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.
The paper extends risk measures to two-step approximations and studies log-concave distributions.
problem Extending classical risk measures to two-step approximations.
method Optimization problem for determining optimal regime thresholds and values for log-concave distributions.
result Conditions for the uniqueness of regime changing in log-concave distributions.
New method improves sampling for weakly log-concave posteriors.
problem Sampling from weakly log-concave posterior distributions.
method Stochastic Langevin Monte Carlo with over-damped diffusion.
result Simulation horizon is ( d log ( n ) 2 ) ( 1 + r ) 2 (d \log(n)^2)^{(1+r)^2} ( d log ( n ) 2 ) ( 1 + r ) 2 with Poisson subsampling. This paper optimizes Bayesian estimation for log-concave models using Langevin Monte-Carlo.
problem Optimizing Bayesian estimators for log-concave models with Langevin Monte-Carlo.
method Quantitative statistical bounds and numerical approximation of Gibbs measures.
result Established optimal numerical strategy and its cost for Bayesian posterior mean approximation.
We consider the problem of sampling from a strongly log-concave density in R d \mathbb{R}^d R d , and prove a non-asymptotic upper bound on the mixing time of the Metropolis-adjusted Langevin algorithm (MALA). The method draws samples by simulating a Markov chain obtained from the discretization of an appropriate Langevin dif…
Paper provides exponential convergence guarantees for Iterative Markovian Fitting.
problem Addressing the Schrödinger Bridge problem in computational optimal transport and generative modeling.
method Develops non-asymptotic exponential convergence guarantees for Iterative Markovian Fitting.
result First non-asymptotic exponential convergence guarantees for IMF under mild structural assumptions.
New method samples from non-log-concave distributions with weak dissipativity.
problem Sampling from distributions that are not log-concave and weakly dissipative.
method Taming scheme tailored to growth and decay properties of the target distribution.
result Explicit non-asymptotic guarantees for KL, TV, and Wasserstein distances.
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.
Unified analysis of KL divergence using shifted composition for sampling.
problem Sampling from target distributions with KL divergence guarantees.
method Shifted composition rule applied to KL divergence, combining local error analysis and Girsanov's theorem.
result Unified KL guarantees for strongly log-concave, weakly log-concave, and log-Sobolev distributions.
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.
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 . 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.
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.
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.
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.
Estimates log-concave densities in graphical models using tent functions.
problem Maximum likelihood estimation of log-concave densities in undirected graphs.
method MLE as product of tent functions corresponding to maximal cliques.
result MLE can be found via convex optimization.
Paper proves super log-concavity of first eigenfunction for certain hyperbolic domains.
problem Proving super log-concavity of first eigenfunction for horo-convex domains in hyperbolic space.
method Analyzes properties of Laplacian eigenfunctions in hyperbolic geometry.
result Optimal proof of super log-concavity for horo-convex domains with constraints.
The paper develops inequalities for log-concave functions and related surface areas.
problem Understanding log-concave functions and their inequalities.
method Establishing new inequalities through f-divergences and functional affine surface areas.
result New inequalities on functional affine surface area and bounds for Kullback-Leibler divergence.
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.
We construct a compact symplectic manifold with a Hamiltonian circle action for which the Duistermaat-Heckman function is not log-concave.
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.
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.
Zigzag sampling algorithm efficiently samples from strongly log-concave distributions with low computational cost.
problem Sampling from strongly log-concave distributions efficiently and with low computational complexity.
method Zigzag sampling algorithm with warm start assumption, focusing on gradient evaluations.
result Achieves ε error in chi-square divergence with computational cost of O(κ²d^(1/2)(log(1/ε))^(3/2)) gradient evaluations.
New lower bounds for sampling from log-concave distributions in higher dimensions.
problem Proving lower bounds for sampling from log-concave distributions in higher dimensions.
method Multiscale construction inspired by geometric measure theory and reduction to block Krylov algorithms.
result Query lower bounds for sampling from log-concave distributions in higher dimensions are established.
Study Langevin Monte Carlo for sampling non-log-concave distributions.
problem Sampling from non-log-concave distributions, especially Gaussian mixtures.
method Discretizations of overdamped Langevin diffusions.
result Numerical simulations compare Langevin Monte Carlo algorithms' performance.
Karshon constructed the first counterexample to the log-concavity conjecture for the Duistermaat-Heckman measure: a Hamiltonian six manifold whose fixed points set is the disjoint union of two copies of T 4 T^4 T 4 . In this article, for any closed symplectic four manifold N N N with b + b+ b + greater than 1, we show that there is a…
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.
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.
Improved sampling for diffusion models and log-concave distributions.
problem Efficient sampling for diffusion models and log-concave distributions.
method Algorithms for sampling with δ δ δ -error in p o l y l o g ( 1 / δ ) \mathrm{polylog}(1/δ) polylog ( 1/ δ ) steps using accurate score estimates. result Exponential improvement in complexity over previous results.
