Since their introduction by Thurston, measured geodesic laminations on hyperbolic surfaces occur in many contexts. In [Mor], we have introduced a notion of flat laminations on surfaces endowed with a half-translation structure (that is a singular flat surface with holonomy {+/-Id}, similar to geodesic laminations on hy…
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48 results for “flatness measure”
In [Mor], we have introduced a notion of flat laminations on surfaces endowed with a flat structure, similar to geodesic laminations on hyperbolic surfaces. Here is a sequel to this article that aims at defining transversal measures on flat laminations similar to transversal measures on hyperbolic laminations, taking i…
Detect spacetime curvature with event causality measurements.
problem Detecting spacetime curvature without rulers and clocks.
method Prove spacetime non-flatness through causal relations.
result Sixteen measurements verify non-flatness of non-conformally flat spacetimes.
New flatness measure for deep networks invariant to scaling.
problem Lack of invariant flatness measures for deep networks under parameter rescaling.
method Introduced a quotient manifold structure and Hessian-based invariant flatness measure.
result Confirms that Large-Batch SGD minima are sharper than Small-Batch SGD minima.
New flatness measure for neural nets is invariant to reparameterizations.
problem Lack of invariance in existing flatness measures to reparameterizations.
method Proposed a reparameterization-invariant flatness measure.
result The new flatness measure correlates with generalization error.
In this paper, we show that the Euler characteristic of an even dimensional closed projectively flat manifold is equal to the total measure which is induced from a probability Borel measure on RP^n invariant under the holonomy action, and then discuss its consequences and applications. As an application, we show that t…
The study bounds Hausdorff measure of flat singular points in area-minimizing currents.
problem Bounding Hausdorff measure of flat singular points in area-minimizing currents.
method Proving locally finite -dimensional Hausdorff measure and Minkowski content bounds.
result The set of flat singular points has locally finite -dimensional Hausdorff measure.
Investigates how flatness of loss curve relates to generalization in machine learning models.
problem Understanding why flatness correlates with generalization in machine learning models.
method Relates flatness to interpolation from representative data, derives notions of representativeness and feature robustness.
result Derives a novel relative flatness measure that correlates with generalization and solves reparameterization issues.
New research challenges the flatness-generalization link in deep neural networks.
problem The correlation between flatness of the loss landscape and generalization in deep neural networks is questioned.
method The study examines various flatness measures and popular SGD variants, finding some break the flatness-generalization link. It proposes using , a global quantity, as a predictor of generalization.
result The log of Bayesian prior upon initialization, , is a significantly more robust predictor of generalization than flatness measures.
Consider a closed marked flat surface of genus and area 1 and its universal covering . We show that the measure class of the Hausdorff measure of the Gromov boundary of uniquely determines .
Flatness After All?cs.LG
Paper proposes a new flatness measure for neural networks to improve generalization.
problem Generalization in deep learning models, especially with overparameterization.
method Soft rank measure of the Hessian to assess flatness and generalization.
result Soft rank flatness measure accurately estimates generalization gaps for various models.
Paper solves tracking control for -flat systems using classical states.
problem Tracking control for -flat systems.
method Quasi-static feedback of classical states.
result Achieves linear, decoupled and asymptotically stable tracking error dynamics.
We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in $\rn$. The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier work of D. Sullivan, our methods also yield an analytic characterization for smo…
The paper extends Siegel-Veech formula to convex flat cone spheres.
problem No formula exists for flat surfaces with irrational cone angles.
method Defined a generalized Siegel-Veech transform and Siegel-Veech measure.
result The Siegel-Veech measure is absolutely continuous and piecewise real analytic.
We compute the Riemannian volume on the moduli space of flat connections on a nonorientable 2-manifold, for a natural class of metrics. We also show that Witten's volume formula for these moduli spaces may be derived using Haar measure, and we give a new proof of Witten's volume formula for the moduli space of flat con…
Formula found for probability of random triangles on flat tori being homotopically trivial.
problem Calculating the probability of random triangles on flat tori being homotopically trivial.
method Reduced problem to new invariant of measurable sets in the plane unchanged by area-preserving affine transformations.
result Probability is minimized on rectangular tori and maximized on regular hexagonal tori.
