Harmonic map flow's singularity properties proven with Lojasiewicz inequalities.
problem Finite-time singularities of harmonic map flow in critical dimensions.
method Proving a weighted Lojasiewicz inequality.
result Continuity of body map and no-neck property for bubble-tree decompositions.
The Lojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanislaw Lojasiewicz (1965) using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman (1988). In this article, we first give an elementary geometric, coordinate-based proof of t…
We prove several abstract versions of the Lojasiewicz-Simon gradient inequality for an analytic functional on a Banach space that generalize previous abstract versions of this inequality, weakening their hypotheses and, in particular, the well-known infinite-dimensional version of the gradient inequality due to Lojasie…
Study proves Lojasiewicz inequalities for harmonic maps near simple bubble trees.
problem Analyzing harmonic maps near simple bubble trees.
method Proves Lojasiewicz inequalities for harmonic maps close to simple bubble trees.
result Obtains new results on the convergence of harmonic map flow and energy spectrum.
We apply our abstract gradient inequalities developed by the authors in arXiv:1510.03817 to prove Lojasiewicz--Simon gradient inequalities for the harmonic map energy function using Sobolev spaces which impose minimal regularity requirements on maps between closed, Riemannian manifolds. Our Lojasiewicz--Simon gradient …
In this sequel to arXiv:1510.03817, we apply our abstract Lojasiewicz-Simon gradient inequality to prove Lojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces which impose minimal regularity requirements on pairs of connections and sections. The Lojasiewicz-Simon gradient …
Uniqueness of nondegenerate blowups for planar networks shown.
problem Uniqueness of nondegenerate blowups for the motion by curvature of planar networks.
method Proof based on Lojasiewicz-Simon gradient inequality applied to stability properties of critical points of the length functional.
result Uniqueness of nondegenerate compact blowups for the motion by curvature of planar networks.
This paper explores geometric and analytic aspects of Lojasiewicz inequalities on vector bundles.
problem Understanding growth and stability conditions for real-analytic functions over vector bundles.
method Outline theory of functionals and variational problems over vector bundles, explore applications to real-analytic functionals.
result Describes the energy functional on $S^{n-1$ as a functional over a vector bundle.
The paper proves a Lojasiewicz inequality for maps from the 2-sphere to itself.
problem Analyzing maps from the 2-sphere to itself using Lojasiewicz inequalities.
method Using Lojasiewicz-Simon inequalities and Topping's repulsion estimates, along with a bubble-tree induction argument.
result Polynomial convergence of weak solutions of harmonic map flow on compact domains.
It is a consequence of the Morse-Bott Lemma on Banach spaces that a smooth Morse-Bott function on an open neighborhood of a critical point in a Banach space obeys a Lojasiewicz gradient inequality with the optimal exponent one half. In this article we prove converses for analytic functions on Banach spaces: If the Loja…
The elastic flow of curves converges smoothly to a critical point.
problem Smooth convergence of elastic flow of curves.
method Application of Lojasiewicz-Simon inequality.
result Smooth convergence to a critical point.
For any compact Lie group G and closed, smooth Riemannian manifold (X,g) of dimension d≥2, we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal G-bundle over X supporting a connection with Lp-small curvature, when p>d/2, to the case of a connection with …
Study on stability of ALE Ricci-flat metrics using a modified Perelman's λ-functional.
problem Stability and instability of ALE Ricci-flat metrics.
method Use of a modified Perelman's λ-functional and Lojasiewicz inequality.
result Demonstrates dynamical instability of ALE Ricci-flat metrics.
The Willmore flow preserves surface volume, leading to convergence to a sphere.
problem Long-term behavior of volume-preserving Willmore flow on surfaces.
method Volume-preserving Willmore flow, blow-up analysis, constrained Lojasiewicz-Simon inequality.
result Smooth solutions exist for spherical surfaces with Willmore energy below 8π and converge to a sphere.
In real algebraic geometry, Lojasiewicz's theorem asserts that any integral curve of the gradient flow of an analytic function that has an accumulation point has a unique limit. Lojasiewicz proved this result in the early 1960s as a consequence of his gradient inequality. Many problems in calculus of variations are que…
New functional proves mass positivity for ALE metrics.
problem Proving mass positivity for ALE metrics with Ricci-flat deformations.
method Introduced a new functional λALE and proved its monotonicity and Lojasiewicz-Simon inequality. result Established that small perturbations of Ricci-flat ALE metrics with nonnegative scalar curvature have nonnegative mass.
