arXiv research

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

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…

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

In recent literature, a general two step procedure has been formulated for solving the problem of phase retrieval. First, a spectral technique is used to obtain a constant-error initial estimate, following which, the estimate is refined to arbitrary precision by first-order optimization of a non-convex loss function. N…

In spite of several notable efforts, explaining the generalization of deterministic non-smooth deep nets, e.g., ReLU-nets, has remained challenging. Existing approaches for deterministic non-smooth deep nets typically need to bound the Lipschitz constant of such deep nets but such bounds are quite large, may even incre…

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

We propose a new proximal, path-following framework for a class of constrained convex problems. We consider settings where the nonlinear---and possibly non-smooth---objective part is endowed with a proximity operator, and the constraint set is equipped with a self-concordant barrier. Our approach relies on the followin…

One of the mysteries in the success of neural networks is randomly initialized first order methods like gradient descent can achieve zero training loss even though the objective function is non-convex and non-smooth. This paper demystifies this surprising phenomenon for two-layer fully connected ReLU activated neural n…

Adaptive data fusion boosts efficiency in multi-task optimization.

The paper challenges the smooth null infinity model by constructing counter-examples and showing non-smoothness of null infinity.

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

Investigates training challenges for morphological neural networks.

This study reduces Willmore flows of tori to simpler problems and finds new conformally constrained Willmore tori.

Study on reproducibility in optimization with bounds on limits.

We consider compact convex hypersurfaces contracting by functions of their curvature. Under the mean curvature flow, uniformly convex smooth initial hypersurfaces evolve to remain smooth and uniformly convex, and contract to points after finite time. The same holds if the initial data is only weakly convex or non-smoot…

The paper proves stability of Ricci flow for certain initial conditions.

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

Traditional dictionary learning methods are based on quadratic convex loss function and thus are sensitive to outliers. In this paper, we propose a generic framework for robust dictionary learning based on concave losses. We provide results on composition of concave functions, notably regarding super-gradient computati…

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

New method tackles non-smooth tensor data for better recovery.

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

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

Gradient descent solves asymmetric low-rank matrix factorization efficiently.

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.

We construct smooth solutions to Ricci flow starting from a class of singular metrics and give asymptotics for the forward evolution. The singular metrics heal with a set of points (of codimension at least three) coming out of the singular point. We conjecture that these metrics arise as final-time limits of Ricci flow…

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…

Characterizes neural kernel and NNGP for various activations.

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.

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

FedProx algorithm improved for non-smooth and heterogeneous data.

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

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.

Efficient algorithms for large Maxent models improve wildfire probability predictions.

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.

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.

Consistency models accelerate generation with theoretical guarantees.

Smooth bundles with rough data maintain Hodge kernel isomorphism.

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

Adaptive NN method improves matrix completion for non-smooth data.

New bounds for online portfolio selection without smoothness assumptions.

A new algorithm improves both computational efficiency and statistical optimality for robust low-rank matrix and tensor estimation.

Paper proves CLT for quantile SGD with constant learning rate.