Monotonicity of saddle mapsmath.DG
The Schoen-Yau theorem is extended to saddle maps.
problem Univalentness of saddle maps between discs.
method Proof of Schoen-Yau theorem for saddle maps.
result Analog of Schoen-Yau theorem proven for saddle maps.
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48 results for “saddle maps”
Heavy-ball algorithms can always avoid saddle points with random initialization.
problem Optimizing nonconvex functions with saddle points.
method Developed a new mapping to interpret heavy-ball algorithms as iterations, proving they can escape saddle points.
result Heavy-ball algorithms can escape saddle points with random initialization.
DLNs dynamics change with variance, leading to saddle-to-saddle training phases.
problem Understanding the dynamics of DLNs with varying initialization variance.
method Analyzing the phase transition of DLNs' dynamics as variance changes.
result Gradient descent visits a sequence of saddles, reaching a sparse global minimum.
The paper introduces horizon saddle connections to study dilation surfaces.
problem Understanding the geometric and dynamical properties of dilation surfaces.
method Introducing horizon saddle connections and quasi-Hopf surfaces.
result Existence of horizon saddle connections restricts the Veech group of dilation surfaces.
Gradient method achieves linear convergence for saddle point problems without strong convexity.
problem Solving saddle point problems with non-strongly convex functions.
method Primal-dual gradient method with a novel analysis technique.
result Linear convergence achieved without strong convexity of .
Horizon saddle connections imply dense hyperbolic geodesics on dilation surfaces.
problem Characterize dilation surfaces with dense hyperbolic geodesics.
method Analyzing saddle connections and directional flow properties.
result Dilation surfaces with horizon saddle connections have dense hyperbolic geodesics.
Study Palais-Smale sequences for Ricci curvature on homogeneous spaces.
problem Characterize Palais-Smale sequences for prescribed Ricci curvature.
method Complete description of divergent sequences on compact homogeneous spaces.
result Existence of saddle points on generalized Wallach and flag manifolds.
This paper develops methods to solve saddle-point problems on Riemannian manifolds with exponential stability.
problem Solving saddle-point problems on Riemannian manifolds with exponential stability.
method Developed a projected dynamical system on a Riemannian manifold to solve saddle-point problems, leveraging the strong monotonicity of the gradient of the Lagrangian function.
result Established exponential stability and convergence of the projected dynamical system to the unique saddle-point.
Paper avoids strict saddles in stochastic optimization without unit excitation assumption.
problem Avoiding strict saddles in stochastic optimization without unit excitation assumption.
method Pathwise Lyapunov-Perron framework, local smoothness, finite-moment assumptions.
result Avoidance of strict saddles for stochastic mirror descent and proximal-type methods.
When a boudnary-parabolic representation of a link group to PSL(2,) is given, Inoue and Kabaya suggested a combinatorial method to obtain the developing map of the representation using the octahedral triangulation and the shadow-coloring of certain quandle. Quandle is an algebraic system closely related wit…
Deep ReLU networks escape from the origin via saddle points with a low-rank bias.
problem Understanding the dynamics of gradient descent in deep ReLU networks.
method Analysis of escape directions and singular values of weight matrices.
result The first singular value of the -th layer weight matrix is at least larger than any other singular value.
Affine equivalence of half-translation surfaces via saddle connection graphs.
problem Understanding affine equivalence of half-translation surfaces.
method Association of saddle connection graphs and investigation of their automorphism groups.
result Every isomorphism between saddle connection graphs is induced by an affine homeomorphism between the underlying half-translation surfaces.
Study flows with isolated non-saddle sets and their region of influence.
problem Understanding the complexity of isolated non-saddle sets in flows.
method Cohomological analysis and construction of flows with prescribed structures.
result Construct flows with specific structures for their region of influence.
Study finds saddle connections on random surfaces follow Poisson distribution.
problem Distribution of saddle connections on random translation surfaces.
method Analysis of saddle connections on surfaces of large genus.
result Number of saddle connections in given lengths converges to Poisson distribution.
Saddle connection complexes are rigid under affine equivalence.
problem Characterizing the rigidity of saddle connection complexes.
method Proving simplicial isomorphisms between saddle connection complexes are induced by affine diffeomorphisms.
result Saddle connection complexes are complete invariants of affine equivalence classes of half-translation surfaces.
