We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs , the Gosset graph and suitable Cartesian …
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48 results for “sharp results”
ASAM: Adaptive Sharpness-Aware Minimization for Scale-Invariant Learning of Deep Neural Networkscs.LG
ASAM improves deep neural network generalization by adapting sharpness to scale.
problem Fixed-radius sharpness measure is sensitive to parameter scaling, weakening its connection to generalization.
method Introduces adaptive sharpness, a scale-invariant measure, and proposes ASAM for deep learning.
result ASAM significantly improves model generalization performance across various datasets.
Sharp log-Sobolev inequalities proved for spaces.
problem Proving log-Sobolev inequalities in noncompact metric measure spaces.
method Sharp isoperimetric inequality, symmetrisation, scaling argument, Hamilton-Jacobi inequality, Sobolev regularity.
result Sharp log-Sobolev inequalities established in spaces.
Sharp bounds on Alexandrov spaces' boundaries with rigidity analysis.
problem Volume bounds on Alexandrov spaces' boundaries.
method Sharp volume bounds and rigidity analysis of Alexandrov spaces.
result New sharp volume bounds and classification of rigidity cases.
Sharp heat equation gradient estimates on compact manifolds.
problem Gradient estimates for positive solutions on weighted manifolds.
method Proving sharp gradient estimates for positive solutions to the weighted heat equation.
result Refined gradient estimates and Liouville theorems for ancient solutions.
Sharpness Minimization Algorithms Do Not Only Minimize Sharpness To Achieve Better Generalizationcs.LG
Sharpness minimization algorithms don't solely improve generalization.
problem Why do overparameterized neural networks generalize?
method Theoretical and empirical investigation of two-layer ReLU networks.
result Sharpness minimization algorithms do not always lead to better generalization.
Sharp spectral gap estimates on manifolds with integral curvature bounds.
problem Proving spectral gap estimates on manifolds with integral curvature bounds.
method Generalizing previous results to include integral curvature bounds.
result Confirms a conjecture about spectral gap estimates on manifolds with integral curvature bounds.
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 chord-arc estimates for curve shortening flow on spheres.
problem Understanding the behavior of curves on spheres under curve shortening flow.
method Proving sharp chord-arc estimates and curvature control.
result Simple spherical curves either contract to points or converge to great circles.
Sharp estimate for flow in any dimension.
problem Interior gradient estimate for graphical mean curvature flow.
method Proving sharp interior gradient estimate for area decreasing graphical mean curvature flow in arbitrary codimension.
result Generalized result in arbitrary codimension.
Generalized Blaschke rolling theorem for curved spaces.
problem Extending classical theorem to curved spaces.
method Generalization to Riemannian manifolds with bounded curvature.
result Sharp results in arbitrary dimensions, new even in constant curvature spaces.
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.
We prove that two homogeneous ultra-metric spaces are coarsely equivalent if and only if where is the so-called sharp entropy of . This classification implies that each homogeneous proper ultra-metric space is coarsely equivalent to the a…
Optimal option portfolios under Sharpe Ratio maximization with skew-elliptical t-distributed returns
problem Optimal option portfolios under Sharpe Ratio maximization
method Formulation for explicit portfolio weights
result Different optimal portfolios for Sharpe Ratio and return-to-Value-at-Risk (VaR) ratio
Sharpe ratio is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the excess return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and need to be esti…
We address the question of determining the eigenvalues (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), …
Sharp criterion for Chern-Gauss-Bonnet integral using Q curvature.
problem Quantifying the Chern-Gauss-Bonnet integral using Q curvature.
method New approach involving singular integral estimation.
result Derivation of asymptotic formula for Q curvature equation.
Sharp comparison theorems for 3D manifolds with scalar curvature bound.
problem Understanding the geometry and topology of 3D manifolds with scalar curvature constraints.
method Sharp comparison results for Green's function and spectrum, derived from scalar curvature bounds.
result Sharp upper and lower bounds for the Green's function and spectrum of 3D manifolds.
Deep linear networks minimize sharpness, avoiding large eigenvalues.
problem Understanding optimization dynamics in deep linear networks for regression.
method Analyzing sharpness (largest eigenvalue of Hessian) of minimizers and gradient flow solutions.
result Gradient flow implicitly regularizes towards flat minima, with sharpness bounded by a constant.
Kronheimer and Mrowka introduced a new knot invariant, called , which is a gauge theoretic analogue of Rasmussen'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 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.
Sharp results link DLN gradient flow to basis pursuit optimization and GHA phase transitions.
problem Understanding implicit regularization in Diagonal Linear Networks.
method Sharp convergence bounds and characterization of minimizers.
result Gradient flow of DLNs with tiny initialization approximates minimizers of basis pursuit optimization problem.
Estimates true Sharpe ratio of selected assets with various methods.
problem Estimating the true Sharpe ratio of a selected asset with high in-sample ratio.
method Polyhedral lemma, James Stein shrinkage, debiasing, thresholding, empirical Bayes.
result James Stein estimator performs best across various parameter values.
Sharp curvature estimates for mean curvature flow in spheres.
problem Understanding the behavior of surfaces evolving under mean curvature flow in spheres.
method Proving asymptotically sharp curvature pinching estimates and using them to derive derivative and convexity estimates.
result Partial classification of singularity models and new rigidity results for ancient solutions.
Sharp Sobolev inequalities on noncompact Riemannian manifolds with via Optimal Transport theorymath.AP
Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.
problem Proving sharp Sobolev inequalities on noncompact Riemannian manifolds with non-negative Ricci curvature.
method Using Optimal Mass Transportation with quadratic distance cost.
result Sharp -Sobolev and -logarithmic Sobolev inequalities established for and .
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.