New algorithms improve neural network generalization by finding flat minima.
problem Finding better generalization in neural networks through flat minima.
method Developed Entropy-SGD and Replicated-SGD algorithms to maximize flatness in the loss function.
result Consistently improved generalization error for various deep learning architectures.
Benedetti and Guadagnini have conjectured that the marked lenght spectrum of the constant mean curvature foliation in a 2+1 dimensional flat spacetime with compact hyperbolic Cauchy surfaces converges, in the direction of the singularity, to that of the marked measure spectrum of the R-tree dual to the measur…
On the Linearization of Flat Two-Input Systems by Prolongations and Applications to Control Designmath.OC
Paper shows how to linearize flat systems with two inputs.
problem Linearizing flat nonlinear control systems with two inputs.
method Using prolongations of a control, the system can be made static feedback linearizable.
result A tracking control can be designed without requiring measurements of a generalized Brunovsky state.
In this paper we consider non-compact non-flat simply connected harmonic manifolds. In particular, we show that the Martin boundary and Busemann boundary coincide for such manifolds. For any finite volume quotient we show that (up to scaling) there is a unique Patterson-Sullivan measure and this measure coincides with …
The paper studies harmonic map heat flow to flat tori, proving ergodic behavior and convergence to hyperbolic measure.
problem Analyzing the behavior of harmonic map heat flow to moduli space of flat tori.
method Investigates stability and ergodic behavior of harmonic map heat flow using hyperbolic structure and relative entropy.
result The flow converges weak-- to the normalized hyperbolic measure on the moduli space.
Study shows tori metrics converging to flat under specific conditions.
problem Understanding convergence of metrics on tori with non-negative scalar curvature.
method Uniformly conformal metrics and controlled geometry sequences.
result Sequence of metrics converges to flat metric in multiple senses.
New measure predicts deep neural network generalization better than existing ones.
problem Existing measures fail to explain generalization in overparameterized deep networks.
method Introduce prunability: smallest fraction of parameters that can be pruned without loss increase.
result Prunability highly correlates with generalization performance across various networks.
Flatness is a False Friendstat.ML
New findings challenge the use of flatness measures in neural networks.
problem The validity of flatness measures in assessing generalization in neural networks.
method Analysis of Hessian-based flatness norms and their relation to generalization.
result Solutions with large weights and low loss are often sharper than expected, contradicting flatness measures.
In this paper we consider flat metrics (semi-translation structures) on surfaces of finite type. There are two main results. The first is a complete description of when a set of simple closed curves is spectrally rigid, that is, when the length vector determines a metric among the class of flat metrics. Secondly, we gi…
Paper shows limits of Heisenberg manifolds are flat tori.
problem Understanding limits of sub-Riemannian Heisenberg manifolds.
method Analyzes collapsed Gromov--Hausdorff limits of compact Heisenberg manifolds.
result Collapsed limits are isometric to flat tori.
In this paper we introduce flat grafting as a deformation of quadratic differentials on a surface of finite type that is analogous to the grafting map on hyperbolic surfaces. Flat grafting maps are generic in the strata structure and preserve parallel measured foliations. We use flat grafting to construct paths connect…
We study the generic invariant probability measures for the geodesic flow on connected complete nonpositively curved manifolds. Under a mild technical assumption, we prove that ergodicity is a generic property in the set of probability measures defined on the unit tangent bundle of the manifold and supported by traject…
New research shows flat minima in robust loss landscapes correlate with good adversarial robustness.
problem Adversarial training leads to robust overfitting, poor robust generalization.
method Average- and worst-case metrics to measure flatness in robust loss landscapes.
result Flatness in robust loss landscapes correlates with good adversarial robustness.
Paper introduces normalized flat minima to address scale dependence in neural network optimization.
problem Scale dependence in existing flat minima definitions affects generalization studies.
method PAC-Bayesian analysis to introduce normalized flat minima, free from scale dependence.
result Normalized flat minima provides better hierarchy in hypothesis class and improved generalization.
The paper examines convergence of currents and forms under smooth diffeomorphisms.
problem Analyzing convergence of currents and forms under -limits of diffeomorphisms.
method Geometric analysis, measure theory, homotopy theory.
result Pushforwards of rectifiable currents converge in the flat norm.
Functions on a swallowtailmath.DG
We classify submersions from to up to diffeomorphisms which preserve the swallowtail and use this classification to study its flat geometry. The flat geometry is derived from the contact of the swallowtail with planes, which is measured by the singularities of the height function.
