FP-BMA improves generalization by encouraging flat posteriors in Bayesian Model Averaging.
problem Lack of flat posterior in approximate Bayesian inference methods hinders effective Bayesian Model Averaging.
method Proposes Flat Posterior-aware Bayesian Model Averaging (FP-BMA) and Flat Posterior-aware Bayesian Transfer Learning schemes.
result FP-BMA successfully captures flat posteriors, improving generalization performance.
Origami creates flat torus models of any size.
problem Creating flat torus models of any size.
method Explicit origami folding instructions.
result Flat torus models of any size created.
EDLP samples flat modes in discrete spaces using entropy.
problem Sampling flat modes in discrete spaces is challenging.
method EDLP uses a continuous auxiliary variable and local entropy to guide sampling.
result EDLP consistently outperforms traditional methods in various tasks.
The paper connects flatness to generalization in learning multi-index models with neural networks.
problem Understanding the generalization of non-convex neural networks using flatness measures.
method Analyzes 2-layer non-convex homogeneous neural networks and their connection to multi-index models.
result Flattest interpolators achieve small population loss and generalize well, establishing a direct link between flatness and generalization.
The notion of flat minima has played a key role in the generalization studies of deep learning models. However, existing definitions of the flatness are known to be sensitive to the rescaling of parameters. The issue suggests that the previous definitions of the flatness might not be a good measure of generalization, b…
Automatically identifies geometric flat outputs for robotic systems.
problem Lack of systematic and practical means to identify flat outputs for arbitrary robotic systems.
method Casts the search for a globally valid, equivariant flat output as an optimization problem using Riemannian geometry, Lie group theory, and differential forms.
result Approximate transcription of continuum formulation to a quadratic program achieves precise agreement with known closed-form flat outputs.
A flat Klein bottle is visualized using origami.
problem Visualizing a Klein bottle's flatness and topology.
method Curved-crease origami with inelastic film.
result The sculpture illustrates both flatness and non-orientability.
We propose a general framework for constructing and describing infinite type flat surfaces of finite area. Using this method, we characterize the range of dynamical behaviors possible for the vertical translation flows on such flat surfaces. We prove a sufficient condition for ergodicity of this flow and apply the cond…
We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two dimensions are intimately related to the minimal surfaces in a flat pseudo Riemannian 3-s…
Flatness of the loss curve is conjectured to be connected to the generalization ability of machine learning models, in particular neural networks. While it has been empirically observed that flatness measures consistently correlate strongly with generalization, it is still an open theoretical problem why and under whic…
New model shows SGD can prefer sharp or flat solutions based on label noise.
problem Understanding SGD's preference for flat or sharp solutions during training.
method Solved an analytically solvable model to explore SGD behavior.
result Data distribution determines sharpness at convergence; isotropic label noise leads to flat minimum preference.
New infinite families of flat spaces found from symmetric spaces.
problem Finding new flat homogeneous spaces.
method Starting from compact symmetric spaces, constructing infinite families of compact homogeneous spaces with invariant Bismut connections.
result Infinite families of compact homogeneous spaces with vanishing Ricci tensor.
Study shows one-dimensional location-scale-shape models are flat in Wasserstein geometry.
problem Investigating curvature in location-scale-shape models under Wasserstein metric.
method Introduced location-scale-shape model and investigated its geometry.
result Location-scale-shape model is intrinsically flat but extrinsically curved in Wasserstein geometry.
It was empirically confirmed by Keskar et al.\cite{SharpMinima} that flatter minima generalize better. However, for the popular ReLU network, sharp minimum can also generalize well \cite{SharpMinimacan}. The conclusion demonstrates that the existing definitions of flatness fail to account for the complex geometry of Re…
Flat minima lead to better generalization in low-rank matrix recovery models.
problem Understanding why flat minima generalize well in overparameterized models.
method Analysis of overparameterized matrix and bilinear sensing, robust PCA, covariance matrix estimation, and neural networks with quadratic activation functions.
result Flat minima, measured by the trace of the Hessian, exactly recover the ground truth in low-rank matrix recovery models under standard statistical assumptions.
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.
