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.
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.
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.
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 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.
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.
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.
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.
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.
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. 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.
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.
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.
Gibbs sampler mixes quickly for certain smooth distributions.
problem Drawing samples from log-smooth log-concave distributions.
method Analyzes Gibbs sampler on log-smooth and strongly log-concave distributions.
result Gibbs sampler mixes in O ⋆ ( κ 2 n 7.5 ) O^{\star}(κ^2 n^{7.5}) O ⋆ ( κ 2 n 7.5 ) steps. 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.
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.
ULA estimates covariance of log-concave distributions efficiently.
problem Estimating covariance matrices of log-concave distributions efficiently.
method Unadjusted Langevin algorithm (ULA) for sampling and covariance estimation.
result Sample complexity of single-chain ULA is smaller than that of parallel ULA by a logarithmic factor.
New algorithm samples from log-concave distributions with high accuracy in polynomial time.
problem Sampling from log-concave distributions with high accuracy in infinity distance.
method Directly converts continuous samples from K K K with total-variation bounds to samples with infinity bounds. result Output a point ε ε ε -close to π π π in infinity distance with runtime bounds that depend on polylogarithmic and polynomial factors of 1 / ε 1/ε 1/ ε . New sampling algorithm for non-log-concave distributions requires many queries.
problem Sampling from non-log-concave distributions with good accuracy.
method Lower bound on query complexity and algorithm for sampling.
result Tight query complexity characterization for sampling from non-log-concave distributions.
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.
We provide new results for noise-tolerant and sample-efficient learning algorithms under s s s -concave distributions. The new class of s s s -concave distributions is a broad and natural generalization of log-concavity, and includes many important additional distributions, e.g., the Pareto distribution and t t t -distribution.…
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.
The paper studies stability of mean-field variational inference for log-concave distributions.
problem Stability of mean-field variational inference for log-concave distributions.
method Novel approach via linearized optimal transport, lifting non-convex problem to convex optimization over transport maps.
result Dimension-free Lipschitz continuity of the MFVI optimizer with respect to the target distribution, measured in 2-Wasserstein distance.
New sampling algorithms for complex distributions without log-concavity.
problem Efficient sampling from complex, high-dimensional distributions.
method Randomized splitting Langevin Monte Carlo (RSLMC) algorithm.
result Uniform-in-time error bounds for RSLMC and RLMC algorithms.
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.
Improved Langevin algorithms with prior diffusion achieve dimension-independent convergence for non-log-concave distributions.
problem Understanding the dimension dependency of computational complexity in high-dimensional sampling.
method Investigation of prior diffusion technique for log-Sobolev inequality target distributions.
result Modified Langevin algorithm achieves dimension-independent KL divergence convergence.
SGLD proves geometric ergodicity via reflection coupling for nonconvex log-concave distributions.
problem Proving geometric ergodicity of SGLD in nonconvex, log-concave settings.
method Reflection coupling technique to handle SGLD's time discretization and minibatch issues.
result SGLD has an invariant distribution and geometric ergodicity in W 1 W_1 W 1 distance. 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.
A new sampling method reduces computational cost for high-dimensional log-concave distributions.
problem High computational cost of ULMC in high dimensions.
method Random Coordinate ULMC (RC-ULMC) selects a single coordinate per iteration.
result RC-ULMC is cheaper than classical ULMC, especially in highly skewed and high-dimensional problems.
Information concentration of probability measures have important implications in learning theory. Recently, it is discovered that the information content of a log-concave distribution concentrates around their differential entropy, albeit with an unpleasant dependence on the ambient dimension. In this work, we prove th…
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…
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.
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 . 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 . Study on Wasserstein distance for numerical approximations of stochastic differential equations.
problem Estimating the Wasserstein distance between stochastic differential equation distributions and their numerical approximations.
method Unified framework for analyzing different integrators and a novel splitting method for underdamped Langevin dynamics.
result A novel splitting method for underdamped Langevin dynamics with optimal complexity.
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. Paper tackles sampling from non-log-concave distributions using denoising diffusion.
problem Sampling from non-log-concave distributions efficiently.
method DDMC framework, Zeroth-Order Diffusion Monte Carlo (ZOD-MC) algorithm.
result ZOD-MC achieves inverse polynomial dependence on sampling accuracy, efficient for low dimensions.
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.
Non-convex SGD learns halfspaces with adversarial label noise efficiently.
problem Agnostically learning halfspaces in adversarial label noise settings.
method Non-convex SGD optimization for halfspace learning.
result Non-convex SGD achieves misclassification error close to optimal with adversarial noise.
Universal tester-learner for halfspaces over structured distributions.
problem Learning halfspaces over a wide class of structured distributions.
method Uses a fully polynomial tester-learner based on hypercontractivity and sum-of-squares (SOS) programs.
result Achieves error O ( o p t ) + ε O(\mathrm{opt}) + ε O ( opt ) + ε on any labeled distribution that the tester accepts. 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. A new sampling method, RC-LMC, reduces computational cost for high-dimensional log-concave distributions.
problem High computational cost of LMC in high dimensions.
method RC-LMC updates only one coordinate at a time, adding noise.
result RC-LMC is more efficient than LMC in high dimensions, especially for skewed distributions.
Strongly log-concave (SLC) distributions are a rich class of discrete probability distributions over subsets of some ground set. They are strictly more general than strongly Rayleigh (SR) distributions such as the well-known determinantal point process. While SR distributions offer elegant models of diversity, they lac…
New algorithm samples from log-concave distributions over polytopes efficiently.
problem Sampling from log-concave distributions over polytopes.
method Generalization of Dikin walk with soft-threshold regularizer.
result Improves sampling efficiency for polytopes.
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 privacy mechanism reduces error in query results.
problem Achieving privacy while minimizing noise in query results.
method Extended sufficient and necessary condition for ( ε , δ ) (ε, δ) ( ε , δ ) -differential privacy for symmetric and log-concave noise densities. result Significantly lower mean squared errors than Laplace and Gaussian mechanisms.
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.