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 proves complex Monge-Ampère equations with new singularity types and confirms log-concavity of volume.
problem Existence and uniqueness of solutions to complex Monge-Ampère equations with prescribed singularities.
method Proves existence and uniqueness of solutions to complex Monge-Ampère equations with general model type singularities in big cohomology classes.
result Log-concavity of volume of closed positive (1,1)-currents is confirmed.
Strict concavity proven for growth indicator function of certain groups.
problem Proving strict concavity of growth indicator function for specific groups.
method Smoothness of Manhattan hypersurface and critical-exponent map.
result Strict concavity of growth indicator function for relatively Anosov groups.
Gibbs sampler contracts entropy under strong log-concavity, improving mixing time.
problem Improving the mixing time of Gibbs sampler under strong log-concavity.
method Analyzing Gibbs sampler contraction under strong log-concavity, providing sharp contraction rate.
result Gibbs sampler contracts entropy linearly with condition number and independent of dimension under strong log-concavity.
CAVI converges for log-concave measures via optimal transport.
problem Finding the closest product measure to a log-concave measure via CAVI.
method Adapting coordinate descent techniques from Euclidean space to optimal transport for log-concave densities.
result Proves convergence of CAVI for log-concave densities and provides rates of convergence under additional conditions.
Optimizes algorithms for non-concave bandit problems.
problem Optimizing algorithms for non-concave bandit problems.
method Unified zeroth-order optimization paradigm.
result Minimax-optimal algorithms in the dimension for low-rank generalized linear bandit problems.
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.
Optimizes Einstein equations using optimal transport.
problem General relativity equations and thermodynamics.
method Optimal transport formulation linking curvature, cosmological constant, and energy-momentum tensor.
result New mathematical connection between general relativity and thermodynamics.
The paper solves Lovelock gravity's constraint equations, generalizing the σ k σ_k σ k -Yamabe problem.
problem Solving Lovelock gravity's constraint equations in a specific manifold setting.
method Using the σ k σ_k σ k -Yamabe problem as a basis, the paper extends solutions to the Lovelock constraint equations and studies concavity properties. result Several cases where conformal solutions exist, including when Lovelock theories are close to General Relativity.
The paper provides tight bounds for improving multi-armed bandits problem.
problem Improving multi-armed bandits problem with concave reward functions.
method Upper and lower bounds for randomized online algorithms, providing an O ( k log k ) O(\sqrt{k} \log k) O ( k log k ) approximation. result Achieved nearly-tight approximation guarantees for the improving multi-armed bandits problem.
In stochastic portfolio theory, a relative arbitrage is an equity portfolio which is guaranteed to outperform a benchmark portfolio over a finite horizon. When the market is diverse and sufficiently volatile, and the benchmark is the market or a buy-and-hold portfolio, functionally generated portfolios introduced by Fe…
Study minimax risk of score estimation for log-concave distributions.
problem Minimizing risk in score estimation for log-concave distributions.
method Developed subclasses of log-concave densities and constructed a locally adaptive, multiscale estimator.
result Established minimax rates for score estimation over specific subclasses of log-concave densities.
MALA improves sampling from log-concave densities with faster mixing times.
problem Sampling from strongly log-concave densities efficiently.
method Discretization of Langevin diffusion with accept-reject step.
result MALA requires O ( κ d log ( 1 / δ ) ) \mathcal{O} \big(κd \log(1/δ) \big) O ( κ d log ( 1/ δ ) ) steps for TV error δ δ δ . New algorithm reduces variance in stochastic gradient estimation.
problem Optimizing the variance of stochastic gradient algorithms for non-log-concave distributions.
method Developed a Multi-index Antithetic Stochastic Gradient Algorithm (MASGA) that is independent of the distribution's structure.
result MASGA achieves performance comparable to Monte Carlo estimators with unbiased samples.
Estimating mean from one-bit samples of symmetric log-concave distributions.
problem Estimating the mean of a symmetric log-concave distribution with limited one-bit measurements.
method Analyzes mean squared error in three settings: centralized, adaptive, and distributed, with and without quantization.
result One round of adaptivity is sufficient to achieve optimal mean-square error in the adaptive setting.
Study shows exact dimensionality and regularity of manifolds for specific groups.
problem Exact dimensionality and regularity of manifolds for relatively Anosov groups.
method Dynamical methods, including finite and mixing of Bowen–Margulis–Sullivan measures.
result Manifolds are C 1 C^1 C 1 -regular and growth indicator is strictly concave. This work proposes new methods for variational inference using gradient flows on Gaussian measures.
problem Developing algorithmic guarantees for variational inference.
method Proposes principled methods for variational inference using gradient flows on the Bures--Wasserstein space of Gaussian measures.
result Strong theoretical guarantees for log-concave posteriors.
New sampling method guarantees approximate first-order stationary points for non-convex functions.
problem Sampling from non-log-concave densities with non-convex potential functions.
method Averaged Langevin Monte Carlo with complexity analysis.
result Langevin Monte Carlo outputs a sample with ε-relative Fisher information after O(L²d²/ε²) iterations.
Optimal convex loss function improves regression coefficient estimation.
problem Asymptotic variance improvement in linear regression estimation.
method Score matching extension for log-concave projection.
result Semiparametric estimator attains minimal asymptotic covariance.
