Extends diffuse interface methods to graphs and hypergraphs with non-smooth potentials.
problem Semi-supervised learning on graphs and hypergraphs.
method Generalizes diffuse interface methods using non-smooth potential functions and hypergraph Laplacians.
result The diffuse interface method can be applied to both graph and hypergraph data.
New sampling algorithm for non-smooth potentials.
problem Sampling from non-smooth potentials.
method Proximal algorithm based on rejection sampling.
result Achieves better complexity than existing methods.
New algorithm samples efficiently from complex composite potentials.
problem Sampling from densities with smooth and non-smooth components.
method Metropolis-Hastings framework with proximal-based proposal.
result Mixes to target density in O ( d log ( d / ε ) ) O(d \log (d/\varepsilon)) O ( d log ( d / ε )) iterations. This work defines mean curvature in non-smooth spaces and proves sharp inequalities.
problem Defining and proving geometric inequalities in non-smooth metric spaces.
method Introducing mean curvature for level sets in non-smooth spaces and proving sharp inequalities.
result Mean curvature vectors in non-smooth spaces satisfy sharp Willmore inequalities.
Sharp ABP estimate on metric spaces via optimal transport.
problem Sharp ABP estimate on metric measure spaces.
method Optimal transport theory.
result Established a sharp ABP estimate on metric measure spaces.
Expanding FCCO to non-smooth weakly-convex problems, improving deep learning performance.
problem Addressing the limitations of current FCCO methods by tackling non-smooth weakly-convex problems.
method Developed a single-loop algorithm for non-smooth weakly-convex FCCO and extended it to tri-level problems.
result Established the complexity for finding ε-stationary points in the Moreau envelop of the objective function.
This work improves SGMs' convergence guarantees for semiconvex distributions with discontinuous gradients.
problem Establishing convergence guarantees for SGMs under weak regularity conditions.
method Developed non-asymptotic Wasserstein-2 convergence analysis for SGMs targeting semiconvex distributions with discontinuous gradients.
result Achieved optimal dependence of O ( d ) O(\sqrt{d}) O ( d ) on data dimension d d d and convergence rate of order one. Investigates training challenges for morphological neural networks.
problem Training morphological neural networks with gradient descent is difficult.
method Examines differentiation and backpropagation for morphological networks, using Bouligand derivative.
result Provides theoretical insights and guidelines for initialization and learning rates.
A new optimization method, BPM, converges linearly in non-convex, non-smooth problems.
problem Non-smooth and non-convex optimization challenges.
method Ball-Proximal Point Method (BPM), inspired by Proximal Point Method (PPM).
result BPM converges linearly and in a finite number of steps in non-convex, non-smooth problems.
New algorithm finds local minima in non-convex, non-smooth problems.
problem Finding local minimizers in non-convex and non-smooth optimization.
method Perturbed Proximal Descent, tailored for non-smooth cases.
result First known results for non-smooth optimization.
Proves convergence of PSGLA for sampling non-convex potentials.
problem Sampling from non-convex potentials with stability.
method Combines ULA and proximal optimization with stability analysis.
result First proof of convergence for PSGLA on non-convex potentials.
MARINA-P improves non-smooth federated optimization with adaptive stepsizes.
problem Non-smooth federated optimization in machine learning applications.
method Extends EF21-P and MARINA-P to non-smooth convex setting, proving optimal convergence rate and communication complexity bounds.
result MARINA-P achieves O ( 1 / T ) O(1/\sqrt{T}) O ( 1/ T ) convergence rate and communication complexity matching classical subgradient methods. 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.
AsylADMM improves gossip-based learning for non-smooth objectives.
problem Efficient and robust decentralized learning on edge devices.
method Asynchronous gossip algorithm for non-smooth optimization.
result AsylADMM converges faster on non-smooth problems.
Survey on preserving curvature bounds for non-smooth Ricci flow.
problem Preserving curvature bounds for non-smooth initial data in Ricci flow.
method Survey of various weak initial data and preservation of curvature bounds.
result Various curvature lower bounds preserved up to a constant for non-smooth initial data.
We introduce non-smooth symplectic forms on manifolds and describe corresponding Poisson structures on the algebra of Colombeau generalized functions. This is achieved by establishing an extension of the classical map of smooth functions to Hamiltonian vector fields to the setting of non-smooth geometry. For mildly sin…
New bounds explain deterministic non-smooth deep nets without large Lipschitz constants.
problem Challenges in explaining generalization of deterministic non-smooth deep nets.
method De-randomized PAC-Bayes margin bounds for deterministic non-convex and non-smooth predictors.
result New bounds avoid large Lipschitz constants, providing generalization guarantees.
