arXiv research

Sharp stability in Almgren problem solved in any dimension.

Sharp Li-Yau equality proven for shrinking Ricci solitons without curvature assumptions.

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 regularity for Pfaff system leads to isometric immersions in arbitrary dimensions.

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…

Bounds on factual and counterfactual distributions under measurement error in discrete models.

Sharp spectral theorem splits certain non-compact manifolds.

New stability framework relaxes boundedness assumptions for generalization bounds.

We prove a sharp Onofri-type inequality and non-existence of extremals for a Moser-Tudinger functional on the sphere in the presence of potentials having positive order singularities. We also investigate the existence of critical points and give some sufficient conditions under symmetry or nondegeneracy assumptions.

In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental soluti…

We obtain sharp quantitative Laplacian upper and lower estimates under no assumption on curvatures. As a result, we derive quantitative Laplacian, area and volume comparison theorems for tubes in Riemannian and Kähler manifolds under weak integral curvature assumptions. We also give some applications, such as a general…

We classify local minimizers of $\intσ_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $4\leq n\leq 5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimiz…

Sharp pinching conditions restrict the geometry and topology of submanifolds.

New flexible confidence sequences for robust statistical inference.

A method to assess sensitivity to unmeasured confounding with sharp bounds.

New criterion for Weyl law on Riemannian manifolds without standard assumptions.

Omega ratio, defined as the probability-weighted ratio of gains over losses at a given level of expected return, has been advocated as a better performance indicator compared to Sharpe and Sortino ratio as it depends on the full return distribution and hence encapsulates all information about risk and return. We comput…

Sharp bounds on ATE with unmeasured confounders, valid even when misspecified.

Sharp estimates for mean curvature flow confirm bounded diameter conjecture.

We give a sharp comparison between the spectra of two Riemannian manifolds (Y,g) and (X,g_0) under the following assumptions: (X,g_0) has bounded geometry, (Y,g) admits a continuous Gromov-Hausdorff ε-approximation onto (X,g_0) of non zero absolute degree, and the volume of (Y,g) is almost smaller than the volume of (X…

Sharp bounds and parabolicity results for 3-manifolds with scalar curvature.

The paper proves inequalities for hypersurfaces in weighted manifolds.

Efficiently estimates sparse linear regression with heavy-tailed and outlier-contaminated data.

Unified analysis improves SAM for non-convex optimization.

We discuss the maximum modulus principle, and weak unique continuation, for CR functions on an abstract almost CR manifold M. We investigate these matters under the assumption of weak pseudoconcavity, and obtain sharp results about propagation along Sussmann leaves.

Study causal inference under specific sampling methods with monotonicity assumptions.

Gradient descent near stability threshold shows sharpness oscillations.

Monge SAM improves deep learning by making sharpness-aware minimization invariant to reparametrizations.

Gradient descent near stability threshold exhibits sharpness oscillations.

Improved mass-capacity bounds for specific 3D manifolds.

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.

Sharp inequality for Lorentzian spaces with timelike Ricci bounds.

Sharp inequalities proved for RCD spaces, showing equality conditions.

Gradient descent trains neural networks to match kernel regression's sharp generalization rate.

Sharp analysis of out-of-distribution error in overparameterized models with importance weights.

Closed hyperbolic manifolds and manifolds with nonpositive sectional curvature are geometrically rigid under certain curvature conditions.

Sharp Lipschitz bounds for flow-matching and diffusion models with optimal sampling rates.

We show that the martingale component in the long-term factorization of the stochastic discount factor due to Alvarez and Jermann (2005) and Hansen and Scheinkman (2009) is highly volatile, produces a downward-sloping term structure of bond Sharpe ratios, and implies that the long bond is far from growth optimality. In…

Overparameterization enhances SAM's effectiveness in minimizing sharpness.

We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly increase the first eigenvalue normalized by area if the initial su…

Improved heat equation estimates without gradient curvature assumption.

Sharp isoperimetric inequality derived from Q-curvature for smooth metrics.

Study shows distance to boundary is always attained on varifolds with bounded curvature.

Sharp generalization of boundary regularity for area minimizing currents with arbitrary multiplicity.

Sharp bounds established for Federated Averaging (FedAvg), improving convergence rates.

New method turns any regression model into a calibrated probabilistic model.

We study generalizations of Reifenberg's Theorem for measures in $\mathbb R^n$ under assumptions on the Jones' $β$-numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which holds for general measures without density assumptions, give effective measure bounds…

Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci…