Sharp characterization of Willmore invariant in higher dimensions.
problem Understanding the Willmore invariant in various dimensions.
method Characterization using conformal fundamental forms and tensors.
result Sharp sufficient condition for vanishing Willmore invariant in even dimensions.
Motivated by a sharp eigenvalue estimate for the Kohn Laplacian, we prove a theorem that characterizes the CR sphere in terms of the existence of a non-trivial complex-valued function satisfying a certain overdetermined system.
Study sharp decay of capacity for subharmonic functions on compact Hermitian manifolds.
problem Sharp decay of capacity of sublevel sets of (ω,m)-subharmonic functions. method Generalizes previous results on Kähler manifolds and obtains full characterizations of polar sets.
result Full characterizations of polar sets and extremal functions.
We show that gradient shrinking, expanding or steady Ricci solitons have potentials leading to suitable reference probability measures on the manifold. For shrinking solitons, as well as expanding soltions with nonnegative Ricci curvature, these reference measures satisfy sharp logarithmic Sobolev inequalities with low…
Sharp bounds on scalar curvature spectrum and rigidity theorems.
problem Understanding scalar curvature bounds and rigidity on manifolds.
method Sharp upper bounds for the bottom spectrum of the Beltrami Laplacian, scalar curvature rigidity theorem.
result Sharp upper bound for the bottom spectrum of the Beltrami Laplacian and scalar curvature rigidity theorem.
Derives a sharp inequality for trace-free matrices with applications to hypersurfaces.
problem Classifying conformally flat hypersurfaces and characterizing rotational hypersurfaces.
method Derives a sharp inequality relating eigenvalues of trace-free matrices and applies it to hypersurfaces.
result New proof of the classification of conformally flat hypersurfaces and construction of a functional for rotational hypersurfaces.
Sharp boundaries for detecting dense subhypergraphs established.
problem Detecting dense subhypergraphs in random hypergraphs.
method Established sharp detection boundaries for known and unknown edge probabilities.
result Sharp detectable regions differ significantly from graph counterparts.
Study energy-minimizing maps in projective spaces, proving sharp bounds.
problem Finding optimal mappings in projective spaces.
method Proving lower bounds and characterizing energy-minimizing maps.
result Sharp lower bounds and characterization of energy-minimizing maps.
Kronheimer and Mrowka introduced a new knot invariant, called s♯, which is a gauge theoretic analogue of Rasmussen's s invariant. In this article, we compute Kronheimer and Mrowka's invariant for some classes of knots, including algebraic knots and the connected sums of quasi-positive knots with non-trivial r…
Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
Sharp inequalities and solitons studied in statistical submersions.
problem Understanding geometric properties of statistical submersions.
method Proving sharp inequalities and establishing geometrical properties of statistical submersions.
result Characterization of fibers as Ricci-Bourguignon solitons with conformal vector field.
We obtain a sharp characterization of the Euclidean ball among all convex bodies K whose boundary has a pointwise k-th mean curvature not smaller than a geometric constant at almost all normal points. This geometric constant depends only on the volume and the boundary area of K. We deduce this characterization from a n…
Sharp thresholds and contiguity for community detection in contextual SBM.
problem Community detection in graphs with high-dimensional node-covariates.
method Contextual Stochastic Block Model, non-rigorous cavity method, information theory.
result Established the sharp threshold for detection and weak recovery in the contextual SBM.
Sharp inequalities and symmetries on Riemannian surfaces quantified.
problem Understanding symmetries and asymmetries in Riemannian surfaces.
method Introducing scattering energy to measure asymmetry and proving isoperimetric inequalities.
result Sharp quantitative isoperimetric inequalities and domains with vanishing scattering energy characterized.
SAM improves neural network generalization by penalizing sharpness, clarifying its exact notion and mechanism.
problem Improving deep neural network generalization for various settings.
method Sharpness-Aware Minimization (SAM) technique that penalizes a notion of sharpness of the model.
result SAM regularizes the third notion of sharpness, most likely preferred for practical performance.
Sharp bounds on quasimode norms on compact space forms.
problem Characterize compact manifolds using quasimode decay rates.
method Analyzes upper and lower bounds of quasimode norms on compact space forms.
result Characterizes compact manifolds of constant curvature using quasimode decay rates.