The paper explores various stationarity concepts in non-smooth optimization.
problem Understanding stationarity in non-smooth optimization problems.
method Introduction and discussion of different stationarity concepts for non-convex non-smooth functions.
result Clarification of the relationship among different stationarity concepts and their relevance in iterative methods.
Smoothness analysis of adversarial training reveals L ∞ L_\infty L ∞ constraints cause more non-smoothness.
problem Non-smoothness of adversarial training loss function.
method Analyzed the smoothness of adversarial training loss function using optimal attacks for model parameters.
result The L ∞ L_\infty L ∞ constraint causes more non-smoothness than L 2 L_2 L 2 constraint. In the framework of Lorentzian warped products, we study the Friedmann-Robertson-Walker cosmological model to investigate non-smooth curvatures associated with multiple discontinuities involved in the evolution of the universe. In particular we analyze non-smooth features of the spatially flat Friedmann-Robertson-Walke…
Improved non-smooth optimization methods achieve faster convergence rates.
problem Non-smooth optimization problems, especially in ℓ ∞ \ell_\infty ℓ ∞ and ℓ 1 \ell_1 ℓ 1 -SVM. method Higher-order accelerated methods, leveraging recent advances in smooth convex optimization.
result Achieved O ( ε − 4 / 5 ) O(ε^{-4/5}) O ( ε − 4/5 ) iteration complexity for ℓ ∞ \ell_\infty ℓ ∞ regression, breaking previous barriers. This work speeds up hyperparameter selection for non-smooth convex models using implicit differentiation.
problem Optimizing hyperparameters of non-smooth convex models.
method Implicit differentiation of proximal gradient and coordinate descent methods.
result Implicit differentiation can speed up hyperparameter optimization, especially for non-smooth problems.
Synthetic framework for null hypersurfaces in non-smooth spacetimes.
problem Analyzing null hypersurfaces in non-smooth spacetimes.
method Develops synthetic null hypersurfaces using optimal transport and Lorentzian geometry.
result Synthetic null energy condition stabilizes under convergence and applies to low-regularity spacetimes.
Deep neural networks excel at learning non-smooth functions.
problem Understanding why deep neural networks perform better for non-smooth functions.
method Theoretical analysis of statistical properties of deep neural networks for non-smooth functions.
result Deep neural networks achieve almost optimal generalization error for non-smooth functions.
New algorithms optimize non-smooth, non-convex objectives with improved complexity.
problem Optimizing non-smooth, non-convex stochastic objectives.
method Reduction to online learning, applying optimistic online learning techniques.
result Improved complexity for finding ( δ , ε ) (δ,ε) ( δ , ε ) -stationary points. In this paper, we develop a novel {\bf ho}moto{\bf p}y {\bf s}moothing (HOPS) algorithm for solving a family of non-smooth problems that is composed of a non-smooth term with an explicit max-structure and a smooth term or a simple non-smooth term whose proximal mapping is easy to compute. The best known iteration compl…
Positive mass theorem for non-smooth metrics on flat manifolds with corners.
problem Proving a positive mass theorem for non-smooth metrics on asymptotically flat manifolds with non-compact boundary.
method Proves a positive mass theorem for metrics that are only continuous across a compact hypersurface.
result Obtains a positive mass theorem on manifolds with non-compact corners.
Paper tackles private optimization for non-smooth objectives efficiently.
problem Private stochastic convex optimization for non-smooth objectives.
method Noisy mirror descent algorithm.
result Achieves optimal rates in statistical complexity and number of queries.
We investigate a generalization of the so-called metric splitting of globally hyperbolic space-times to non-smooth Lorentzian manifolds and show the existence of this metric splitting for a class of wave-type space-times. Our approach is based on smooth approximations of non-smooth space-times by families (or sequences…
Advances smooth over-parameterization for solving non-smooth optimization problems.
problem Non-smooth optimization with structural constraints in imaging and machine learning.
method Smooth over-parameterization of non-smooth problems, using gradient descent and mirror descent.
result Gradient descent on the reformulated smooth problem converges efficiently without parameter tuning.
