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…
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48 results for “Non-Smooth Functions”
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.
Perturbed Proximal Descent to Escape Saddle Points for Non-convex and Non-smooth Objective Functionscs.LG
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.
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.
Smoothness analysis of adversarial training reveals 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 constraint causes more non-smoothness than constraint.
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.
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.
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.
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…
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.
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 iterations.
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.
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.
Theoretical Limits of Pipeline Parallel Optimization and Application to Distributed Deep Learningstat.ML
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.
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.
New Morse theory for shapes at distances.
problem Understanding shapes at distances from a reference point.
method Defining Morse functions and using non-smooth analysis, geometric measure theory.
result Homotopy type changes at critical values, with one cell added per critical point.
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.
In this paper, we discuss the problem of minimizing the sum of two convex functions: a smooth function plus a non-smooth function. Further, the smooth part can be expressed by the average of a large number of smooth component functions, and the non-smooth part is equipped with a simple proximal mapping. We propose a pr…
Injectivity of X-ray transform proven for non-smooth metrics.
problem Injectivity of X-ray transform on non-smooth metrics.
method Microlocal analysis of the normal operator, establishing ellipticity and smoothing properties.
result Injectivity of X-ray transform on for metrics with finitely differentiable tensor.
Novel method for shape optimization of non-smooth PDEs.
problem Optimizing shapes governed by non-smooth PDEs.
method Functional variational approach and sensitivity analysis.
result Necessary conditions for locally optimal shapes.
The paper proposes a method to model non-smooth functions using clustering, classification, and Gaussian process modeling.
problem Modeling discontinuities and non-smoothness in expensive computational models.
method Three-stage approach combining clustering, classification, and Gaussian process modeling.
result The approach successfully models discontinuities and non-smoothness in various functions.
Extending Itô's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer-Itô, applies to one dimensional semimartingales and convex functions. There are also satisfactory generalizations of Itô's formula for diffusion processes whe…
Paper proposes ADMM algorithms for non-smooth optimization under RDP.
problem Optimizing composite functions with non-smooth penalties under privacy constraints.
method Developed ssADMM and mpADMM algorithms for non-smooth optimization problems with RDP guarantees.
result Both ssADMM and mpADMM outperform baseline methods in high privacy settings.
This work improves polynomial approximations for functions with asymmetric behavior.
problem Efficiently approximating functions with asymmetric behavior, especially those growing unbounded on one side.
method Introduces weighted deep polynomial approximants that combine learnable deep polynomials with one-sided weights.
result Weighted deep polynomial approximants outperform existing methods in approximating functions with asymmetric behavior.
New iterative regularization method tackles non-smooth, non-strongly convex functionals.
problem Tackles non-smooth, non-strongly convex functionals in regularization problems.
method Primal-dual algorithm with convergence and stability analysis.
result First iterative regularization procedure for non-smooth, non-strongly convex functionals.
Bayesian Probabilistic Integration uses BART for high-dimensional, non-smooth functions.
problem Bayesian quadrature's limitations in high-dimensional or non-smooth functions.
method Bayesian Additive Regression Trees (BART) priors for numerical integration.
result Explicit convergence rates can be obtained in various settings.
Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required non-trivial smoothness assumptions, which do not apply to many modern applications of SGD with non-smooth object…
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.
Optimal private ERM and SCO with subquadratic gradient complexity.
problem Private optimization of non-smooth convex functions.
method Subquadratic gradient complexity algorithm using subsampling and smoothing.
result Achieved optimal excess empirical risk and population loss.
The study extends curvature bounds to non-smooth spaces and proves stability of mean curvature.
problem Proving curvature bounds in non-smooth spaces.
method Extending results from smooth Riemannian manifolds to non-smooth RCD spaces.
result Stability of mean curvature bounds under uniform convergence.
Stochastic gradient descent's long-term fluctuations are described by a diffusion limit.
problem Long-term behavior of stochastic gradient descent in non-smooth settings.
method Functional central limit theorem applied to rescaled trajectory of SGD.
result Characterization of long-term fluctuations around the minimizer.
Algorithms for solving optimization problems arising from deep neural net models: nonsmooth problemsmath.OC
Paper tackles optimization challenges in deep neural nets with nonconvex and non-smooth objectives.
problem Optimization of deep neural net models with nonconvex and non-smooth objectives.
method Summarizes challenges, state of the art, and presents numerical results on a specific class of problems.
result Numerical results on non-convex and non-smooth optimization problems.
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 convergence rate and communication complexity matching classical subgradient methods.
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.
Develops wavelet-based neural network approximation theory.
problem Analyzing neural network approximation capabilities over various activation functions.
method Wavelet frame theory on spaces of homogeneous type, sufficient conditions for approximation, error estimates.
result Derives sufficient conditions for neural networks to approximate any functions in a given space, including non-smooth activations.
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.
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.
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.
Researchers relax the CVF's smoothness requirement to create more flexible flow models.
problem Challenges in constructing flexible density models due to the CVF's smoothness requirement.
method Introduce -diffeomorphisms as generalized transformations that may violate smoothness on zero Lebesgue-measure sets.
result The relaxation allows for the use of non-smooth activation functions like ReLU in residual flows.
Parallel optimization limits are tight for non-smooth convex functions.
problem Limiting parallel acceleration in convex optimization.
method Information-theoretic measure of adaptivity, lower bounds for parallel runtime.
result No randomized algorithm can achieve better convergence rates than a one-query-per-round algorithm with adaptivity better than .
Develops nonparametric regression for non-smooth functions using fractional Laplacian.
problem Non-smooth regression functions in high dimensions.
method Fractional Laplacian eigenmaps for -fractional Sobolev spaces.
result Upper bound on estimation error of $n^{- rac{2s}{2s+d}}$.
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.
Diffuse interface methods have recently been introduced for the task of semi-supervised learning. The underlying model is well-known in materials science but was extended to graphs using a Ginzburg--Landau functional and the graph Laplacian. We here generalize the previously proposed model by a non-smooth potential fun…
New algorithm for federated learning with non-smooth regularizers.
problem Federated Learning with non-smooth composite optimization problems.
method Proposed Federated Dual Averaging (FedDualAvg) algorithm to overcome convergence issues.
result FedDualAvg outperforms other algorithms in federated composite optimization.
Improved stochastic gradient descent analysis for non-smooth convex functions.
problem Minimizing non-smooth, non-differentiable convex functions.
method Stochastic gradient descent with suffix averaging method analysis.
result Error rate of final iterate is with high probability.
Adaptive NN method improves matrix completion for non-smooth data.
problem Matrix completion with non-smooth non-linear functions under high missingness.
method Two-sided nearest neighbors with \Holder function class non-linearity.
result NN error rate matches oracle's for latent factors, non-trivial for wide range of missingness.
FedProx algorithm improved for non-smooth and heterogeneous data.
problem Theoretical understanding of FedProx for non-convex federated optimization.
method Local dissimilarity invariant convergence theory through algorithmic stability.
result Convergence guarantees for non-smooth FL problems and minibatch size.
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…