Proposes NRS to find flat minima in deep neural networks.
problem Finding optimal solutions in deep neural networks with overparameterization.
method NRS leverages the concept of flat minima and uses Kullback-Leibler divergence to regularize the neighborhood region in weight space.
result NRS drives optimizers towards flat minima, improving generalization ability across various model architectures.
The article constructs strong Carrollian geometries at infinity for Ricci flat Einstein manifolds.
problem Understanding projective and Carrollian geometries at infinity for Ricci flat Einstein manifolds.
method Developed a new type of Cartan geometry based on non-effective homogeneous models for projective geometry.
result Carrollian geometries are determined by the projective compactification data of Ricci flat Einstein manifolds.
In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. A recent example constructed by the author is projectively flat with zero curvature. In this paper, we introduce a techn…
Study how past eon's matter affects present eon in Penrose's cyclic cosmology.
problem Determining present eon's matter content from past eon's matter.
method Use Penrose's reciprocity hypothesis to link past and present eons' matter.
result Perfect fluid matter content of past eon influences present eon's matter content.
In a recent paper Donaldson explains how to use an older construction of Joyce to obtain four dimensional local models for scalar-flat Kahler metrics with a 2-torus symmetry. Using this idea, he recovers and generalizes the Taub-NUT metric by including it in a new family of complete scalar-flat toric Kahler metrics. In…
Around 2007, A. Chang, J. Qing, and P. Yang proved a conformal gap theorem for Bach-flat metrics with round sphere as the model case. In this article, we extend this result to prove conformally invariant gap theorems for Bach-flat 4-manifolds with (CP2,gFS) and $(\mathbb{S}^2\times\mathbb{S}^2,g_{prod…
Geodesic descent optimizes likelihood in dually flat spaces.
problem Maximum likelihood estimation in exponential families.
method m-geodesic and e-geodesic updates on dually flat spaces.
result Geodesic updates can reach maximum likelihood estimator in one step.
Study shows mass-capacity inequality for specific geometric manifolds.
problem Establishing mass-capacity inequality for certain geometric manifolds.
method Using conformally flat manifolds with nonnegative scalar curvature.
result Equality implies harmonically conformal to a specific subset of Euclidean space.
Zeroth-order methods favor flat minima in machine learning.
problem Finding solutions with small Hessian trace in optimization.
method Zeroth-order optimization with two-point estimator.
result Zeroth-order optimization converges to flat minima.
We exhibit several transformations of surfaces in R^4. First, one that takes a flat surface and gets a surface with flat normal bundle; then, one that takes a surface with flat normal bundle and gets a flat surface; finally, a one-parameter family of transformations on a flat surface with flat normal bundle and gives a…
Minkowski space is the local model of 3 dimensionnal flat spacetimes. Recent progress in the description of globally hyperbolic flat spacetimes showed strong link between Lorentzian geometry and Teichm{ü}ller space. We notice that Lorentzian generalisations of conical singularities are useful for the endeavours of desc…
Study shows smooth convergence of round surfaces in flat space-time models.
problem Volume preserving mean curvature flow of round surfaces in asymptotically flat spaces.
method Volume preserving mean curvature flow in asymptotically flat 3-manifolds.
result The flow converges smoothly to a stable CMC surface.
We define a partition of the space of projectively flat metrics in three classes according to the sign of the Chern scalar curvature; we prove that the class of negative projectively flat metrics is empty, and that the class of positive projectively flat metrics consists precisely of locally conformally flat-Kähler met…
Study flat manifolds' collapsed limits as flat orbifolds.
problem Understanding collapsed limits of flat manifolds.
method Analyzing totally geodesic foliations and Gromov-Hausdorff limits.
result Identify collapsed limits as flat orbifolds and provide criteria for singularity.
New exact spherically symmetric vacuum solutions found in Finsler gravity.
problem Finding exact vacuum solutions in Finsler gravity.
method Spherically symmetric, asymptotically flat Berwald spacetimes solved for Finsler gravity vacuum equation.
result Only one class of spherically symmetric Berwald spacetimes is compatible with asymptotic flatness and a well-defined causal structure.
