The paper establishes conditions for strict power concavity in convolutions.
problem Conditions for strict power concavity in convolutions.
method Analyzes sufficient conditions for strict parabolic power concavity of convolutions.
result Establishes sufficient conditions for strict power concavity of convolutions.
In this paper, we prove the concavity of p-entropy power of probability densities solving the p-heat equation on closed Riemannian manifold with nonnegative Ricci curvature. As applications, we give new proofs of Lp-Euclidean Nash inequality and Lp-Euclidean Logarithmic Sobolev inequality, moreover, an improv…
Estimates self- and cross-impact concavity and decay patterns in financial markets.
problem Understanding the impact of financial transactions on market dynamics.
method Nonparametric estimation of concave multi-asset propagator models using metaorders and order flow data.
result Concave self-impact with shifted power-law decay, significant gain from cross-impact, and improved predictive accuracy.
We prove the concavity of p-Rényi entropy power for positive solutions to the doubly nonlinear diffusion equations on Rn or compact Riemannian manifolds with nonnegative Ricci curvature. As applications, we give new proofs of the sharp Lp-Sobolev inequality and Lp-Gagliardo-Nirenberg inequalities on…
The paper examines how nonlinear transformations affect ridge sets in manifold learning.
problem Understanding the impact of nonlinear transformations on ridge sets in manifold learning.
method Examined the effects of nonlinear transformations on ridge sets using mathematical proofs and numerical experiments.
result The inclusion relationship $\cR(f\circ p)\subseteq \cR(p)$ holds for strictly increasing and concave transformations, and the Hausdorff distance between transformed and non-transformed ridge sets is smaller.
Paper proves generalized Talagrand inequality for Sinkhorn distance.
problem Proving a generalized Talagrand inequality for Sinkhorn distance.
method Using entropy power inequality and infinitesimal displacement convexity of optimal transport map.
result Extends previous results of Gaussian Talagrand inequality for Sinkhorn distance to strongly log-concave case.
By using asymptotic Morse inequalities we give a lower bound for the space of holomorphic sections of high tensor powers in a positive line bundle over a q-concave domain. The curvature of the positive bundle induces a hermitian metric on the manifold. The bound is given explicitely in terms of the volume of the domain…
Study concavity of solutions to elliptic equations under conformal deformations.
problem Establish concavity estimates for the principle eigenfunction of weighted Schrödinger operators.
method Analyzing the Dirichlet problem for the weighted Schrödinger operator \[-Δu + Vu = λρu\] with conformal connections.
result Partial resolution of Nguyen's conjecture on fundamental gap of horoconvex domains and power convexity estimate for solutions in spherical geometry.
This paper concerns the evolution of complete noncompact locally uniformly convex hypersurface in Euclidean space by curvature flow, for which the normal speed Φ is given by a power β≥1 of a monotone symmetric and homogeneous of degree one function F of the principal curvatures. Under the assumption that F …
In this paper, we prove the concavity of the Renyi entropy power for nonlinear diffusion equation (NLDE) associated with the Laplacian and the Witten Laplacian on compact Riemannian manifolds with non-negative Ricci curvature or CD(0,m)-condition and on compact manifolds equipped with time dependent metrics and poten…
Optimal trading strategy under market resistance and concave price impact model.
problem Optimal trading in a market with endogenous resistance and concave price impact.
method Modeling market resistance, deriving a stochastic Fredholm equation, proving existence and uniqueness, proposing an iterative scheme.
result Existence and uniqueness of optimal control under certain conditions, exponential convergence of iterative scheme.
Two new methods improve block-sparse signal recovery from noisy data.
problem Recovering block-sparse signals with unknown partitions.
method LogLOP-l2/l1 and AdaLOP-l2/l1 methods using log-sum penalty and MCP.
result Our methods outperform existing techniques in estimation accuracy.
The study finds that only round spheres shrink self-similarly under certain curvature flows.
problem Investigating self-similar solutions to curvature flows by high powers of curvature.
method Analyzing closed strictly convex hypersurfaces in Rn+1 under specific curvature flows. result Only round spheres shrink self-similarly under the studied curvature flows.
Study shows cliff-learning in transfer learning from foundation models.
problem Data-scaling of transfer learning from foundation models in low data regimes.
method Investigation of cliff-learning phenomenon through foundation-model analysis and toy models.
result Cliff-learning reflects compatibility between priors and tasks.
