating discrete conditional probability distributions efficiently.
method Smoothed dyadic partitioning and graph-based smoothing.
result Significantly reduces error in conditional distribution estimation.
Study learns mixtures of smooth product distributions from samples.
problem Learning mixtures of non-parametric product distributions.
method Two-stage approach using identifiability properties of tensor decomposition and signal processing techniques.
result Recovery of component distributions under a smoothness condition.
Paper analyzes KSG mutual information estimator for smooth distributions.
problem Analyzing the convergence rate of KSG estimator for smooth distributions.
method Adaptive recombination of KL entropy estimators analysis.
result Convergence rate of KSG estimator for smooth distributions is analyzed.
New bounds for agnostic learning with average smoothness.
problem Distribution-free nonparametric regression with average smoothness.
method Distribution-free uniform convergence bounds and agnostic learning algorithm.
result Distribution-free uniform convergence bounds for average-smoothness classes in the agnostic setting.
A new method for generating samples without training, using smoothed score matching.
problem Generating samples efficiently and without training.
method Moment-matched score-smoothed overdamped Langevin dynamics (MM-SOLD).
result The method enables fast, robust, training-free sampling with competitive sample fidelity and diversity.
New method samples from piecewise smooth distributions using Hamiltonian Monte Carlo.
problem Sampling from distributions with discontinuous gradients.
method Generalized Randomized Hamiltonian Monte Carlo (GRHMC) for piecewise smooth targets.
result GRHMC processes sample from piecewise smooth target distributions with the desired distribution as the invariant distribution.
FLUID uses flows to unify filtering and smoothing for complex systems.
problem Bayesian filtering and smoothing for high-dimensional nonlinear systems.
method FLUID encodes observation histories into a fixed summary statistic, using flows for filtering and smoothing.
result FLUID provides accurate approximations of filtering and smoothing distributions.
Paper proves smoothness for variational inference, giving convergence guarantees.
problem Proving convergence guarantees for black-box variational inference.
method Describes gradients in an inner-product space, using Bessel's inequality.
result Objective is M-Lipschitz smooth if target is, excluding entropy.
Optimizes deep learning pipelines with novel algorithms for smooth and non-smooth functions.
problem Optimizing deep learning pipelines for smooth and non-smooth functions.
method Provided matching lower and upper bounds for smooth convex and non-convex functions, and developed PPRS for non-smooth convex functions.
result PPRS achieves near-linear speed-up and convergence time for non-smooth non-convex problems.
Optimal inference in distributed quantile regression without stringent scaling conditions.
problem Challenges in achieving optimal inference in distributed quantile regression due to the non-smooth nature of the QR loss function.
method Double-smoothing approach applied to local and global objective functions, with a trade-off between communication cost and statistical error.
result Established a finite-sample theoretical framework for distributed QR estimators, showing a trade-off between communication cost and statistical error.
The study quantifies deep learning generalization error using data distribution and network smoothness.
problem Understanding the generalization error in deep learning models.
method Introducing cover complexity (CC) to measure data difficulty and using the inverse of the modulus of continuity to quantify neural network smoothness. A bound for expected accuracy/error is derived considering both CC and neural network smoothness.
result The expected error of trained neural networks scales with the square root of the number of classes and has a linear relationship with respect to the cover complexity.
New Langevin algorithm works well even for rough distributions.
problem Sampling from non-smooth distributions.
method Simple Langevin algorithm without smoothness assumptions.
result Algorithm performs well even with discontinuous gradients.
New findings show learning deeper neural networks is hard even with Gaussian inputs and non-degenerate weights.
problem The computational complexity of learning neural networks, especially deeper ones.
method Smoothed analysis framework and local pseudorandom generators.
result Learning depth-3 ReLU networks under Gaussian input distribution is hard even if weight matrices are non-degenerate.
New findings on boosting sample complexity and implications for hardcore theorem.
problem Understanding the sample complexity of smooth boosting and its implications.
method Analyzing the sample complexity of smooth boosting and relating it to the hardcore theorem.
result The sample complexity of smooth boosting matches existing overhead and provides a separation from distribution-independent boosting.
New polynomial convergence guarantees for SGM on general data distributions.
problem Efficient guarantees for multimodal and non-smooth distributions in SGM.
method Polynomial convergence guarantees for denoising diffusion models on general data distributions, with no assumptions on functional inequalities or smoothness.
result Wasserstein distance guarantees for distributions of bounded support or decaying tails, and TV guarantees for further smoothness assumptions.
Quantifies polynomial approximation rates for smooth functions under various distributions.
problem Approximating smooth functions with polynomials under different distributional constraints.
method Develops a quantitative analogue of Carleman's theorem using complex analysis.
result Establishes superexponential rates of approximation for certain function classes over general distributions.