arXiv research

Regularity results for geodesic X-ray transform on nonsmooth manifolds

We construct some nonsmoothable actions of Z2 * Z2 on spin four-manifolds by using an equivariant version of Furuta' s 10/8inequality. The examples satisfy following property: any proper subgroup of Z2 * Z2 is smoothable for some smooth structure.

We show that every closed, simply connected, spin topological 4-manifold except $S^4$ and $S^2\times S^2$ admits a homologically trivial, pseudofree, locally linear action of $\mathbb{Z}_p$ for any sufficiently large prime number $p$ which is nonsmoothable for any possible smooth structure.

Two new methods solve nonsmooth optimization on Riemannian Stiefel manifold.

We construct a nonsmoothable Z\times Z-action on the connected sum of an Enriques surface and S^2\times S^2, such that each of generators is smoothable. We also construct a nonsmoothable self-homeomorphism on an Enriques surface.

Let $X$ be a closed, simply-connected, smooth, spin 4-manifold whose intersection form is isomorphic to $n(-E_8)\bigoplus mH$, where $H$ is the hyperbolic form. In this paper, we prove that for $n$ such that $n\equiv 2 ~{\rm mod} ~4$, there exists a locally linear pseudofree $\mathbb{Z}_2$-action on $X$ which is nonsmo…

New theory for nonsmooth systems helps optimize and control complex functions.

PPGD solves nonconvex nonsmooth optimization problems without KL property.

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold $M$. We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the distance function to a closed subset $C$ of $M$; 2) solvability and implicit func…

S.Bauer and M.Furuta defined a stable cohomotopy refinement of the Seiberg-Witten invariants. In this paper, we prove a vanishing theorem of Bauer-Furuta invariants for 4-manifolds with smooth Z/2-actions. As an application, we give a constraint on smooth Z/2-actions on homotopy K3#K3, and construct a nonsmoothable loc…

Study on Adam-family methods for nonsmooth optimization with convergence guarantees.

Much of the vast literature on the integral during the last two centuries concerns extending the class of integrable functions. In contrast, our viewpoint is akin to that taken by Hassler Whitney [{\it Geometric integration theory}, Princeton Univ. Press, Princeton, NJ, 1957] and by geometric measure theorists because …

This paper tackles nonsmooth optimization in machine learning.

In this work we consider the stochastic minimization of nonsmooth convex loss functions, a central problem in machine learning. We propose a novel algorithm called Accelerated Nonsmooth Stochastic Gradient Descent (ANSGD), which exploits the structure of common nonsmooth loss functions to achieve optimal convergence ra…

Study on nonsmooth contractive SA with constant stepsize and Q-learning.

A new method solves convex optimization problems on manifolds efficiently.

Study compares nonsmooth spaces with integrable Ricci bounds.

New stability bounds for SGD on nonsmooth convex losses.

New methods help escape strict saddle points in nonsmooth optimization.

New method improves PCA robustness using Wasserstein distances.

We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existence and uniqueness of viscosity solutions to Hamilton-Jacobi equations defined on Riemannian manifolds.

New Langevin Monte Carlo algorithms for sampling from nonsmooth distributions.

We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonconvex part is smooth and the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem is very limited. For example, it is not known whether the proximal stochastic gra…

New method solves nonsmooth low-rank matrix optimization problems efficiently.

Study nonholonomic systems with collisions using variational principles.

Study on numerical reliability of AD for MaxPool in neural nets.

Paper develops privacy-preserving federated learning for nonsmooth objectives.

New approach to gravity theory sacrifices smoothness for ellipticity.

Paper develops algorithms for nonsmooth, nonconvex statistical learning problems.

Overview of recent results on isoperimetric inequalities on manifolds with Ricci lower bounds.

The paper proves isoperimetric regions on Riemannian manifolds with Ricci bounded below.

In regularized risk minimization, the associated optimization problem becomes particularly difficult when both the loss and regularizer are nonsmooth. Existing approaches either have slow or unclear convergence properties, are restricted to limited problem subclasses, or require careful setting of a smoothing parameter…

Paper develops an online covariance estimator for nonsmooth stochastic approximation problems.

Paper proposes a new method for sparse spectral clustering on Stiefel manifold.

Unified Lagrangian-based methods for nonsmooth nonconvex optimization.

We consider in this paper a class of composite optimization problems whose objective function is given by the summation of a general smooth and nonsmooth component, together with a relatively simple nonsmooth term. We present a new class of first-order methods, namely the gradient sliding algorithms, which can skip the…

Nonconvex and nonsmooth problems have recently attracted considerable attention in machine learning. However, developing efficient methods for the nonconvex and nonsmooth optimization problems with certain performance guarantee remains a challenge. Proximal coordinate descent (PCD) has been widely used for solving opti…

The paper guarantees global stability for stochastic subgradient methods in nonsmooth nonconvex optimization.

We consider a class of nonconvex nonsmooth optimization problems whose objective is the sum of a smooth function and a finite number of nonnegative proper closed possibly nonsmooth functions (whose proximal mappings are easy to compute), some of which are further composed with linear maps. This kind of problems arises …

Nesterov's extrapolation improves convergence in nonsmooth optimization.

Improved shuffling gradient methods converge faster for nonsmooth convex optimization.

On a Riemannian manifold, lower Ricci curvature bounds are known to be characterized by geodesic convexity properties of various entropies with respect to the Kantorovich-Rubinstein-Wasserstein square distance from optimal transportation. These notions also make sense in a (nonsmooth) metric measure setting, where they…

We consider optimization problems over the Stiefel manifold whose objective function is the summation of a smooth function and a nonsmooth function. Existing methods for solving this kind of problems can be classified into three classes. Algorithms in the first class rely on information of the subgradients of the objec…

Study Langevin Monte Carlo for sampling non-log-concave distributions.

Develops a mathematical model for automatic differentiation in machine learning.

Adam's convergence rate in nonsmooth nonconvex optimization is analyzed.

This paper concerns dictionary learning, i.e., sparse coding, a fundamental representation learning problem. We show that a subgradient descent algorithm, with random initialization, can provably recover orthogonal dictionaries on a natural nonsmooth, nonconvex $\ell_1$ minimization formulation of the problem, under mi…

Decomposes smooth manifolds into algebraic submanifolds.