Paper establishes lower bounds for finite-sum optimization problems using novel construction methods.
problem Lower complexity bounds for finite-sum optimization problems with various component functions.
method Developed novel approach to construct hard instances and analyzed PIFO algorithms.
result Established lower complexity bounds for convex-concave and nonconvex-strongly-concave objectives.
In this paper, we prove the concavity of the Shannon entropy power for the heat equation associated with the Laplacian or the Witten Laplacian on complete Riemannian manifolds with suitable curvature-dimension condition and on compact super Ricci flows. Under suitable curvature-dimension condition, we prove that the ri…
We consider inverse curvature flows in the (n+1)-dimensional Euclidean space, n≥2, expanding by arbitrary negative powers of a 1-homogeneous, monotone curvature function F with some concavity properties. We obtain asymptotical roundness, meaning that circumradius minus inradius of the flow hypersurfaces decay…
In this paper, we first investigate the flow of convex surfaces in the space form R3(κ) (κ=0,1,−1) expanding by F−α, where F is a smooth, symmetric, increasing and homogeneous of degree one function of the principal curvatures of the surfaces and the power α∈(0,1] for κ=0,−1 and α=1 for κ=1…
The study provides guarantees for diffusion-based models under log-concave data, offering best-known convergence rates.
problem Theoretical guarantees for convergence of diffusion-based generative models under log-concave data distributions.
method Assumption of strongly log-concave data distributions, Lipschitz continuous functions for score estimation, and novel auxiliary process.
result Best known upper bounds for Wasserstein-2 distance between Gaussian distribution and sampling algorithm.
A novel decentralized algorithm improves minimax optimization in federated learning.
problem Minimax optimization in federated learning with data heterogeneity.
method Decentralized Gradient Tracking (K-GT-Minimax) for nonconvex-strongly-concave optimization.
result Demonstrates superior convergence rate for NC-SC minimax optimization.
Minimal graph level sets are concave if boundary is concave.
problem Understanding curvature of minimal graph level sets.
method Proved an inequality and showed geometric properties.
result Level sets of minimal graphs are concave if boundary is concave.
Proves log-concavity of cluster algebra coefficients for type An.
problem Log-concavity of cluster algebra coefficients.
method Introduced atomic theta basis and proved log-concavity for type An. result Proved log-concavity of coefficients for cluster algebra variables of type An. Study improves sampling from non-log-concave distributions using Fisher information.
problem Sampling from non-log-concave distributions with high Fisher information guarantees.
method Proximal sampler with RGO implementation, leveraging log-concave sampling results.
result Improved complexity guarantee in relative Fisher information for non-log-concave sampling.
The paper studies price impacts in asset liquidation markets.
problem Understanding price impacts in asset liquidation markets.
method Equilibrium formulation and analysis of price impacts.
result Existence and uniqueness of clearing prices for portfolio liquidation.
Established concavity principle for curved spaces.
problem Solving equations on curved spaces with nonnegative curvature.
method Applied concavity principle to elliptic and parabolic equations on locally symmetric spaces with nonnegative curvature.
result First general concavity principle on spaces with non-constant sectional curvature.
We consider a discrete-time, generically incomplete market model and a behavioural investor with power-like utility and distortion functions. The existence of optimal strategies in this setting has been shown in a previous paper under certain conditions on the parameters of these power functions. In the present paper w…
Establishes log-concavity estimates for convex domains' first Dirichlet eigenfunctions.
problem Quantifying the Hessian of log-concave eigenfunctions on convex domains.
method Analyzes log-concavity properties of the first Dirichlet eigenfunction on convex domains.
result Obtains quantitative estimates for the Hessian of logu. New saddle network architectures preserve convex-concave geometry in optimization problems.
problem Optimization models with convex x and concave y components.
method Structured separable decomposition and saddle network architectures.
result Proven one-dimensional approximation theorem and high accuracy on various test functions.
Heat flow fails to preserve concavity in curved spaces.
problem Non-preservation of concavity properties in curved spaces.
method Analysis of Dirichlet heat flow on Riemannian manifolds.
result No concavity properties are preserved unless curvature is zero.
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.
problem Understanding the concavity of spacetimes in Finsler geometry.
method Analyzes flag curvature and future capsules to characterize concavity.
result Berwald spacetimes are locally concave if and only if their flag curvature is nonnegative in timelike directions.
