arXiv research

Abstracts a theorem for non-smooth maps in infinite dimensions.

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…

Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölde…

Survey on harmonic maps in non-smooth spaces, focusing on rigidity.

Recent research connects Hörmander's old work to modern boundary Laplacian analysis.

Novel method for shape optimization of non-smooth PDEs.

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…

Study partial derivatives on non-smooth metric measure structures.

Using tools and results from geometric measure theory, we give a simple new proof of the main result (Theorem 1.3) in K. Kondo and M. Tanaka, Approximation of Lipschitz Maps via Immersions and Differentiable Exotic Sphere Theorems, \textit{Nonlinear Anal.} \textbf{155} (2017), 219--249, as well as the converse statemen…

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…

The paper studies moduli spaces of non-smooth metric structures with non-negative Ricci curvature.

Consistency models accelerate generation with theoretical guarantees.

The notion of topological degree is studied for mappings from the boundary of a relatively compact strictly pseudo-convex domain in a Stein manifold into a manifold in terms of index theory of Toeplitz operators on the Hardy space. The index formalism of non-commutative geometry is used to derive analytic integral form…

MARINA-P improves non-smooth federated optimization with adaptive stepsizes.

Lipschitz maps on metric surfaces are rigid if they preserve area.

Extends curve theory to non-smooth data with finite curvature and torsion.

AsylADMM improves gossip-based learning for non-smooth objectives.

This paper shows that the time $t$ map of the averaged Euler equations, with Dirichlet, Neumann, and mixed boundary conditions is canonical relative to a Lie-Poisson bracket constructed via a non-smooth reduction for the corresponding diffeomorphism groups. It is also shown that the geodesic spray for Neumann and mixed…

Survey on preserving curvature bounds for non-smooth Ricci flow.

Quantile Temporal-Difference learning proved convergent with proof.

The paper explores various stationarity concepts in non-smooth optimization.

New bounds explain deterministic non-smooth deep nets without large Lipschitz constants.

New example of non-smooth isotopy after stabilization in 4-manifolds.

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…

Smoothness analysis of adversarial training reveals $L_\infty$ constraints cause more non-smoothness.

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…

This work speeds up hyperparameter selection for non-smooth convex models using implicit differentiation.

New algorithms optimize non-smooth, non-convex objectives with improved complexity.

Positive mass theorem for non-smooth metrics on flat manifolds with corners.

Paper tackles private optimization for non-smooth objectives efficiently.

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.

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains $C$, with non-smooth boundary, in possibly non-compact manifolds. Assuming $C$ is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restrictio…

New methods improve convergence in non-convex non-smooth learning problems.

The study analyzes convergence rates for sparse pivotal estimators in high-dimensional regression.

Adaptive data fusion boosts efficiency in multi-task optimization.

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.

Proves isometric embeddings in Euclidean spaces for RCD spaces.

New SPS variant improves non-smooth optimization without small gradients.

Stochastic approximation proves asymptotic normality for non-smooth problems.

Bayesian optimization tackles non-smooth tuning problems.

We provide improved convergence rates for various \emph{non-smooth} optimization problems via higher-order accelerated methods. In the case of $\ell_\infty$ regression, we achieves an $O(ε^{-4/5})$ iteration complexity, breaking the $O(ε^{-1})$ barrier so far present for previous methods. We arrive at a similar rate fo…

Safe-EF improves federated learning for non-smooth, constrained optimization.

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.…

Modified perturbation method removes non-smoothness in solving Black-Scholes equations.

New method for efficient proximal mapping of 1-path-norm in shallow networks.

The study extends curvature bounds to non-smooth spaces and proves stability of mean curvature.

Expanding FCCO to non-smooth weakly-convex problems, improving deep learning performance.