arXiv research

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

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

Study partial derivatives on non-smooth metric measure structures.

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.

Injectivity of X-ray transform proven for non-smooth metrics.

We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of $L^2$, Quillen and BCOV metrics on Hodge and determinant bundles. The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown to be related to well-known topological invarian…

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

In this paper we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, d…

Sharp ABP estimate on metric spaces via optimal transport.

This paper shows moduli spaces of RCD(0,2) structures are contractible.

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 constructs Ricci flow solutions for non-smooth metrics in four dimensions.

Study solves Gel'fand's inverse problem in non-smooth spaces with Ricci curvature bounds.

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

Metrics are semipositively curved if they meet a specific asymptotic condition.

Smooth bundles with rough data maintain Hodge kernel isomorphism.

We characterize the convexity of functions and the monotonicity of vector fields on metric measure spaces with Riemannian Ricci curvature bounded from below. Our result offers a new approach to deal with some rigidity theorems such as `splitting theorem' and `volume cone implies metric cone theorem' in non-smooth conte…

Study shows Bergman metric is non-Einstein for certain domains.

In this paper, we extend the Hamilton's gradient estimates \cite{har93} and a monotonicity formula of entropy \cite{ni04} for heat flows from smooth Riemannian manifolds to (non-smooth) metric measure spaces with appropriate Riemannian curvature-dimension condition.

Study rigidity in low-regularity Riemannian and semi-Riemannian metrics.

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…

We produce complete bounded curvature solutions to Kähler-Ricci flow with existence time estimates, assuming only that the initial data is a smooth \K metric uniformly equivalent to another complete bounded curvature \K metric. We obtain related flow results for non-smooth as well as degenerate initial conditions. We a…

New inequality for eigenfunctions on curved spaces.

We present a new relation between the short time behavior of the heat flow, the geometry of optimal transport and the Ricci flow. We also show how this relation can be used to define an evolution of metrics on non-smooth metric measure spaces with Ricci curvature bounded from below.

We propose a variable metric framework for minimizing the sum of a self-concordant function and a possibly non-smooth convex function, endowed with an easily computable proximal operator. We theoretically establish the convergence of our framework without relying on the usual Lipschitz gradient assumption on the smooth…

We study the positive mass theorem for certain non-smooth metrics following P. Miao's work. Our approach is to smooth the metric using the Ricci flow. As well as improving some previous results on the behaviour of the ADM mass under the Ricci flow, we extend the analysis of the zero mass case to higher dimensions.

Defines mass for non-smooth hyperbolic spaces using a modified flow.

The book explores Lie groups and Carnot-Carathéodory spaces, highlighting their applications in metric geometry and geometric group theory.

Improved model for non-smooth signals with complex spectra.

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

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

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

The paper develops bounds and regularity for minimal boundaries in non-smooth spaces with Ricci curvature.

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

Optimal transport for measures on noisy tree metrics is solved with robust approach.

We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of definin…

Study generalizes Möbius energy to non-smooth sets in arbitrary dimensions.

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

We study decreasing rearrangements of functions defined on (possibly non-smooth) metric measure spaces with Ricci curvature bounded below by $K>0$ and dimension bounded above by $N\in (1,\infty)$ in a synthetic sense, the so called $CD(K,N)$ spaces. We first establish a Polya-Szego type inequality stating that the $W^{…

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

Many popular statistical models, such as factor and random effects models, give arise a certain type of covariance structures that is a summation of low rank and sparse matrices. This paper introduces a penalized approximation framework to recover such model structures from large covariance matrix estimation. We propos…

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.

For non-smooth surfaces, the measure of Brownian loops is derived using the Polyakov-Alvarez formula.

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…

Sharp uncertainty principle for nodal sets in singular spaces.

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