We define a class of L-convex-concave subsets of RPn, where L is a projective subspace of dimension l in RPn. 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-…
New algorithm proves convergence for MAP estimation with denoisers.
problem Proving convergence of MAP estimation methods using pretrained denoisers.
method A simple gradient descent algorithm on smoothed proximal objectives.
result Algorithm provably converges to the proximal operator under log-concavity.
Geodesic concavity and hypersymplectic structures in G2-structures space.
problem Analyzing the geodesic concavity and hypersymplectic structures in the space of closed G2-structures. method Utilising the geodesic constructed in the previous article, we show geodesic concavity and decrease in length of G2 Laplacian flow. result Hitchin's volume functional is geodesically concave and the G2 Laplacian flow decreases the length. Gradient methods converge exponentially in concave network games.
problem Finding Nash equilibria in concave network zero-sum games.
method Gradient Ascent and Optimistic Gradient Ascent analyses.
result Exponential convergence rates in various game settings.
This study examines how earnings announcements affect option volatility and pricing.
problem The impact of earnings announcements on option volatility and pricing.
method Analysis of extremely short-term options data to study bimodality and concavity in IV curves.
result Investors pay a premium to hedge against extreme volatility during earnings announcements in the presence of concave IV smiles.
Log-concavity of eigenfunctions on curved surfaces is proven, leading to fundamental gap estimates.
problem Proving log-concavity of eigenfunctions on curved surfaces.
method Analyzing the Laplacian eigenfunctions on positively curved surfaces.
result Strong log-concavity of the first eigenfunction on positively curved surfaces.
Improved sampling guarantees for weakly log-concave distributions.
problem Sampling from distributions that are not strongly log-concave.
method Proximal sampler with convergence guarantees under weaker assumptions.
result New state-of-the-art sampling guarantees for various target distributions.
New flow expands hypersurfaces in hyperbolic space, showing round limiting shape for certain powers.
problem Understanding the limiting shape of hypersurfaces expanding in hyperbolic space.
method Introduced shifted inverse curvature flow with positive power p for a smooth curvature function. result For 0<p≤1, limiting shape is always round as maximal existence time is approached. 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.
Establishes a concavity property for positive Hessian quotient operators.
problem Analyzing positive Hessian quotient operators on Riemannian manifolds.
method Proves a special concavity property and a Jacobi inequality.
result Proves a Jacobi inequality for symmetric tensors.
This paper concerns the evolution of a closed hypersurface of dimension n(≥2) in the Euclidean space Rn+1 under a mixed volume preserving flow. The speed equals a power β(≥1) of homogeneous, either convex or concave, curvature functions of degree one plus a mixed volume preserving term, incl…
The paper extends entropy formulas to super Ricci flows on metric measure spaces.
problem Entropy formulas for super Ricci flows on metric measure spaces.
method Extending Perelman's W-entropy and Shannon entropy power to super Ricci flows. result Equivalence between volume non-local collapsing property and lower boundedness of W-entropy on RCD(0,N) spaces. We numerically study an Asset Liability Management problem linked to the decommissioning of French nuclear power plants. We link the risk aversion of practitioners to an optimization problem. Using different price models we show that the optimal solution is linked to a de-risking management strategy similar to a concav…
Despite the success of single-agent reinforcement learning, multi-agent reinforcement learning (MARL) remains challenging due to complex interactions between agents. Motivated by decentralized applications such as sensor networks, swarm robotics, and power grids, we study policy evaluation in MARL, where agents with jo…
Unified routing and arbitrage with concave continuation.
problem Combining routing and arbitrage in financial markets.
method Extending AMM trade functions to negative inputs via concave continuation.
result Unified approach unifies routing and arbitrage.
The study proves non-existence of concave functions on specific metric spaces.
problem Proving the non-existence of concave functions on certain metric spaces.
method Analogue theorems for Alexandrov spaces and Cα-Hölder Riemannian manifolds. result Proves non-existence of concave functions on complete manifolds with finite volume and specific metric spaces.
Log-concavity proven for multinomial likelihoods under specific constraints.
problem Log-concavity of multinomial likelihoods under interval censoring constraints.
method Proved log-concavity by showing M-convex subsets of the discrete simplex.
result Likelihood function is completely log-concave.