We consider (flat) Cauchy-complete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes - model spacetimes - including four subfamilies: translation sp…
In this paper, we consider an equivalence problem of second order partially differential equations (PDE) and a duality of the flat differential equation. For the equivalence problem, explicit form of invariants (curvatures) are given. We also investigate a duality associated with the flat equation using double fibratio…
Improved flatness in annuli using PDE methods.
problem Flatness improvement in annuli.
method PDE-based approach adapted to exterior domains.
result Alternative proof of minimal surface end-structure and asymptotics.
New homogeneous manifolds with invariant Bismut Ricci flat connections are constructed.
problem Constructing homogeneous manifolds with invariant Bismut Ricci flat connections.
method Classification and construction of homogeneous spaces with specific properties.
result Examples of compact homogeneous Riemannian manifolds with invariant Bismut Ricci flat connections are provided.
Four dimensional simply connected Lie groups admitting a pseudo Kähler metric are determined. The corresponding Lie algebras are modelized and the compatible pairs (J,ω) are parametrized up to complex isomorphism (where J is a complex structure and ω is a symplectic structure). Such structure gives rise to a pseu…
We summarize the global geometric formulation of Einstein-Scalar-Maxwell theories twisted by flat symplectic vector bundle which encodes the duality structure of the theory. We describe the scalar-electromagnetic symmetry group of such models, which consists of flat unbased symplectic automorphisms of the flat symplect…
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.
Defines new bi-flat structures from integrable systems and flat coordinates.
problem Creating new bi-flat structures from integrable systems.
method Combining Frölicher-Nijenhuis bicomplex with Lauricella bi-flat structures.
result Defines multi-parameter families of Lauricella bi-flat structures.
We find a normal form for two-input flat discrete-time systems.
problem No comparable normal form exists for flat continuous-time systems.
method State- and input transformations to achieve a triangular structure.
result A systematic parameterization of system variables by the flat output and its shifts.
This paper deals essentially with affine or projective transformations of Lie groups endowed with a flat left invariant affine or projective structure. These groups are called flat affine or flat projective Lie groups. Our main results determine Lie groups admitting flat bi-invariant affine or projective structures. Th…
The paper proves that linearization along trajectories preserves flatness in discrete-time systems.
problem The relation between nonlinear and linear time-varying systems.
method Linearization along trajectories of a flat discrete-time system.
result The linearized system is flat, and a flat output can be derived.
The Yamabe flow on flat manifolds converges to a scalar flat metric.
problem Analyzing the convergence of Yamabe flow on asymptotically flat manifolds.
method Yamabe flow starting from an asymptotically flat manifold, convergence analysis.
result The flow converges to an asymptotically flat, scalar flat metric under certain conditions.
Study of flat metrics on orbifolds and their moduli spaces.
problem Understanding flat metrics on orbifolds and their moduli spaces.
method Analysis of Teichmüller spaces and mapping class groups.
result Moduli space of flat metrics on orbifolds is a very good orbifold under certain conditions.
Study of Riemann solitons and η-hyperbolic Ricci solitons on Bochner-flat Lorentzian Kähler spacetime manifolds.
problem Analyzing soliton behaviors on Bochner-flat Lorentzian Kähler spacetime manifolds.
method Deriving explicit formulas for soliton parameters and analyzing their behaviors.
result Criteria for shrinking, steady, and expanding behaviors of solitons.
The paper finds compact symbolic approximations for Ricci-flat metrics using Calabi-Yau hypersurfaces.
problem Finding explicit constructions of Ricci-flat metrics on Calabi-Yau manifolds remains challenging.
method Analysis of machine learning approximations and formalisation of symmetries.
result Ricci-flat metrics have more symmetries than the underlying manifold, leading to compact representations.
Study of symplectically flat connections and their functionals on smooth manifolds.
problem Understanding symplectically flat connections and their functionals on smooth manifolds.
method Extend symplectically flat connections to ζ-flat connections, introduce functionals with zeroes as symplectically flat connections, study critical points of these functionals, describe characteristic classes of ζ-flat bundles. result Novel geometric flows and characteristic classes of ζ-flat bundles are described. Study links' flat-virtual diagrams to create link invariants.
problem Equivalence and invariants of flat-virtual diagrams.
method Maps from links in thickened surfaces to flat-virtual links.
result Investigation of flat-virtual diagrams' equivalence and invariants.