Study shows saddle connection graph's geometry and quasi-isometry properties.
problem Characterize the geometry and quasi-isometry of saddle connection graphs.
method Proved 4-hyperbolicity and uniform quasi-isometry to a tree, used generalised unicorn paths.
result Saddle connection graph is not quasi-isometrically rigid and its boundary is straight foliations.
Paper analyzes algorithms for nonstationary saddle-point optimization problems.
problem Nonstationary saddle-point optimization problems in game theory, reinforcement learning, and machine learning.
method Proposes extragradient and Frank-Wolfe algorithms for online and bandit settings.
result Establishes sub-linear regret bounds for the proposed algorithms.
Algorithm classifies saddle-focus singularities in Hamiltonian systems.
problem Classifying nondegenerate saddle-focus singularities in integrable Hamiltonian systems.
method Developed an algorithm based on semi-local equivalence to represent singularities as almost direct products.
result Obtained complete lists of saddle-focus singularities of complexities 1, 2, and 3.
Proximal methods avoid local minima in weakly convex problems.
problem Weakly convex optimization problems with strict saddle properties.
method Proximal methods on nonsmooth functions with strict saddle guarantees.
result Proximal methods converge to local minimizers only, when initialized randomly.
Classifies Morse flows on 3-sphere with specific saddle connections.
problem Classifying Morse-Smale flows on a 3-sphere with specific saddle connections.
method Used generalized Heegaard diagrams (Pr-diagrams) to classify flows.
result Found all possible, up to homeomorphism, ways to embed two circles in a 2-sphere with no more than 10 points of transversal intersection.
Stable saddle solutions found for specific dimensions of the Allen-Cahn equation.
problem Stability of saddle solutions for the Allen-Cahn equation in specific dimensions.
method Analyzing the Simons cone and energy functional to confirm saddle solutions' stability.
result Stable saddle solutions found for .
Study saddle connections on hyperelliptic surfaces, finding growth rates.
problem Count saddle connections on hyperelliptic surfaces without interior intersections.
method Used horocycle renormalization to prove lower bound growth rate.
result Found saddle connections satisfy growth rate.
SGD in DLNs reveals feature learning dynamics.
problem Understanding SGD dynamics in DLNs during saddle-to-saddle training.
method Stochastic Langevin dynamics with anisotropic, state-dependent noise; one-dimensional per-mode SDEs; Boltzmann distribution approximation.
result SGD noise encodes feature learning progression but does not alter saddle-to-saddle dynamics.
We extend asymptotic formulas for saddle connections on translation surfaces.
problem Counting saddle connections on translation surfaces with large genus.
method Recursive formulas and asymptotic analysis for all strata and multiplicities.
result Asymptotics for all saddle connections on translation surfaces of growing genus.
Develops algorithm to escape saddle points in Byzantine settings.
problem Byzantine workers create fake local minima near saddle points.
method ByzantinePGD, a robust first-order algorithm.
result Converges to approximate true local minimizer with low iteration complexity.
Study precise rates of horizontal gap shrinkage on generic translation surfaces.
problem Understanding precise decay rates of horizontal gaps in translation surfaces.
method Analyzing saddle connections and their angles on translation surfaces.
result Obtained precise decay rates for the difference in angle between almost horizontal saddle connections.
Gradient-based methods struggle with saddle points; curvature exploitation helps.
problem Gradient-based methods struggle with saddle points, leading to undesired stable stationary points.
method Exploits curvature information to escape undesired stationary points.
result Different optimization methods, including gradient and Adagrad, can escape non-optimal stationary points when curvature exploitation is used.
FeDualEx tackles saddle point optimization in federated learning with composite objectives.
problem Saddle point optimization with constraints and non-smooth regularization in federated learning.
method Federated Dual Extrapolation (FeDualEx) algorithm for saddle point optimization and composite objectives.
result FeDualEx effectively solves saddle point optimization problems with composite objectives in federated learning.
Golden L surface has unbounded bunching of saddle connections
problem Unbounded bunching of saddle connections on translation surfaces
method Translation surface with golden ratio
result Every positive integer K has a ball containing at least K saddle connection periods
Gradient descent can be very slow in escaping saddle points.
problem Gradient descent's slowness in escaping saddle points.
method Gradient descent with natural random initialization and non-pathological functions.
result Gradient descent can take exponential time to escape saddle points.