Continuous metrics on ample bundles lie in infinite-dimensional cones.
problem Understanding the structure of positive metrics on ample line bundles.
method Analyzing bounded graded filtrations and embedding into Mabuchi-flat cones.
result Continuous metrics embed isometrically into the space of positive metrics.
Study shows dynamics of rank 1 orbifolds in flat surfaces.
problem Characterize dynamics of rank 1 affine invariant orbifolds.
method Analyzes M-isoperiodic foliations and their ergodic properties.
result Leaves of the isoperiodic foliation are either all closed or all dense.
What are appropriate geometric conditions ensuring that a complete Riemannian 2-cylinder without conjugate points is flat? Examples with nonpositive curvature show that one has to assume that the ends of the cylinder open sublinearly. We show that sublinear growth of the ends is indeed sufficient if it is measured by t…
We focus on the topology and dynamics of minimal sets and Levi-flats in surfaces of general type. Our method relies on the ergodic theory of Riemann surfaces laminations: we use harmonic measures and Lyapunov exponents. Our first result establishes that minimal sets have large Hausdorff dimension when a leaf is simply …
We give an alternative proof of a result of Cantat and Dupont, showing that any automorphism of a K3 surface with measure of maximal entropy in the Lebesgue class must be a Kummer example. Our method exploits the existence of Ricci-flat metrics on K3s and also covers the non-projective case.
Neural nets approximate Ricci flat metrics for Calabi-Yau manifolds.
problem Lack of analytical Ricci flat metrics for Calabi-Yau threefolds.
method Employed neural network approximations for several Calabi-Yau manifolds of dimensions two and three.
result Measures of Ricci flatness improved by three orders of magnitude after training.
Establish a unified framework for negative results in Fourier analysis.
problem Fourier restriction, -improving, and Fourier decay problems
method Quantitative understanding of geometric properties of measures
result Explicit obstructions to measure satisfying Fourier restriction, -improving, or Fourier decay estimates
New proof of Kondo-Tanaka theorem using geometric measure theory.
problem Existence of special systems of Whitney flat 1-forms on homology manifolds.
method Geometric measure theory and tools from non-smooth analysis.
result Simple new proof of Kondo-Tanaka theorem and its converse.
Proves spectral gap bounds for Teichmüller geodesics on flat surfaces.
problem Quantify spectral gaps for Teichmüller geodesics.
method Bounding spectral gaps in terms of geometric quantities on flat surfaces.
result Quantitative non-uniform hyperbolicity of Teichmüller geodesic flow.
New insights into Markov chain geometry via positive transition measures.
problem Lack of statistical meaning in the space of transition probabilities.
method Constructing an extension of the space of transition probabilities using Amari's theory of positive measures.
result Introduction of a new dually flat structure for the space of positive transition measures.
Study on how reparametrization affects neural nets' parameter spaces from a geometric perspective.
problem Inconsistencies in flatness measures, optimization, and probability densities under reparametrization.
method Riemannian geometry to study invariance of neural nets under reparametrization.
result Invariance of neural nets is an inherent property if the metric is explicitly represented and transformation rules are correct.
A piecewise flat Finsler metric on a triangulated surface is a metric whose restriction to any triangle is a flat triangle in some Minkowski space with straight edges. One of the main purposes of this work is to study the properties of geodesics on a piecewise flat Finsler surface, especially when it meets a vertex…
In this paper we consider strata of flat metrics coming from quadratic differentials (semi-translation structures) on surfaces of finite type. We provide a necessary and sufficient condition for a set of simple closed curves to be spectrally rigid over a stratum with enough complexity, extending a result of Duchin-Lein…
One of the basic aims of this paper is to study the relationship between the geometry of ``hypersurface like'' subsets of Euclidean space and the properties of the measures they support. In this context we show that certain doubling properties of a measure determine the geometry of its support. A Radon measure is said …
We study adiabatic limits of Ricci-flat Kahler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Ampere equation, we show that the Ricci-flat metrics collapse (away f…
Proposes a new method for evaluating and constructing hierarchical topic models.
problem Evaluation and construction of hierarchical topic models.
method Represent HTM as layers and edges, introduce quality measures, and develop a heterogeneous algorithm.
result The proposed heterogeneous algorithm significantly outperforms baseline approaches.