Sharp gradient estimates for positive Ricci curvature manifolds.
problem Understanding geometric properties of manifolds with positive Ricci curvature.
method Proving sharp gradient estimates and monotonicity formulae.
result Sharp gradient estimates and monotonicity formulae for positive Ricci curvature manifolds.
We prove a comparison theorem on the first Neumann eigenvalue on Bakry-Emery manifolds. Examples are constructed to illustrate the sharpness of the result. A linear explicit lower bound is also proved. We also discuss the asymptotic sharpness of such a result.
We derive a sharp cusp count for finite volume complex hyperbolic surfaces which admit smooth toroidal compactifications. We use this result, and the techniques developed in [DiC12], to study the geometry of cusped complex hyperbolic surfaces and their compactifications.
Sharp inequality found on three-balls for fourth order Sobolev traces.
problem Fourth order Sobolev trace inequality on three-balls.
method Established through equivalence to a third order Sobolev inequality on two-spheres.
result Sharp fourth order Sobolev trace inequality on three-balls.
Sharp-MAML improves MAML by reducing saddle points in few-shot learning.
problem Challenges in optimizing MAML due to complex loss landscape.
method Sharpness-aware minimization applied to MAML.
result Sharp-MAML and its variant outperform plain MAML on few-shot learning tasks.
We discuss - in what is intended to be a pedagogical fashion - generalized "mean-to-risk" ratios for portfolio optimization. The Sharpe ratio is only one example of such generalized "mean-to-risk" ratios. Another example is what we term the Fano ratio (which, unlike the Sharpe ratio, is independent of the time horizon)…
The sharp peak-flat trough pattern and critical speculationcond-mat.stat-mech
We find empirically a characteristic sharp peak-flat trough pattern in a large set of commodity prices. We argue that the sharp peak structure reflects an endogenous inter-market organization, and that peaks may be seen as local ``singularities'' resulting from imitation and herding. These findings impose a novel strin…
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.
Sharp long distance upper bounds for solutions of Leibenson's equation on Riemannian manifoldsmath.AP
Sharp upper bounds found for solutions of a specific equation on Riemannian manifolds.
problem Finding upper bounds for solutions of a specific equation on Riemannian manifolds.
method Proved sharp upper estimates of weak subsolutions to the Leibenson equation on Riemannian manifolds with non-negative Ricci curvature.
result Improved and proved a conjecture about upper bounds for solutions of the Leibenson equation.
Noise in linear networks minimizes sharpness and leads to shrinkage-thresholding.
problem Minimizing sharpness in diagonal linear networks.
method Stochastic sharpness-aware minimization (SAM) with isotropic noise.
result Noise forces shrinkage-thresholding of true parameters.
Sharp estimates for mean curvature flow of graphs are shown and examples are given to illustrate why these are sharp. The estimates improves earlier (non-sharp) estimates of Klaus Ecker and Gerhard Huisken.
Sharpe ratio (sometimes also referred to as information ratio) is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the (excess) net return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and …
We prove the two theorems of the title, settling two long standing questions in the local theory of singular minimal hypersurfaces. The sharpness of either result is with respect to its hypothesis on the size of the allowable singular sets. The proofs of both theorems rely heavily on the author's recent regularity and …
The paper develops new methods to study sharp isoperimetric properties on complex spaces.
problem Sharp isoperimetric comparison on non-collapsed spaces with lower Ricci bounds.
method Original argument to estimate first and second variation of the area for isoperimetric sets, avoiding regularity theory.
result Generalizes results for smooth and non-compact manifolds, Alexandrov spaces, and convex bodies.
Sharp concentration results for sums of heavy-tailed random variables.
problem Analyzing sums of independent heavy-tailed random variables.
method Using concentration inequalities and large deviation principles for distributions satisfying specific tail bounds.
result Sharp concentration inequalities and large deviation results for sums of heavy-tailed random variables.
Sharp estimates for parabolic equations on manifolds using symmetrization.
problem Estimating solutions to parabolic equations on manifolds.
method Symmetrization techniques and isoperimetric inequalities.
result Generalization of Bandle's comparison to Riemannian setting.
The paper calculates graph Ricci curvature and finds properties of specific graph types.
problem Understanding Ricci curvature on irregular graphs.
method Developed a formula for graph Ricci curvature based on optimal bijections.
result Derived structural and theorem results for specific graph types.
Sharp pinching conditions restrict the geometry and topology of submanifolds.
problem Understanding submanifolds under pinching conditions in arbitrary Riemannian manifolds.
method Analyzing submanifolds with pinching conditions involving second fundamental form and mean curvature.
result The pinching condition imposes strong geometric and topological restrictions on submanifolds.
Doubly-Valid/Doubly-Sharp Sensitivity Analysis for Causal Inference with Unmeasured Confoundingstat.ME
Sharp bounds on ATE with unmeasured confounders, valid even when misspecified.
problem Bounding average treatment effects with unmeasured confounders.
method Distributionally robust optimization, double sharpness, double validity.
result Proposes estimators with robustness properties for valid bounds.
Sharp bounds found for minimal surface solutions.
problem Finding bounds for minimal surface solutions.
method Analyzing minimal surface equation with specific boundary conditions.
result Sharp bounds established for solutions over certain domains.
The Sharpe ratio is a way to compare the excess returns (over the risk free asset) of portfolios for each unit of volatility that is generated by a portfolio. In this paper we introduce a robust Sharpe ratio portfolio under the assumption that the risk free asset is unknown. We propose a robust portfolio that maximizes…
DGSAM improves domain generalization by minimizing individual sharpness.
problem Improving domain generalization models that perform well on unseen target domains.
method Shifts DG paradigm toward minimizing individual sharpness across source domains.
result DGSAM reduces performance variance across domains with less computational overhead.