arXiv research

New single-loop algorithm tackles weakly convex constraints in stochastic optimization.

Neural network approximates weakly efficient frontier of convex vector optimization problems.

Paper tackles efficient learning of non-convex hypotheses in metric spaces.

In this paper, we show that if the optimization function is restricted-strongly-convex (RSC) and restricted-smooth (RSM) -- a rich subclass of weakly submodular functions -- then a streaming algorithm with constant factor approximation guarantee is possible. More generally, our results are applicable to any monotone we…

Weakly convex polyhedra which are star-shaped with respect to one of their vertices are infinitesimally rigid. This is a partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of convex caps.

Proximal methods avoid local minima in weakly convex problems.

In this work we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem. The idea is based on the finding that a weakly convex function as an approximation of the $\ell_0$ pseudo norm is able to better induce sparsity than the commonly used $\ell_1$ norm. For a cl…

Smooth analog of Gromov's dihedral rigidity for 3D weakly convex domains.

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

On R^n endowed with a riemannian metric of bounded nonpositive curvature, the weakly convex closed subsets are topologically trivial. The stability of such subsets under intersection characterizes the euclidean spaces.

Adaptive algorithm AMSGrad converges for weakly convex constrained optimization problems.

This paper improves inverse problem solving with weakly convex regularisers and proves convergence.

SGD avoids critical points on weakly convex functions.

New algorithm solves complex non-convex problems efficiently.

New adaptive methods solve weakly convex stochastic optimization problems.

Submodular functions are a broad class of set functions, which naturally arise in diverse areas. Many algorithms have been suggested for the maximization of these functions. Unfortunately, once the function deviates from submodularity, the known algorithms may perform arbitrarily poorly. Amending this issue, by obtaini…

Paper analyzes convergence of stochastic methods under heavy-tailed noise.

The paper proves a Schwarz lemma for weakly Kähler-Finsler manifolds.

In this paper, we consider first-order convergence theory and algorithms for solving a class of non-convex non-concave min-max saddle-point problems, whose objective function is weakly convex in the variables of minimization and weakly concave in the variables of maximization. It has many important applications in mach…

Study on polyhedra rigidity, finding non-existence of flexible weakly convex decomposable polyhedra.

In this paper, we study the problem of monotone (weakly) DR-submodular continuous maximization. While previous methods require the gradient information of the objective function, we propose a derivative-free algorithm LDGM for the first time. We define $β$ and $α$ to characterize how close a function is to continuous D…

The study compares spectral volumes of manifolds with weakly convex boundaries.

A submanifold $M^m$ of a Euclidean space $R^{m+p}$ is said to have harmonic mean curvature vector field if $Δ\vec{H}=0$, where $\vec{H}$ is the mean curvature vector field of $M\hookrightarrow R^{m+p}$ and $Δ$ is the rough Laplacian on $M$. There is a conjecture named after Bangyen Chen which states that submanifolds o…

The goal of this paper is to study weakly Einstein critical metrics of the volume functional on a compact manifold $M$ with smooth boundary $\partial M$. Here, we will give the complete classification for an $n$-dimensional, $n=3$ or $4,$ weakly Einstein critical metric of the volume functional with nonnegative scalar …

Develops a new SPP algorithm with variance reduction for weakly convex optimization.

Gradient descent performs well on weakly convex losses, offering generalization guarantees.

New approach tackles resource constraints in bandit problems with weakly adaptive algorithms.

In this paper we consider three-manifolds with weakly umbilic boundary (the Second Fundamental form of the boundary is a constant multiple of the metric). We show that if the initial manifold has positive Ricci curvature and the boundary is convex (nonnegative Second Fundamental form), its metric can be deformed via th…

Recent years have witnessed the rapid development of block coordinate update (BCU) methods, which are particularly suitable for problems involving large-sized data and/or variables. In optimization, BCU first appears as the coordinate descent method that works well for smooth problems or those with separable nonsmooth …

Paper extends SMM to weakly convex and multi-convex surrogates for non-convex optimization.

Paper proposes an algorithm for sampling from complex mixture distributions without requiring smoothness.

Unified approach for multicalibration in weakly supervised learning.

The paper proves optimal estimates and inequalities for spectral functions on certain manifolds.

Study on Yang-Mills fields on $S^4$ proving self-duality constraints.

Framework for designing nonlinearities in neural networks with slope constraints.

We establish the regularity theory for certain critical elliptic systems with an anti-symmetric structure under inhomogeneous Neumann and Dirichlet boundary constraints. As applications, we prove full regularity and smooth estimates at the free boundary for weakly Dirac-harmonic maps from spin Riemann surfaces. Our met…

Strict convexity proven for certain self-expanders in high dimensions.

Embeds 3-manifolds in symplectic 4-manifolds with constraints.

Properties of two classes of generally convex sets in the n-dimentional real Euclidean space, called m-semiconvex and weakly m-semiconvex, 1<=m<n, are investigated in the present work. In particular, it is established that an open set with smooth boundary in the plan which is weakly 1-semiconvex but not 1-semiconvex co…

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…

We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to…

We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be interpreted as ideal polyhedra, if we regard $\mathbb{R}\mathbb{P}^3$ as a combinati…

This paper introduces a general multi-class approach to weakly supervised classification. Inferring the labels and learning the parameters of the model is usually done jointly through a block-coordinate descent algorithm such as expectation-maximization (EM), which may lead to local minima. To avoid this problem, we pr…

Unified approach tackles high-dimensional tensor bandits with convex optimization and weakly decomposable regularizers.

We consider Blackwell approachability, a very powerful and geometric tool in game theory, used for example to design strategies of the uninformed player in repeated games with incomplete information. We extend this theory to "generalized quitting games" , a class of repeated stochastic games in which each player may ha…

The study classifies weakly almost Fuchsian manifolds and proves geometric properties.

A distributed subgradient method tackles non-convex optimization problems in networks.

The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a convex polyhedron b…