Gradient descent with biased rounding errors converges faster under certain conditions.
problem Stagnation or negative impact of rounding errors in neural network training with low precision.
method Analysis of gradient descent with stochastic fixed-point rounding errors under the Polyak-Lojasiewicz inequality.
result Biased rounding errors can improve convergence rates, especially when the Polyak-Lojasiewicz inequality holds.
Paper proves strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
problem Proving strong uniqueness of cylindrical tangent flows near singularity in Ricci flow.
method Established Lojasiewicz inequality for pointed W-entropy under cylindrical geometry assumption. result Strong uniqueness of cylindrical tangent flows at first singular time of Ricci flow proved.
Unique cylindrical tangent cone for Simons' hypersurface found.
problem Uniqueness of cylindrical tangent cones for area-minimizing hypersurfaces.
method Developed a new Lojasiewicz inequality for non-isolated singularities.
result Cylindrical tangent cone for Simons' hypersurface is unique.
The article analyzes the stability of a curve shortening flow for planar networks.
problem Stability analysis of anisotropic curve shortening flow for planar networks.
method Used Lojasiewicz-Simon gradient inequality to derive stability results.
result For initial data close to an energy minimizer, the flow exists globally and converges to a different energy minimum.
Level set flow's singularities are type I under 2-convexity, leading to specific curvature blow-up rates.
problem Understanding the nature and behavior of singularities in level set flow.
method Analytical approach using Lojasiewicz inequality and curvature blow-up rates.
result The arrival time is C2 near a critical point if and only if it satisfies a Lojasiewicz inequality. We study the Riemannian quantiative isoperimetric inequality. We show that direct analogue of the Euclidean quantitative isoperimetric inequality is--in general--false on a closed Riemannian manifold. In spite of this, we show that the inequality is true generically. Moreover, we show that a modified (but sharp) versio…
The paper proves stability and convergence of minimal networks under curvature motion.
problem Stability and convergence of minimal networks under curvature motion.
method Proved Lojasiewicz-Simon gradient inequalities for minimal networks.
result Motion by curvature starting from networks close to minimal ones exists for all times and smoothly converges.
In this note, we prove an L2n-energy gap result for Yang-Mills connections on a principal G-bundle over a compact manifold without using Lojasiewicz-Simon gradient inequality (arXiv:1502.00668).
The paper proves smooth convergence of evolving hypersurfaces to critical points.
problem Analyzing the convergence of evolving hypersurfaces.
method Gradient flow of a functional with a Lojasiewicz-Simon inequality.
result Asymptotic convergence to critical points of the functional.
The paper shows mean curvature flow keeps diameter bounded under certain conditions.
problem Proving the bounded diameter of hypersurfaces under mean curvature flow.
method Use of Lojasiewicz inequalities and solution of mean-convex neighbourhood conjecture.
result The intrinsic diameter stays uniformly bounded as the flow approaches the first singular time.
The paper proves strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
problem Proving strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
method Establishing a Lojasiewicz inequality for the pointed W-entropy in Ricci flow under the assumption of geometry near the base point being close to a generalized cylinder. result Proves strong uniqueness of generalized cylindrical tangent flows and shows that the subset of points with rectifiable Sqck(N) is horizontally parabolic. Unified framework for complex, split-complex, and dual numbers.
problem Analytic and geometric scope of real-analytic functions.
method Generalized Cauchy-Riemann structure and unified real algebra family.
result Milnor-Le type fibration theorem for nondegenerate algebras.
Study of non-convex potential functions in deep learning with Poincaré inequality.
problem Understanding convergence of stochastic dynamics in non-convex potential landscapes.
method Introduced log-Polyak-Lojasiewicz (log-PL) measures and analyzed their convergence properties.
result Langevin dynamics converges at a rate of O~(1/ε) for sufficiently small ε. AdamL optimizes deep learning models by incorporating loss function information.
problem Adaptive optimizers can suffer from poor generalization due to nonuniform gradient scaling.
method AdamL is a novel adaptive optimizer that considers loss function information for better generalization.
result AdamL achieves faster convergence or lower objective function values compared to other optimizers.
New analysis shows GMD can converge linearly under PL-like conditions.
problem Establishing linear convergence for generalized mirror descent.
method PL-based analysis for time-dependent mirrors, Taylor-series approach for stochastic GMD.
result Linear convergence of stochastic GMD under PL-like conditions.
Study on Wasserstein gradient flow for MMD between Coulomb measures.
problem Analyzing the long-time behavior of MMD between probability and target measures using Coulomb kernels.
method Existence of global weak solutions, ultracontractive estimate, regularity analysis, exponential decay proof, defective Polyak-Lojasiewicz inequality.
result Exponential decay of squared MMD toward a uniformly positive target measure on flat torus.