In this paper, we give a new sharp generalization bound of lp-MKL which is a generalized framework of multiple kernel learning (MKL) and imposes lp-mixed-norm regularization instead of l1-mixed-norm regularization. We utilize localization techniques to obtain the sharp learning rate. The bound is characterized by the d…
GD monotonically decreases GFS sharpness in neural networks and scalar models.
problem Oscillatory behavior of loss in GD training.
method Analysis of GFS sharpness and empirical validation.
result GFS sharpness decreases monotonically during GD training.
Explains optimal functional inequalities, focusing on Sobolev and fractional Sobolev.
problem Optimal functional inequalities and their stability.
method Compactness theorems, characterization of optimizers, and quantitative stability analysis.
result Characterization and stability of optimizers for Sobolev inequalities and their fractional generalizations.
Let Mn(n≥3) be an n-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that Mn satisfies some integral pinching conditions. We give some rigidity theorems on compact manifolds with harmonic curvature and positive scalar curvature. In particular, Theorem 1.4,…
Sharp estimates for Finsler metrics in convex domains.
problem Estimating distances in Finsler metrics near convex points.
method Sharp estimates for intrinsic distances of Finsler metrics.
result Characterization of k-quasi hyperbolic metric in convex geometry. Truncated SGD with heavy-tailed noise eliminates sharp local minima.
problem Avoiding sharp local minima in deep learning models.
method Truncated SGD with heavy-tailed gradient noise.
result Truncated SGD can eliminate sharp local minima entirely from its training trajectory.
We refine Theorem A due to Gursky \cite{G3}. As applications, we give some rigidity theorems on four-manifolds with postive Yamabe constant. In particular, these rigidity theorems are sharp for our conditions have the additional properties of being sharp. By this we mean that we can precisely characterize the case of e…
The paper proves inequalities for hypersurfaces in weighted manifolds.
problem Willmore-type inequalities for closed hypersurfaces in weighted manifolds.
method Analyzes weighted manifolds with nonnegative Bakry-Émery Ricci curvature, proving sharp inequalities and characterizing equality cases.
result Derives sharp Willmore-type and Willmore-like inequalities in steady and shrinking gradient Ricci solitons.
Sharp estimates for p-capacity on manifolds with Ricci curvature bounds.
problem Estimating p-capacity on manifolds with Ricci curvature constraints.
method Sharp comparison inequalities, warped-product model ends, and scale-invariant quantities.
result Characterization of equality cases and optimal ranges for normalization parameters.
We will discuss some sharp estimates for CMC graphs in a Riemannian 3-manifold MxR whose boundary is contained in a slice. We will start by giving sharp lower bounds for the geodesic curvature of the boundary and improve these bounds when assuming additional restrictions on the maximum height that such a surface reache…
Sharpness of actions on reductive homogeneous spaces proven for various groups.
problem Proving proper and cocompact actions on reductive homogeneous spaces.
method Using quasi-isometric embedding and Anosov representations.
result Characterization and proof of non-compactness for certain homogeneous spaces.
We obtain sharp lower bounds on the radii of inscribed balls for strictly convex isoperimetric domains lying in a 2-dimensional Alexandrov metric space of curvature bounded below. We also characterize the case when such bounds are attained.
Sharp bounds found for energy in projective space mappings.
problem Finding bounds for energy in mappings of real projective spaces.
method Sharp lower and upper bounds for energy in homotopy classes of mappings from real projective space to Riemannian manifolds.
result Characterization of maps that achieve the lower bound for energy and determination of the infimum of energy in a homotopy class.
Work of Ni and Zhang has shown that for the torus knot Tr,s with r>s>1 every surgery slope p/q≥6730(r2−1)(s2−1) is a characterizing slope. In this paper, we show that this can be lowered to a bound which is linear in rs, namely, p/q≥443(rs−r−s). The main technical ingredient in…
We consider a Riemannian cylinder endowed with a closed potential 1-form A and study the magnetic Laplacian with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product…
In this paper we provide a sharp characterization of the smooth four-dimensional sphere. The assumptions of the theorem are conformally invariant, and can be reduced to an L^2 inequality of the Weyl tensor and positivity of the Yamabe invariant.
