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.
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.
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.
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.
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. 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. New method tackles non-smooth tensor data for better recovery.
problem Non-smooth changes in tensor data degrade traditional t-SVD methods.
method Learnable tensor nuclear norm, Alternating Proximal Multiplier Method (APMM), multi-objective tensor recovery framework.
result The proposed method effectively recovers tensor data with non-smooth changes.
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.
In high dimensional sparse regression, pivotal estimators are estimators for which the optimal regularization parameter is independent of the noise level. The canonical pivotal estimator is the square-root Lasso, formulated along with its derivatives as a "non-smooth + non-smooth" optimization problem. Modern technique…
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…
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.
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.
We consider the problem of finding local minimizers in non-convex and non-smooth optimization. Under the assumption of strict saddle points, positive results have been derived for first-order methods. We present the first known results for the non-smooth case, which requires different analysis and a different algorithm…
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…
EnCF improves data assimilation for implicit, non-smooth observations.
problem Data assimilation challenges with implicit, many-to-one observations.
method EnCF uses a stochastic controlled flow to update forecast distributions.
result EnCF outperforms Kalman filters for non-Gaussian, implicit observations.
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.
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.
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.
Efficient algorithms for large Maxent models improve wildfire probability predictions.
problem Training large-scale, non-smooth Maxent models efficiently for big data.
method First-order optimization algorithms using Kullback-Leibler divergence.
result Our algorithms outperform state-of-the-art methods by one order of magnitude.
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.
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.
Consistency models accelerate generation with theoretical guarantees.
problem Empirical success of consistency models without theoretical justification.
method Theoretical analysis of consistency models mapping inputs to arbitrary points.
result Achieve KL divergence of order O ( ε 2 ) O(\varepsilon^2) O ( ε 2 ) with $ O\left(\log\left(\frac{d}{\varepsilon}
ight)
ight) $ iterations. Smooth bundles with rough data maintain Hodge kernel isomorphism.
problem Maintaining Hodge kernel isomorphism for smooth bundles with non-smooth geometric data.
method Analyzing nilpotent differential operators and Hodge-Dirac-type operators under perturbations of geometric data.
result Kernels of Hodge-Dirac operators remain isomorphic under uniform perturbations of geometric data.
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.
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.
New bounds for online portfolio selection without smoothness assumptions.
problem Online portfolio selection with non-Lipschitz, non-smooth losses.
method Data-dependent bounds using novel smoothness characterizations and FTRL with self-concordant regularizers.
result Achieves logarithmic regrets when data is 'easy' and sublinear worst-case regrets.
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.
We provide improved convergence rates for various \emph{non-smooth} optimization problems via higher-order accelerated methods. In the case of ℓ ∞ \ell_\infty ℓ ∞ regression, we achieves an O ( ε − 4 / 5 ) O(ε^{-4/5}) O ( ε − 4/5 ) iteration complexity, breaking the O ( ε − 1 ) O(ε^{-1}) O ( ε − 1 ) barrier so far present for previous methods. We arrive at a similar rate fo…
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.
We theoretically discuss why deep neural networks (DNNs) performs better than other models in some cases by investigating statistical properties of DNNs for non-smooth functions. While DNNs have empirically shown higher performance than other standard methods, understanding its mechanism is still a challenging problem.…
Improved model for non-smooth signals with complex spectra.
problem Current models struggle with non-smooth signals and complex spectral structures.
method CGPCM and RGPCM models with causality and Bayesian nonparametric interpretations, improved variational inference.
result Proposed models show better performance on synthetic and real-world data.
Modified perturbation method removes non-smoothness in solving Black-Scholes equations.
problem Non-smoothness in solving Black-Scholes equations.
method Variable transformations and homotopy perturbation method.
result Excellent agreement with exact solutions for Black-Scholes and multi-asset options.
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.
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.
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…
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.
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 L 2 L^2 L 2 for metrics with finitely differentiable tensor. Timelike curvature and Brunn-Minkowski inequality linked in non-smooth spacetimes.
problem Equivalence between timelike Ricci curvature and Brunn-Minkowski inequality in synthetic Lorentzian spaces.
method Introducing strong q q q -timelike Brunn-Minkowski condition and proving equivalence to curvature conditions. result Timelike curvature dimension condition equivalent to timelike Brunn-Minkowski inequality in specific 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…
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.
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.
We consider the problem of sampling from a density of the form p ( x ) ∝ exp ( − f ( x ) − g ( x ) ) p(x) \propto \exp(-f(x)- g(x)) p ( x ) ∝ exp ( − f ( x ) − g ( x )) , where f : R d → R f: \mathbb{R}^d \rightarrow \mathbb{R} f : R d → R is a smooth and strongly convex function and g : R d → R g: \mathbb{R}^d \rightarrow \mathbb{R} g : R d → R is a convex and Lipschitz function. We propose a new algorithm based on the Metropolis-Has…