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 ) Δ(t) Δ ( t ) associated to Christoffel words and proving its log-concavity. result Strong Fox conjecture for two-bridge knots proved.
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 n n -braids, providing explicit formulas and verifying log-concavity. result Explicit formulas for signature and first 4 coefficients of Alexander polynomials, showing log-concavity.
Fox's trapezoidal conjecture for four-strand Turk's head knots is proven.
problem Proving log-concavity of the coefficient sequence of D n ( z ) D_n(z) D n ( z ) for four-strand Turk's head knots. method Four-block smoothing theorem for products of reciprocal quartics.
result The coefficient sequence of D n ( z ) D_n(z) D n ( z ) is log-concave. The study explains the concavity of price impact in markets.
problem The asymptotic concavity of price impact in meta-orders.
method A model with linear local price impact and co-directional trades.
result Volumes at best bid and ask prices favor the executor.
Investigates probability of error in structured thresholding bandit problems.
problem Probability of misclassifying arms in structured thresholding bandit problems.
method Analyzes two shape constraints: monotonic increasing and concave sequences of arm means.
result Upper and lower bounds for the probability of error match up to constants in the problem dependent regime.
Paper analyzes complexity of solving nonconvex-strongly-concave problems.
problem Finding approximate stationary points of nonconvex-strongly-concave minimax problems.
method Introduces a generic acceleration scheme to solve crafted subproblems.
result Algorithm nearly matches lower complexity bounds in general setting.
A new sampling method using log-concave Markov chains.
problem Sampling from unnormalized densities efficiently.
method Decomposes sampling into log-concave Markov chains with noisy measurements.
result Shows remarkable capacity to 'tunnel' between modes of a distribution.
SA algorithms control dynamic regret in non-stationary settings with strong convexity or exp-concavity.
problem Non-stationary Online Convex Optimization with dynamic regret control.
method Strongly Adaptive (SA) algorithms view dynamic regret as path variation of the comparator sequence.
result SA algorithms achieve i l d e O ( T V T ∨ log T ) ilde O(\sqrt{TV_T} \vee \log T) i l d e O ( T V T ∨ log T ) and i l d e O ( d T V T ∨ d log T ) ilde O(\sqrt{dTV_T} \vee d\log T) i l d e O ( d T V T ∨ d log T ) dynamic regret for strongly convex and exp-concave losses, respectively. RHMC accelerates sampling from log-concave distributions.
problem Sampling from log-concave probability distributions efficiently.
method RHMC uses simulated Hamiltonian dynamics with random integration times.
result RHMC converges exponentially fast in KL divergence for log-concave distributions.
Near-logarithmic regret per switch achieved for mixable/exp-concave losses.
problem Online optimization of mixable loss functions with dynamic environments.
method Online mixture framework using static solvers and hyper-expert creations.
result Near-logarithmic regret per switch with sub-polynomial complexity.
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.
Decision maker's preferences are often captured by some choice functions which are used to rank prospects. In this paper, we consider ambiguity in choice functions over a multi-attribute prospect space. Our main result is a robust preference model where the optimal decision is based on the worst-case choice function fr…
The goal of online prediction with expert advice is to find a decision strategy which will perform almost as well as the best expert in a given pool of experts, on any sequence of outcomes. This problem has been widely studied and O ( T ) O(\sqrt{T}) O ( T ) and O ( log T ) O(\log{T}) O ( log T ) regret bounds can be achieved for convex losses (\cite{zin…
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…
We consider the convex-concave saddle point problem min x max y f ( x ) + y ⊤ A x − g ( y ) \min_{x}\max_{y} f(x)+y^\top A x-g(y) min x max y f ( x ) + y ⊤ A x − g ( y ) where f f f is smooth and convex and g g g is smooth and strongly convex. We prove that if the coupling matrix A A A has full column rank, the vanilla primal-dual gradient method can achieve linear convergence even if f f f is not stron…
New algorithm reduces dynamic regret for exp-concave losses.
problem Minimizing dynamic regret in online learning with exp-concave losses.
method Integrates KKT conditions to achieve optimal dynamic regret.
result Achieves dynamic regret of i l d e O ∗ ( n 1 / 3 C n 2 / 3 ∨ 1 ) ilde O^*(n^{1/3}C_n^{2/3} \vee 1) i l d e O ∗ ( n 1/3 C n 2/3 ∨ 1 ) . 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.