The study analyzes methods for solving non-convex, non-smooth optimization problems.
problem Finding critical points of non-convex and non-smooth functions.
method Gradient descent, proximal update, Frank-Wolfe update methods for general and continuous sub-analytic functions.
result Established rates of convergence and faster rates for specific function classes.
New methods improve convergence in non-convex non-smooth learning problems.
problem Sparse learning from high-dimensional data with non-convex, non-smooth regularizers.
method Stochastic proximal gradient methods with arbitrary sampling.
result Independent sampling improves performance over uniform sampling.
Proposes a differentiable LSE-ICNN for modeling multi-well potentials.
problem Modeling multi-well potentials in various scientific domains.
method Log-sum-exponential (LSE) mixture of input convex neural network (ICNN) modes.
result Smooth surrogate that retains convexity within basins and allows gradient-based learning.
The study analyzes convergence rates for sparse pivotal estimators in high-dimensional regression.
problem Sparse pivotal estimation in high-dimensional regression problems.
method Theoretical analysis and comparison of non-smooth + non-smooth optimization problems, including smoothing techniques.
result Minimax sup-norm convergence rates for square-root Lasso-type estimators are derived.
Adaptive data fusion boosts efficiency in multi-task optimization.
problem Multi-task non-smooth optimization in various fields.
method Adaptive data fusion approach leveraging commonalities among objectives.
result Significant improvements in sample efficiency with sharp statistical guarantees.
New method smooths optimization for sparse regularization.
problem Non-smooth, non-convex optimization problems for sparsity.
method Overparameterization and smooth surrogate penalties.
result Surrogate objective has identical global and local minima.
Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in non-smooth differential geometry. This article presents an introductory survey of recent developments in this field and highlights some applications in mathematical physics.
Abstracts a theorem for non-smooth maps in infinite dimensions.
problem Generalizing inverse mapping theorem for non-smooth maps.
method Introduces property A and applies it to non-smooth maps.
result Generalized inverse mapping theorems for non-smooth maps.
New SPS variant improves non-smooth optimization without small gradients.
problem Improving non-smooth optimization without small gradients.
method Safeguarded Stochastic Polyak Step Size (SPS s a f e _{safe} s a f e ) for non-smooth optimization. result Rigorous convergence guarantees for non-smooth convex optimization without strong assumptions.
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.
Stochastic approximation proves asymptotic normality for non-smooth problems.
problem Solving non-smooth stochastic approximation problems.
method Stochastic approximation algorithms for solving smooth equations, extended to non-smooth problems.
result Asymptotic normality and optimality in non-smooth stochastic approximation is proven.
Bayesian optimization tackles non-smooth tuning problems.
problem Optimizing black-box functions with non-smoothness and limited samples.
method Proposed a clustered Gaussian process (cGP) model for non-smooth optimization.
result Improvement of up to 90% in performance for repetitive experiments.
Study of spectral gaps in non-smooth spaces with bounded Ricci curvature.
problem Analyzing spectral gaps in non-smooth metric measure spaces.
method Establishing a Polya-Szego type inequality and applying it to show spectral gaps for the p-Laplace operator.
result Sharp spectral gap results for the p-Laplace operator on various non-smooth spaces.
Study weak Frenet frame for non-smooth curves with finite curvature and torsion.
problem Defining weak binormal and normal for non-smooth curves with finite total curvature and torsion.
method Piecewise linear methods and density argument applied to polygonal curves.
result Weak binormal and normal are rectifiable curves agreeing with total absolute torsion and vector product of tangent indicatrix and weak binormal.
Gradient descent can optimize penalty parameters for non-smooth penalties.
problem Optimizing penalty parameters for non-smooth penalties in regression problems.
method Modified gradient descent algorithm for non-smooth penalty functions.
result Decreased generalization error with tuned penalty parameters.
We consider first order gradient methods for effectively optimizing a composite objective in the form of a sum of smooth and, potentially, non-smooth functions. We present accelerated and adaptive gradient methods, called FLAG and FLARE, which can offer the best of both worlds. They can achieve the optimal convergence …
Safe-EF improves federated learning for non-smooth, constrained optimization.
problem Federated learning's communication bottlenecks with high-dimensional model updates.
method Error feedback (EF) for non-smooth convex optimization with safety constraints.
result Safe-EF matches lower complexity bounds and ensures safety constraints.
Inertial methods solve non-convex non-smooth optimization problems efficiently.
problem Non-convex non-smooth optimization problems.
method Inertial block proximal methods for solving these problems.
result The methods converge globally under certain conditions and perform well in applications like NMF.