arXiv research

The paper proves constant rank theorems for special Lagrangian equations.

Researchers find entire solutions to magnetic Ginzburg-Landau equations in 4D.

We are concerned with the saddle solutions of the Allen-Cahn equation constructed by Cabré and Terra \cite{C,C2} in $\mathbb{R}^{2m}$. These solutions vanish precisely on the Simons cone. The existence and uniqueness of saddle solution are shown in \cite{C,C2,C1}. Regarding the stab…

We reformulate LIPs as min-max problems for easier solution.

Algorithm classifies saddle-focus singularities in Hamiltonian systems.

PWGF escapes saddle points in nonconvex optimization.

We consider the problem of varying conformally the metric of a four dimensional manifold in order to obtain constant $Q$-curvature. The problem is variational, and solutions are in general found as critical points of saddle type. We show how the problem leads naturally to consider the set of formal barycenters of the m…

GenFlow optimizes faster, avoiding saddle points in fixed time.

Study on scalar curvature minimizability loss and saddle point solutions.

Equivalence of convex optimization, saddle-point problems, and variational inequalities is a well-established concept. The variational inequality (VI) is a static problem which is studied under dynamical settings using a framework called the projected dynamical system, whose stationary points coincide with the static s…

Extends saddle-point method for large-time volatility smiles.

New saddle network architectures preserve convex-concave geometry in optimization problems.

WSFN overcomes saddle points for non-convex functionals in Wasserstein space.

Convergence to a saddle point for convex-concave functions has been studied for decades, while recent years has seen a surge of interest in non-convex (zero-sum) smooth games, motivated by their recent wide applications. It remains an intriguing research challenge how local optimal points are defined and which algorith…

This paper extends Newton's method to distributed learning, avoiding saddle points and handling Byzantine workers.

We extend the Frank-Wolfe (FW) optimization algorithm to solve constrained smooth convex-concave saddle point (SP) problems. Remarkably, the method only requires access to linear minimization oracles. Leveraging recent advances in FW optimization, we provide the first proof of convergence of a FW-type saddle point solv…

Given k>=2, we construct a (2k-2)-parameter family of properly embedded minimal surfaces in H^2 x R invariant by a vertical translation T, called Saddle Towers, which have total intrinsic curvature 4 pi(1-k), genus zero and 2k vertical Scherk-type ends in the quotient by T. As limits of those Saddle Towers, we obtain J…

We revisit the landscape of the simple matrix factorization problem. For low-rank matrix factorization, prior work has shown that there exist infinitely many critical points all of which are either global minima or strict saddles. At a strict saddle the minimum eigenvalue of the Hessian is negative. Of interest is whet…

Under appropriate cooperation protocols and parameter choices, fully decentralized solutions for stochastic optimization have been shown to match the performance of centralized solutions and result in linear speedup (in the number of agents) relative to non-cooperative approaches in the strongly-convex setting. More re…

Study Palais-Smale sequences for Ricci curvature on homogeneous spaces.

Gradient flow in a potential energy (or Euclidean action) landscape provides a natural set of paths connecting different saddle points. We apply this method to General Relativity, where gradient flow is Ricci flow, and focus on the example of 4-dimensional Euclidean gravity with boundary S^1 x S^2, representing the can…

In 1988, Karcher generalized the family of singly periodic Scherk minimal surfaces by constructing, for each natural $n\geq 2$, a $(2n-3)$-parameter family of singly periodic minimal surfaces with genus zero and $2n$ Scherk-type ends in the quotient, called {\it saddle towers}. They have been recently classified by Pér…

Active learning reduces SP calculations by 90%.

The alternating gradient descent (AGD) is a simple but popular algorithm which has been applied to problems in optimization, machine learning, data ming, and signal processing, etc. The algorithm updates two blocks of variables in an alternating manner, in which a gradient step is taken on one block, while keeping the …

A distributed optimization method solves saddle point problems with strong concavity and convexity.

Two new algorithms improve reinforcement learning by avoiding saddle points.

Deep ReLU networks escape from the origin via saddle points with a low-rank bias.

Oracle-efficient algorithm for offline RL with partial data coverage.

We propose a doubly stochastic primal-dual coordinate optimization algorithm for empirical risk minimization, which can be formulated as a bilinear saddle-point problem. In each iteration, our method randomly samples a block of coordinates of the primal and dual solutions to update. The linear convergence of our method…

DLNs dynamics change with variance, leading to saddle-to-saddle training phases.

Study finds saddle connections on random surfaces follow Poisson distribution.

Optimal privacy-preserving algorithm for solving saddle point problems.

Study shows saddle connection graph's geometry and quasi-isometry properties.

In this paper we propose a primal-dual proximal extragradient algorithm to solve the generalized Dantzig selector (GDS) estimation problem, based on a new convex-concave saddle-point (SP) reformulation. Our new formulation makes it possible to adopt recent developments in saddle-point optimization, to achieve the optim…

Classifies Morse flows on 3-sphere with specific saddle connections.

We solve a complex optimization problem for Wasserstein barycenters using stochastic methods.

Study saddle connections on hyperelliptic surfaces, finding growth rates.

SGD in DLNs reveals feature learning dynamics.

We extend asymptotic formulas for saddle connections on translation surfaces.

In this paper we study flows $\varphi:M\times\mathbb{R}\longrightarrow M$ having an isolated non-saddle set. We see that the complexity of the region of influence of an isolated non-saddle set $K$ depends on the way in which $K$ sits on the phase space at the cohomological level. We construct flows in surfaces having i…

To every half-translation surface, we associate a saddle connection graph, which is a subgraph of the arc graph. We prove that every isomorphism between two saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces. We also investigate the automorphism group of the …

Study precise rates of horizontal gap shrinkage on generic translation surfaces.

For a half-translation surface (S,q), the associated saddle connection complex A(S,q) is the simplicial complex where vertices are the saddle connections on (S,q), with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. …

Golden L surface has unbounded bunching of saddle connections

FeDualEx tackles saddle point optimization in federated learning with composite objectives.

Let $M$ be a smooth manifold and let $\F$ be a codimension one, $C^\infty$ foliation on $M$, with isolated singularities of Morse type. The study and classification of pairs $(M,\F)$ is a challenging (and difficult) problem. In this setting, a classical result due to Reeb \cite{Reeb} states that a manifold admitting a …

Saddle-point optimization problems are an important class of optimization problems with applications to game theory, multi-agent reinforcement learning and machine learning. A majority of the rich literature available for saddle-point optimization has focused on the offline setting. In this paper, we study nonstationar…

Bounds on saddle connections on flat spheres with conical singularities.