In this paper, we consider first-order convergence theory and algorithms for solving a class of non-convex non-concave min-max saddle-point problems, whose objective function is weakly convex in the variables of minimization and weakly concave in the variables of maximization. It has many important applications in mach…
Investigates how rebalancing frequency and transaction costs affect log-optimal portfolios.
problem Impact of rebalancing frequency and transaction costs on log-optimal portfolios.
method Proved equivalence to concave program, derived optimality conditions, tested using intraday and daily data.
result Transaction costs can cause bankruptcy for frequency-dependent log-optimal portfolios, approximating to quadratic concave program.
Paper solves globally optimal k-means for low dimensional data.
problem Finding globally optimal k-means solutions for low dimensional data.
method Formulates as a concave assignment problem, iteratively solving small concave and large linear programming problems.
result Solves k-means to global optimality for large data sets with several clusters.
New algorithms minimize dynamic regret in non-stationary online learning.
problem Universal dynamic regret minimization under exp-concave and smooth losses.
method Strongly Adaptive algorithms with a path variational based on second order differences of the comparator sequence.
result Achieve a dynamic regret of i l d e O ( d 2 n 1 / 5 C n 2 / 5 ∨ d 2 ) ilde O(d^2 n^{1/5} C_n^{2/5} \vee d^2) i l d e O ( d 2 n 1/5 C n 2/5 ∨ d 2 ) , optimal modulo dependencies. Reduces dynamic regret to static problem in RKHS.
problem Minimizing cumulative loss in online convex optimization.
method Reduces dynamic regret to static regret problem in RKHS.
result Optimal dynamic regret guarantees for linear losses and new bounds for exp-concave and improper linear regression.
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.
In this article, we study the relationship between the weak limit of a sequence of integral currents in a metric space and the possible Hausdorff limit of the sequence of supports. Due to cancellation, the weak limit is in general supported in a strict subset of the Hausdorff limit. We exhibit sufficient conditions in …
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.
Improved cumulative regret for sequence prediction with limited expert advice.
problem Minimizing cumulative regret in sequence prediction with limited information.
method Convex combination of experts with limited observation, achieving constant regret.
result Strategies achieve constant regret independent of the horizon T, improving over standard bounds.
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 . 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.
Random extrapolation speeds up coordinate descent for sparse and dense data.
problem Efficiently solving primal-dual coordinate descent for sparse and dense data.
method Adapts to sparsity and uses large step sizes for dense data, proving linear convergence under metric subregularity.
result Linear convergence under metric subregularity and optimal sublinear convergence rates in general convex-concave problems.
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.
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.
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 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.
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.
Structured learning is appropriate when predicting structured outputs such as trees, graphs, or sequences. Most prior work requires the training set to consist of complete trees, graphs or sequences. Specifying such detailed ground truth can be tedious or infeasible for large outputs. Our main contribution is a large m…
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.
A new method simplifies sampling from complex distributions without using diffusions.
problem Sampling from complex, high-dimensional distributions efficiently.
method Reduces sampling to solving a sequence of 'nice' sampling problems using SLC distributions.
result Shows how to traverse backwards paths using high-accuracy routines for SLC distributions.
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 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.
We consider the problem of minimizing a difference-of-convex (DC) function, which can be written as the sum of a smooth convex function with Lipschitz gradient, a proper closed convex function and a continuous possibly nonsmooth concave function. We refine the convergence analysis in [38] for the proximal DC algorithm …
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 C α C^α C α -Hölder Riemannian manifolds. result Proves non-existence of concave functions on complete manifolds with finite volume and specific metric spaces.