We show that the local equivalence problem for second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also show that, modulo Cartan duality and point transformations, the Painlevé-I equati…
Paper finds exact Hessian sharpness in deep matrix factorization.
problem Understanding the geometry of loss landscapes in deep matrix factorization.
method Presented the first exact expression for Hessian maximum eigenvalue.
result Spectral-norm balance is a sufficient condition for flatness in deep matrix factorization.
Averaged SGD optimizes a smoothed objective, leading to better generalization.
problem Improving generalization performance in machine learning models.
method Analyzed the smoothed objective function of SGD and proved that averaged SGD can optimize this smoothed function efficiently.
result Averaged SGD can efficiently optimize a smoothed objective, leading to better generalization.
Sharp estimates for heat flow on nonconvex domains.
problem Quantitative estimates for heat flow on nonconvex domains.
method Sharp gradient and transport estimates with novel dependence on time.
result Equivalent characterization of lower bound on second fundamental form.
Study community detection in multi-view data with various types of information.
problem Community detection in multi-view data with different types of information.
method Unified theoretical framework, mutual information analysis, sharp thresholds, iterative algorithms.
result Sharp thresholds for community recovery in various multi-view settings.
Sharp inequalities in nonnegative Ricci curvature spaces using mass transport.
problem Proving sharp isoperimetric and Sobolev inequalities in nonnegative Ricci curvature spaces.
method Optimal mass transport theory, symmetrization techniques, and volume non-collapsing properties.
result Sharp isoperimetric and Sobolev inequalities established in Riemannian manifolds with nonnegative Ricci curvature.
Sharp inequality between TV and Hellinger distances for Gaussian mixtures.
problem Understanding the relationship between total variation and Hellinger distances for Gaussian mixtures.
method Established a general upper bound on Hellinger distance in terms of TV distance raised to a power, demonstrating sharpness with specific examples.
result The Hellinger distance between two Gaussian mixtures is bounded by the TV distance raised to a power 1−o(1), where o(1) is of order 1/loglog(1/TV). Sharp analysis of out-of-distribution error in overparameterized models with importance weights.
problem Understanding and quantifying the degradation of performance in overparameterized models when faced with underrepresented data.
method Sharp analysis of an overparameterized Gaussian mixture model with spurious features and cost-sensitive interpolating solutions incorporating importance weights.
result Characterization of a novel tradeoff between worst-case robustness and average accuracy as a function of importance weight magnitude.
We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbilic estimate of Huisken, the convexity estimates of Huisken--Sinestrari and the cylindrical estimate of Huisken--Sinestrari. Namely, we show …
SGD in linear regression overfits but performs well due to bias-variance trade-off.
problem Understanding overfitting in SGD for linear regression.
method Constant-stepsize SGD with iterate averaging or tail averaging, analyzing full eigenspectrum of data covariance matrix.
result Sharp excess risk bounds revealing bias-variance decomposition for SGD in linear regression.
Characterizes diagrams achieving Morton-Franks-Williams inequality for positive knots and links.
problem Understanding when the Morton-Franks-Williams inequality holds for positive knots and links.
method Combinatorial characterisation and generating examples.
result Examples of diagrams achieving crossing number, braid index, and maximal self-linking number.
Study characterizes compact Einstein-type manifolds with boundary.
problem Characterize compact Einstein-type manifolds with nonempty boundary.
method Proved a sharp boundary estimate, obtained Hawking mass bounds, and provided a topological classification for the boundary.
result Obtained a gap result for compact Einstein-type manifolds with boundary.
In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a sharp lower bound on the hilbert area of ideal triangles, independant of the con…
Study knot invariants and surgeries, proving properties of instanton homologies.
problem Characterize 3-manifolds as Dehn surgery on knots.
method Establish conjugation symmetry, use cobordism maps, analyze homology cobordism group.
result Prove ν♯(K) is always zero or odd. The paper studies LCAK metrics on complex manifolds and their properties.
problem Characterizing and understanding LCAK metrics on complex manifolds.
method Analyzes the geometric structures induced by LCAK metrics and their properties.
result Pluricanonical LCAK metrics have parallel Lee form on compact manifolds.
Sharp sample complexity for multiclass PAC learning with bandit feedback.
problem Multiclass PAC learning with bandit feedback.
method Characterization of optimal sample complexity using a new combinatorial dimension (bandit DS dimension) and a learning algorithm (ListCascade).
result Sharp sample complexity characterization for every concept class up to logarithmic factors.