arXiv research

Kronheimer and Mrowka introduced a new knot invariant, called $s^\sharp$, 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 inequality found on three-balls for fourth order Sobolev traces.

Sharp-MAML improves MAML by reducing saddle points in few-shot learning.

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 …

ASAM improves deep neural network generalization by adapting sharpness to scale.

Sharp bounds found for minimal surface solutions.

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…

Sharpness minimization algorithms don't solely improve generalization.

DGSAM improves domain generalization by minimizing individual sharpness.

SharpBalance improves deep ensemble performance by balancing sharpness and diversity.

SAM improves neural network generalization by penalizing sharpness, clarifying its exact notion and mechanism.

Sharp upper diameter limit found for Ricci solitons.

We prove a sharp estimate for the inscribed radius under certain fully nonlinear curvature flows. This estimate is asymptotically sharp on cylinders.

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…

Sharp Gaussian bounds derived for Schrödinger kernel on Ricci solitons.

The study analyzes sharpness dynamics in neural networks, revealing mechanisms and conditions.

Sharp bounds for curve isoperimetric deficit derived.

Sharp bounds on Alexandrov spaces' boundaries with rigidity analysis.

Sharp inequality linking interior and boundary Yamabe invariants on specific manifolds.

Sharp chord-arc estimates for curve shortening flow on spheres.

Sharp spectral gap estimates on manifolds with integral curvature bounds.

Optimal option portfolios under Sharpe Ratio maximization with skew-elliptical t-distributed returns

Sharp inequalities for matrix means with unknown variance.

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 $J(2n,n)$, the Gosset graph and suitable Cartesian …

Sharp lower bound found for geodesic ball eigenvalues.

Sharp bounds derived for the first two Steklov eigenvalues of exterior domains.

Sharp log-Sobolev inequalities proved for ${\sf CD}(0,N)$ spaces.

Sharp spectral theorem splits certain non-compact manifolds.

We prove sharp bounds for the growth rate of eigenfunctions of the Ornstein-Uhlenbeck operator and its natural generalizations. The bounds are sharp even up to lower order terms and have important applications to geometric flows.

Sharp heat kernel and Green's function estimates on Euclidean volume growth manifolds.

Sharp fractional Sobolev inequalities on closed manifolds identified.

Sharp stability in Almgren problem solved in any dimension.

Estimates true Sharpe ratio of selected assets with various methods.

We address the question of determining the eigenvalues $λ\_n$ (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 $n$ nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), …

When the in-sample Sharpe ratio is obtained by optimizing over a k-dimensional parameter space, it is a biased estimator for what can be expected on unseen data (out-of-sample). We derive (1) an unbiased estimator adjusting for both sources of bias: noise fit and estimation error. We then show (2) how to use the adjust…

We prove that two homogeneous ultra-metric spaces $X,Y$ are coarsely equivalent if and only if $\mathrm{Ent}^\sharp(X)=\mathrm{Ent}^\sharp(Y)$ where $\mathrm{Ent}^\sharp(X)$ is the so-called sharp entropy of $X$. This classification implies that each homogeneous proper ultra-metric space is coarsely equivalent to the a…

Truncated SGD with heavy-tailed noise eliminates sharp local minima.

Sharp upper bound found for Steklov spectrum on revolution submanifolds.

We discuss sharp Sobolev inequalities for vector valued maps.

We introduce a scalable measure of curvature for analyzing training dynamics of large language models.

Sharp upper bound found for stable minimal surfaces.

We give some sharp lower bounds of the first eigenvalue for the Hodge Laplacian acting on differential forms on the boundary of a Riemannian manifold. We also give some sharp estimates for the first nonzero Steklov eigenvalue for differential forms.

Let $M$ be a smooth closed $4k$-manifold whose Yamabe invariant $Y(M)$ is nonpositive. We show that $Y(M\sharp l \Bbb HP^k\sharp m \bar{\Bbb HP^k})=Y(M),$ where $l,m$ are nonnegative integers, and $\Bbb HP^k$ is the quaternionic projective space. When $k=4$, we also have $$Y(M\sharp l CaP^2\sharp m \bar{CaP^2})=Y(M),…

LSAM optimizes deep learning training with improved efficiency.

A sharp quantitative polygonal isoperimetric inequality is obtained.

Sharp spectral gap estimates for higher-order operators on hyperbolic spaces.

Sharp boundaries for detecting dense subhypergraphs established.