Dynamic pricing policy converges to Nash equilibrium with low regret.
problem Sequential price competition among sellers over multiple periods.
method Semi-parametric least-squares estimation of s-concave demand functions.
result Prices converge to Nash equilibrium with rate O ( T − 1 / 7 ) O(T^{-1/7}) O ( T − 1/7 ) and sellers incur regret O ( T 5 / 7 ) O(T^{5/7}) O ( T 5/7 ) . Optimal iterative thresholding algorithms improve upon hard and soft thresholding.
problem Optimizing sparsity or rank constraints in optimization problems.
method Developed the notion of relative concavity for thresholding operators, finding a new class of operators that are optimal.
result A new class of thresholding operators, including ℓ q \ell_q ℓ q thresholding and reciprocal thresholding, achieves the strongest convergence guarantee. Concave elliptic operators yield concave functions on cohomology.
problem Understanding concave functions on cohomology.
method General construction of concave elliptic operators.
result Generalized Khovanskii-Teissier inequalities.
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.
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.
The study bounds the utility of empirically optimal portfolios using stock return data.
problem Maximizing expected ratio of portfolio utility to best asset utility.
method High probability utility bounds derived from Lipschitz or Hölder continuous utility functions.
result Utility bounds depend on utility function, number of assets, and observations.
Simple connection between Harnack inequalities and concavity of arrival time functions.
problem Proving differential Harnack inequalities for various flows.
method Directly proving concavity properties of time-of-arrival functions for a class of flows using a concavity maximum principle.
result Short proof of Hamilton's and Andrews' differential Harnack inequalities.
Proves log-concavity of cluster algebra coefficients for type A n A_n A n .
problem Log-concavity of cluster algebra coefficients.
method Introduced atomic theta basis and proved log-concavity for type A n A_n A n . result Proved log-concavity of coefficients for cluster algebra variables of type A n A_n A n . A pricing principle is introduced for non-attainable claims in incomplete markets.
problem Pricing non-attainable contingent claims in incomplete markets.
method Distorted Radon-Nikodym derivative and Tsallis relative entropy over a family of equivalent martingale measures.
result The pricing principle is closely related to backward stochastic differential equations and is arbitrage-free and time-consistent.
Paper optimizes DC pension fund management with VaR and relative performance constraints.
problem Optimizing DC pension fund performance under VaR and relative performance constraints.
method Introduced an auxiliary process to transform the problem into a self-financing problem, combined linearization, Lagrange dual, martingale, and concavification methods.
result Explicit investment strategies obtained for certain penalty and reward functions.
In this paper we consider the isoperimetric profile of convex cylinders K × R q K\times\mathbb{R}^q K × R q , where K K K is an m m m -dimensional convex body, and of cylindrically bounded convex sets, i.e, those with a relatively compact orthogonal projection over some hyperplane of R n + 1 \mathbb{R}^{n+1} R n + 1 , asymptotic to a right convex cylind…
Study optimal consumption for loss-averse agents considering past spending peaks.
problem Optimal consumption for loss-averse agents with reference to past spending maximum.
method Adopted S-shaped utility, concave envelope, HJB variational inequality, dual transform, and smooth-fit conditions.
result Obtained piecewise closed-form solutions for optimal consumption and investment control.
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.
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 log u \log u log u . 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.
This work improves information concentration for exp-concave distributions, making it dimension-independent.
problem Challenges in information concentration for log-concave distributions with dimension dependence.
method Proves exp-concavity leads to dimension-independent information concentration using a novel variance Brascamp-Lieb inequality.
result Information concentration depends only on the exp-concavity parameter, not the dimension.
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.
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.
The paper constructs random concave functions on the unit simplex.
problem Understanding probability measures on spaces of concave functions.
method Constructing random concave functions via a scaled minimum of random hyperplanes.
result There is a transition from deterministic to non-trivial limiting distributions as the number of hyperplanes increases.
We define a class of L-convex-concave subsets of R P n \Bbb{R}P^n R P n , where L is a projective subspace of dimension l in R P n \Bbb{R}P^n 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-…
Geodesic concavity and hypersymplectic structures in G 2 G2 G 2 -structures space.
problem Analyzing the geodesic concavity and hypersymplectic structures in the space of closed G 2 G2 G 2 -structures. method Utilising the geodesic constructed in the previous article, we show geodesic concavity and decrease in length of G 2 G2 G 2 Laplacian flow. result Hitchin's volume functional is geodesically concave and the G 2 G2 G 2 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.
New stability bounds for Sinkhorn's algorithm in entropic optimal transport.
problem Stability and convergence of Sinkhorn's algorithm for entropic optimal transport.
method Semiconcavity approach to analyze stability and convergence.
result Exponential convergence of Sinkhorn's algorithm under semiconcavity conditions.
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.
We reformulate LIPs as min-max problems for easier solution.
problem Recovering signals from few linear measurements.
method Proposed a min-max reformulation of LIPs.
result Saddle points characterize solutions to LIPs.
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.
Improved sampling from complex distributions with reduced bias.
problem Reducing bias in high-dimensional sampling algorithms.
method Hierarchical entropy analysis to weaken assumptions and expand scope.
result Bias reduction in low-dimensional marginals scales with lower dimension, not full dimension.