The paper provides convergence bounds for approximating a distribution using point clouds.
problem Approximating a distribution using discrete points with minimal Wasserstein distance.
method Lloyd's algorithm with Power cells, analyzed using gradient descent.
result Explicit upper bounds for the convergence speed of the Lloyd-type algorithm.
SGD with machine learning noise converges to global minimum exponentially fast.
problem Optimizing machine learning models with stochastic gradient descent.
method Analysis of SGD with machine learning noise, focusing on energy landscapes and gradient noise.
result SGD converges to the global minimum exponentially fast under certain conditions.
The paper proves a margin inequality for separating hyperplanes, useful for analyzing algorithmic bias.
problem Analyzing the implicit bias of algorithms in machine learning.
method Proves a nonsmooth Kurdyka-Lojasiewicz inequality for margin function.
result The bias of algorithm iterates converges at least as fast as the square-root of the margin convergence rate.
In this monograph, we develop results on global existence and convergence of solutions to abstract gradient flows on Banach spaces for a potential function that obeys the Lojasiewicz-Simon gradient inequality. We prove a Lojasiewicz-Simon gradient inequality for the Yang-Mills energy functional over closed, smooth Riem…
Quantized Stochastic Primal-Dual Methods for Distributed Optimization
problem Distributed optimization with stochastic gradients and finite-bit communication
method q-PDGD, a quantized stochastic primal-dual method
result Linear contraction to an explicit neighborhood under RSI, O(1/k) convergence under PL inequality
We establish a new uniqueness theorem for the three dimensional Schwarzschild-de Sitter metrics. For this some new or improved tools are developed. These include a reverse Lojasiewicz inequality, which holds in a neighborhood of the extremal points of any smooth function. We further prove smoothness of the set of maxim…
We prove that the norm version of the adaptive stochastic gradient method (AdaGrad-Norm) achieves a linear convergence rate for a subset of either strongly convex functions or non-convex functions that satisfy the Polyak Lojasiewicz (PL) inequality. The paper introduces the notion of Restricted Uniform Inequality of Gr…
New function class characterizes loss landscape of deep neural networks without over-parametrization.
problem Complex loss landscape of deep neural networks without over-parametrization.
method Proposed a novel class of functions to characterize loss landscape without over-parametrization.
result Gradient-based optimizers possess theoretical guarantees of convergence under the new function class assumption.
In this article, we introduce a new method (based on Perelman's lambda-functional) to study the stability of compact Ricci-flat metrics. Under the assumption that all infinitesimal Ricci-flat deformations are integrable we prove: (A) a Ricci-flat metric is a local maximizer of lambda in a C^2,alpha-sense iff its Lichne…
Study curves evolving by gradient flow of elastic energy, proving existence, smoothing, and convergence.
problem Evolution of curves with fixed length and clamped boundary conditions.
method Negative L2-gradient flow of elastic energy, existence, parabolic smoothing, constrained Lojasiewicz-Simon gradient inequality. result Convergence to a critical point as time tends to infinity.
The asymptotic behavior of the stochastic gradient algorithm with a biased gradient estimator is analyzed. Relying on arguments based on the dynamic system theory (chain-recurrence) and the differential geometry (Yomdin theorem and Lojasiewicz inequality), tight bounds on the asymptotic bias of the iterates generated b…
Unified framework for analyzing neural networks trained by gradient descent.
problem Lack of generalizable guarantees for neural networks trained by gradient descent.
method Proxy convexity and proxy Polyak-Lojasiewicz inequalities.
result Unified guarantees for neural networks trained by gradient descent.
The paper studies the convergence of elastic flows of curves into manifolds, proving smooth convergence under certain conditions.
problem The convergence of elastic flows of curves into manifolds.
method Parabolic estimates and Lojasiewicz-Simon gradient inequality.
result Smooth convergence of the flow to critical points under specific conditions.
On a Fano manifold, we prove that the Kahler-Ricci flow starting from a Kahler metric in the anti-canonical class which is sufficiently close to a Kahler-Einstein metric must converge in a polynomial rate to a Kahler-Einstein metric. The convergence can not happen in general if we study the flow on the level of Kahler …
We characterize the rate of convergence of a converging volume-normalized Yamabe flow in terms of Morse theoretic properties of the limiting metric. If the limiting metric is an integrable critical point for the Yamabe functional (for example, this holds when the critical point is non-degenerate), then we show that the…
SGD and stochastic gradient descent converge at optimal rates for certain non-convex functions.
problem Optimal convergence rates for non-convex functions under gradient noise.
method Geometric interpretation of the PL-condition to analyze convergence rates.
result Convergence rates of SGD and stochastic gradient descent match those of strongly convex quadratics.