arXiv research

Establishes a concavity property for positive Hessian quotient operators.

Study estimates eigenvalues for concave Hessian operators on convex domains.

A new method solves a complex optimization problem efficiently.

We explain a general construction through which concave elliptic operators on complex manifolds give rise to concave functions on cohomology. In particular, this leads to generalized versions of the Khovanskii-Teissier inequalities.

Estimates for complex Hessian equations on Hermitian manifolds.

Paper introduces $\ell$-DER for regression tasks using morphological operators and convex-concave procedure.

Derives concavity inequality and estimates for $k$-Hessian equations.

We explore the tan-concavity of the Lagrangian phase operator for the study of the deformed Hermitian Yang-Mills (dHYM) metrics. This new property compensates for the lack of concavity of the Lagrangian phase operator as long as the metric is almost calibrated. As an application, we introduce the tangent Lagrangian pha…

Paper derives estimates for Hessian equations under concavity assumptions.

We provide new results for noise-tolerant and sample-efficient learning algorithms under $s$-concave distributions. The new class of $s$-concave distributions is a broad and natural generalization of log-concavity, and includes many important additional distributions, e.g., the Pareto distribution and $t$-distribution.…

Paper estimates curvature of semi-convex solutions in hyperbolic space.

Study concavity of solutions to elliptic equations under conformal deformations.

The study improves fundamental gap estimates for surfaces with non-constant positive curvature.

New algorithm proves convergence for MAP estimation with denoisers.

Algorithm samples from composite log-concave distributions using gradient evaluations and restricted Gaussian oracles.

The paper proves extension theorems for complex manifolds with Levi $q$-concave domains.

Study on critical Lagrangian phase singularities in mean curvature flow.

Stein's method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds are readily available for a diversity of univariate targets, few multivariate targ…

The paper develops methods for sampling from log-concave distributions with constraints.

Study shows splitting schemes can approximate WFR flows faster than the exact flow.

A new algorithm reduces online exp-concave optimization runtime.

We study the Dirichlet problem for the Lagrangian phase operator, in both the real and complex setting. Our main result states that if $Ω$ is a compact domain in $\mathbb{R}^{n}$ or $\mathbb{C}^n$, then there exists a solution to the Dirichlet problem with right-hand side $h(x)$ satisfying $|h(x)| > (n-2)\fracπ{2}$ and…

A new sampling method reduces computational cost for high-dimensional log-concave distributions.

We consider variational inequalities coming from monotone operators, a setting that includes convex minimization and convex-concave saddle-point problems. We assume an access to potentially noisy unbiased values of the monotone operators and assess convergence through a compatible gap function which corresponds to the …

Study proves estimate for Hessian quotient equations on 2D Riemannian manifolds.

Solves curvature problems on manifolds with negative curvature.

New findings extend rigidity results to broader classes of manifolds.

New algorithms solve monotone inclusions and convex-concave minimax problems.

New algorithm speeds up sampling from log-concave distributions over polytopes.

The paper establishes conditions for strict power concavity in convolutions.

The Gursky-Streets equation are introduced as the geodesic equation of a metric structure in conformal geometry. This geometric structure has played a substantial role in the proof of uniqueness of $σ_2$ Yamabe problem in dimension four. In this paper we solve the Gursky-Streets equations with uniform $C^{1, 1}$ estima…

Minimal graph level sets are concave if boundary is concave.

This paper approaches the definition and properties of dynamic convex risk measures through the notion of a family of concave valuation operators satisfying certain simple and credible axioms. Exploring these in the simplest context of a finite time set and finite sample space, we find natural risk-transfer and time-co…

This paper tackles bandit optimization with a new pairwise comparison oracle for unknown strongly concave functions.

Proves log-concavity of cluster algebra coefficients for type $A_n$.

Study improves sampling from non-log-concave distributions using Fisher information.

New algorithm samples from log-concave distributions over polytopes efficiently.

Proposes a neural network framework for feature selection in high-dimensional settings.

Established concavity principle for curved spaces.

Establishes log-concavity estimates for convex domains' first Dirichlet eigenfunctions.

A new sampling method, RC-LMC, reduces computational cost for high-dimensional log-concave distributions.

New saddle network architectures preserve convex-concave geometry in optimization problems.

Heat flow fails to preserve concavity in curved spaces.

We present a simple connection between differential Harnack inequalities for hypersurface flows and natural concavity properties of their time-of-arrival functions. We prove these concavity properties directly for a large class of flows by applying a concavity maximum principle argument to the corresponding level set f…

Investigates concavity of spacetimes, showing conditions for local concavity.

We define a class of L-convex-concave subsets of $\Bbb{R}P^n$, where L is a projective subspace of dimension l in $\Bbb{R}P^n$. These are sets whose sections by any (l+1)-dimensional space L' containing L are convex and concavely depend on L'. We introduce an L-duality for these sets, and prove that the L-dual to an L-…

We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the martingale inequality is determined by a fixed point of a simple nonlinear operato…

In this paper, we give an easy proof of the main results of Andrews and Clutterbuck's paper [J. Amer. Math. Soc. 24 (2011), no. 3, 899--916], which gives both a sharp lower bound for the spectral gap of a Schröinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We…