The main results are two characterisations of log-concave densities in terms of the collection of lift zonoids corresponding to a peacock. These notions are recalled and connected to arbitrage-free asset pricing in financial mathematics.
Improved sampling guarantees for underdamped Langevin Monte Carlo without restrictive assumptions.
problem Sampling from unnormalized densities with improved guarantees and acceleration.
method Novel analysis relaxing assumptions on log-Sobolev inequality and Hessian smoothness, using Rényi discretization bounds.
result First KL divergence guarantees for ULMC without Hessian smoothness under strong log-concavity.
This work studies the location estimation problem for a mixture of two rotation invariant log-concave densities. We demonstrate that Least Squares EM, a variant of the EM algorithm, converges to the true location parameter from a randomly initialized point. We establish the explicit convergence rates and sample complex…
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.
Algorithm samples from composite log-concave distributions using gradient evaluations and restricted Gaussian oracles.
problem Sampling from composite log-concave distributions with limited gradient evaluations.
method Proximal gradient algorithm with RGO for g g g and strong/strongly convex conditions for f f f . result Achieves ε ε ε error in total variation distance in O ~ ( κ d log 4 ( 1 / ε ) ) \widetilde{\mathcal O}(κ\sqrt d \log^4(1/ε)) O ( κ d log 4 ( 1/ ε )) iterations. Random scan CAVI converges linearly under log-concave assumptions.
problem Analyzing the convergence rate of random scan Coordinate Ascent Variational Inference (CAVI) under log-concave conditions.
method Building on previous work, we analyze the random scan version of CAVI using optimal transport geometry.
result We obtain tight linear convergence rates for the random scan version of CAVI.
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.
New algorithms improve convergence rates for non-log-concave sampling and log-partition estimation.
problem Efficiently sampling from non-log-concave distributions and estimating their log-partition function.
method Analysis of information-based complexity, study of polynomial-time sampling algorithms.
result Optimal rates for sampling and log-partition estimation sometimes exceed those for optimization.
New bounds for generative models under weaker assumptions.
problem Establishing convergence guarantees for generative models under weak assumptions.
method Non-asymptotic 2-Wasserstein distance bounds for probability flow ODEs under weak log-concavity and Lipschitz continuity.
result Concrete convergence rates for generative models, including non-log-concave distributions.
New algorithms sample from log concave distributions without gradient Lipschitz continuity.
problem Sampling from log concave distributions without gradient Lipschitz continuity.
method Two algorithms based on monotone polygonal (tamed) Euler schemes.
result Non-asymptotic 2-Wasserstein distance bounds between the process and target measure.
MALA mixes optimally in κ√d steps for log-concave sampling.
problem Sampling from log-concave distributions efficiently.
method Metropolis-Adjusted Langevin Algorithm (MALA) with warm start.
result Optimal minimax mixing time of κ√d iterations for log-concave distributions.
Paper proposes new Langevin samplers for sampling from log-concave distributions with superlinear gradient growth.
problem Sampling from log-concave distributions with superlinear gradient growth.
method Proposes two novel discretizations of kinetic Langevin SDEs, showing contractivity and log-Sobolev inequality.
result Establishes non-asymptotic bounds in 2-Wasserstein distance between sampled distributions and target measures.
Improves SGM convergence bounds in W2-distance without strict assumptions.
problem Convergence bounds for SGMs in W2-distance require stringent assumptions.
method Novel framework using the OU process and PDE analysis.
result Log-concavity evolves from weak to strong over time.
Improved log-concave sampling to O ( d 1 / 2 ) O(d^{1/2}) O ( d 1/2 ) with warm starts.
problem Sampling from strongly log-concave distributions efficiently.
method Warm starts and discretized underdamped Langevin diffusion.
result Achieved O ( d 1 / 2 ) O(d^{1/2}) O ( d 1/2 ) complexity for high-accuracy sampling. New algorithm samples neural network posteriors efficiently.
problem Challenges of sampling multimodal Bayesian posteriors for neural networks.
method Greedy Bayes method using log-concave coupling of posterior and auxiliary random variable.
result Log-concave coupling facilitates efficient sampling of neuron weights.
The paper develops methods for sampling from log-concave distributions with constraints.
problem Sampling from log-concave distributions with constraints.
method Randomized midpoint discretization of Langevin diffusions with various projections.
result New convergence guarantees for constrained Langevin algorithms.
Solves learning halfspaces with Massart noise for log-concave distributions.
problem Learning halfspaces with Massart noise in distribution-specific PAC model.
method Identifies a smooth non-convex surrogate loss and uses SGD to solve the learning problem.
result First computationally efficient algorithm for learning halfspaces with Massart noise for a broad family of distributions.
Improved convergence for non-log-concave sampling.
problem Sampling from non-log-concave distributions.
method Novel conductance analysis of SGLD with auxiliary Markov Chain.
result SGLD achieves ε-sampling error with fewer evaluations.