Log-concave coefficient sequences for two-bridge knots proved.
problem Proving log-concavity of Alexander polynomial coefficient sequences for alternating knots.
method Introducing a polynomial Δ(t) associated to Christoffel words and proving its log-concavity. result Strong Fox conjecture for two-bridge knots proved.
The paper optimizes wealth with concave coefficients in a recursive utility maximization problem.
problem Optimizing wealth with concave coefficients in a recursive utility maximization problem.
method Equivalent backward formulation, Fenchel-Legendre transform, convex duality method.
result Derives the optimal terminal wealth for investors with ambiguity aversion.
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 proves Alexander polynomials of certain 4-braid knots satisfy a conjecture and gives formulas for log-concave sequences.
problem Proving the Alexander polynomials of certain 4-braid knots satisfy Fox's Trapezoidal Conjecture.
method Analyzes families of alternating 4-braids and n-braids, providing explicit formulas and verifying log-concavity. result Explicit formulas for signature and first 4 coefficients of Alexander polynomials, showing log-concavity.
The Links-Gould polynomial of alternating knots is shown to be log-concave and positive.
problem Verifying the positivity and log-concavity of the Links-Gould polynomial for alternating knots.
method Formulated a conjecture and verified it computationally for all 51.3 million knots with up to 19 crossings.
result All but 544 knots satisfy a stronger log-concavity condition.
New algorithms solve complex minimax problems efficiently.
problem Nonconvex-strongly concave minimax problems in machine learning.
method Gradient norm regularized trust-region (GRTR) and Levenberg-Marquardt (LMNegCur) algorithms.
result Proved iteration complexities matching best known results.
The Links-Gould invariant of alternating links has log-concave coefficients.
problem Log-concavity of Links-Gould coefficients for alternating links.
method Experimental and computational evidence.
result The Links-Gould coefficients of alternating links are log-concave.
Fox's trapezoidal conjecture for four-strand Turk's head knots is proven.
problem Proving log-concavity of the coefficient sequence of Dn(z) for four-strand Turk's head knots. method Four-block smoothing theorem for products of reciprocal quartics.
result The coefficient sequence of Dn(z) is log-concave. 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.
Analyzes alternating minimization for nonconvex sets in high-dimensional statistics.
problem Optimizing loss functions over nonconvex sets in high-dimensional statistics.
method Local concavity coefficients for nonconvex sets, alternating minimization, inexact algorithms.
result Reveals distinctions between alternating and non-alternating methods, provides convergence conditions.
New algorithms for learning under s-concave distributions, including Pareto and t-distributions.
problem Learning under broad and natural generalizations of log-concave distributions, including fat-tailed ones.
method Introduce new convex geometry tools to study s-concave distributions and use these properties to provide bounds on learning quantities. result Significantly generalize prior results for margin-based, disagreement-based, and passive learning of intersections of halfspaces.
We solve a century-old conjecture about Alexander polynomials of special alternating links.
problem Fox's conjecture about unimodality of Alexander polynomial coefficients.
method Proving a multivariate generalization of the Alexander polynomial is Lorentzian.
result Alexander polynomial coefficients of special alternating links form a log-concave sequence.
Unified framework connects two market-making models, revealing their underlying equivalence.
problem Independent calibration of two market-making frameworks (Avellaneda-Stoikov and Cartea-Jaimungal).
method Axiomatic approach to market preference functional, showing equivalence under specific conditions.
result Avellaneda-Stoikov and Cartea-Jaimungal frameworks are equivalent under certain conditions.
Explicit robust hedging strategies for convex or concave payoffs under a continuous semimartingale model with uncertainty and small transaction costs are constructed. In an asymptotic sense, the upper and lower bounds of the cumulative volatility enable us to super-hedge convex and concave payoffs respectively. The ide…
A generalized optimistic method for saddle point problems with improved complexity.
problem Solving convex-concave saddle point problems efficiently.
method Proposes a generalized optimistic method that includes the optimistic gradient method as a special case, handling constrained saddle point problems with composite objective functions and arbitrary norms.
result Best-known global iteration complexity bounds for first-, second-, and higher-order methods.
New proof of trapezoidal property for Alexander polynomials of special alternating links.
problem Proving trapezoidal property of Alexander polynomials for special alternating links.
method Analyzing vector configurations from matroids and totally positive matrices.
result Alexander polynomials of special alternating links exhibit log-concavity and trapezoidal properties.
This research accelerates sampling methods using Nesterov's Acceleration.
problem Improving sampling efficiency in MCMC methods.
method Developed a Hessian-Free High-Resolution ODE reformulation of NAG-SC, injected noise, and discretized the diffusion process.
result Quantified acceleration beyond underdamped Langevin in W2 distance for log-strongly-concave targets. 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.
SCOPE fuses categorical variable levels to estimate high-dimensional linear models.
problem Estimating high-dimensional linear models with nominal categorical data.
method SCOPE uses nonconvex concave penalties to fuse levels and achieve efficient computation.
result SCOPE achieves oracle least squares solution under certain conditions.
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.
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.
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.
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 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.
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.
Log-concave densities characterized using peacock and zonoid concepts.
problem Characterizing log-concave densities.
method Characterization using peacock and zonoid concepts.
result Two characterizations of log-concave densities.
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 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-…
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.
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.
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.
New information geometry from exponentially concave functions.
problem Optimal transport and information geometry.
method Information geometry tools applied to exponentially concave functions.
result New information geometry on the unit simplex.
The paper transforms a convex hull into a concave surface around a point cloud.
problem Creating a concave surface that encloses all points in a point cloud.
method Iterative facet replacement and expansion of the convex hull.
result A method to evolve a convex hull into a concave surface that fits the point cloud.
Introduces CSLC models to bridge deep generative models and classical algorithms.
problem Mode collapse and memorization issues in deep generative models and restrictive assumptions in classical algorithms.
method Introduces conditionally strongly log-concave (CSLC) models, factorizing data distribution into strongly log-concave conditional distributions.
result Efficient parameter estimation and sampling algorithms with theoretical guarantees for non-log-concave data distributions.
Study links and quivers, proving polynomial equality conjecture.
problem Link and quiver invariants and their relations.
method Cluster algebra invariants, point count polynomials, skein relations.
result Equality conjecture between plabic graph link polynomial and quiver point count polynomial proved for specific cases.
Paper finds convexity in translating solitons for concave flows.
problem Understanding convexity in translating solitons for concave extrinsic flows.
method Analyzes convexity estimates for translating solitons evolving under concave functions in Rn+1. result Establishes convexity estimates for translating solitons of concave extrinsic geometric flows.
ICCNLS models complex relationships as convex and concave components.
problem Complex input-output relationships with affine ambiguity.
method Sub-gradient constrained affine functions, global orthogonality constraints, L1, L2, and elastic net regularisation.
result Improved predictive accuracy and model simplicity compared to conventional methods.
Optimizes exp-concave losses with a new risk bound.
problem Optimizing exp-concave losses with stochastic convex optimization.
method Empirical Risk Minimization with a unified geometric assumption and local norms.
result Provides an O(d/n+log(1/δ)/n) excess risk bound.