Paper analyzes Transformer learning dynamics, proving benign landscape for in-context learning.
problem Understanding how Transformers learn in context with nonlinear features.
method Mean-field and two-timescale analysis of Transformer dynamics, proving nonconvex but benign landscape.
result Proves mean-field dynamics avoid saddle points, leading to improved optimization.
Bounds on saddle connections on flat spheres with conical singularities.
problem Counting saddle connections on flat spheres with conical singularities.
method Geometry of immersed disks and explicit upper bounds.
result Explicit upper bounds on the number and lengths of saddle connections.
New research suggests deep networks converge to saddle points with high degeneracy.
problem Understanding the convergence of deep neural networks to saddle points.
method Empirical studies and theoretical findings to validate the hypothesis.
result Deep neural networks converge to saddle points with high degeneracy, not just local minima.
Random walks on mapping class group lead to recurrent geodesics in quadratic differentials.
problem Understanding recurrence of geodesics in quadratic differentials.
method Analyzing random walks on mapping class group with specific properties.
result Recurrence of geodesics in the thick part of the principal stratum of quadratic differentials.
First-order methods avoid saddle points for most initializations.
problem Avoiding saddle points in optimization problems.
method First-order methods, including gradient descent and variants, analyzed using dynamical systems and the Stable Manifold Theorem.
result First-order methods avoid saddle points for almost all initializations.
Study dynamics and topology of flows near non-saddle sets or W-sets.
problem Understanding the dynamics and topology of flows near specific invariant sets.
method Cohomological relations and global properties analysis.
result Dynamical classification of surfaces and robustness of non-saddle-sets.
The study shows how to measure translation surfaces with short saddle connections.
problem Measuring the probability of surfaces with short saddle connections.
method Using the multi-scale compactification of strata and algebraicity results.
result Proves strong regularity for invariant measures on translation surfaces.
We show that area minimizing polyhedral surfaces are saddle.
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…
Researchers create new triply periodic minimal surfaces by gluing saddle towers.
problem Creating triply periodic minimal surfaces without symmetry constraints.
method Gluing Karcher-Scherk saddle towers with phase differences and balancing under vertical interaction.
result Expands known triply periodic minimal surfaces into new 5-parameter families.
Gradient descent can take exponentially long to escape saddle points in 2D.
problem Worst-case inefficiency of gradient descent in non-convex optimization.
method Analysis of gradient descent's performance on 2D functions.
result Gradient descent can take exponentially long to escape saddle points.
WSFN overcomes saddle points for non-convex functionals in Wasserstein space.
problem Minimizing non-convex functionals over the Wasserstein space with saddle point avoidance.
method WSFN is a second-order method that preconditions the Wasserstein gradient to avoid saddle points.
result WSFN escapes saddle regions and reaches a global minimizer in polynomial time.
New methods help escape strict saddle points in nonsmooth optimization.
problem Escaping strict saddle points in nonsmooth optimization.
method An inexact stochastically perturbed gradient method applied to the Moreau envelope.
result A variety of algorithms for nonsmooth optimization can efficiently escape strict saddle points of the Moreau envelope.
New insights into matrix factorization show strict saddles have bounded eigenvalues.
problem Understanding the nature of critical points in matrix factorization.
method Analyzing orbits of critical points under the general linear group and identifying canonical points.
result Minimum eigenvalue of strict saddles is not uniformly bounded below zero.
The paper proves convergence properties for a continuous gradient descent method.
problem Analyzing the convergence of a continuous gradient descent method for smooth functions.
method Develops a continuous version of the Backtracking Gradient Descent method and proves convergence properties.
result The method converges to critical points of a function, avoiding saddle points under certain conditions.
A new method avoids saddle points in training machine learning models.
problem Training machine learning models efficiently in the presence of saddle points.
method Modified Laplacian smoothing gradient descent (mLSGD).
result The attraction region for mLSGD is significantly smaller than for gradient descent, avoiding saddle points.
Study systoles in translation surfaces, finding maxima and saddle connections.
problem Characterize maxima of systole function on translation surfaces.
method Define systole, characterize maxima, provide examples, study saddle connections.
result Found local but nonglobal maxima on strata of genus at least three.
A new algorithm trains deep neural networks by adding neurons greedily.
problem Training deep neural networks efficiently and effectively.
method Neuron Pursuit (NP) algorithm, which alternates between neuron addition and loss minimization.
result The algorithm can train deep neural networks efficiently and effectively.