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.
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.
We construct a compact symplectic manifold with a Hamiltonian circle action for which the Duistermaat-Heckman function is not log-concave.
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 T4. In this article, for any closed symplectic four manifold N with b+ greater than 1, we show that there is a…
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.
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.
The paper extends localisation technique to multiple constraints in Euclidean spaces.
problem Proving log-concavity of conditional measures in decomposed convex sets.
method Defining partitions of maximal closed convex sets and proving log-concavity of conditional measures.
result Existence of a partition and log-concavity of conditional measures for almost every set of the partition.
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.
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.
Enhances SGLD for log-concave posteriors with asynchronous computation.
problem Sampling log-concave posterior distributions efficiently.
method Integrates asynchronous computation into SGLD with delayed gradients.
result Convergence in measure is not significantly affected by delayed gradient information.
Researchers found counterexamples to conjectures about optimal transport maps on curved spaces.
problem Extending Caffarelli's contraction theorem to curved spaces.
method Constructing counterexamples to precise conjectures.
result Found counterexamples to Milman's conjectures about optimal transport maps on curved spaces.
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.
Unified study of Brunn-Minkowski conjectures for log-concave measures.
problem Understanding the role of symmetry in inequalities of Brunn-Minkowski type.
method Unified framework, new results for conjectures, improved estimates for Lebesgue and Gaussian measures.
result Unified framework and new results for Brunn-Minkowski conjectures.
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.
New weighted surface area measures for convex bodies with applications.
problem Generalizing surface area measures to weighted Borel measures.
method Formulating and analyzing weighted surface area measures, proving integral formula and Bézout-type inequality.
result New integral formula for mixed measure of three bodies, generalizing Bézout-type inequality.
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.
We study the problem of sampling from a distribution $\target$ using the Langevin Monte Carlo algorithm and provide rate of convergences for this algorithm in terms of Wasserstein distance of order 2. Our result holds as long as the continuous diffusion process associated with the algorithm converges exponentially fa…
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.
This work extends stochastic localization to joint probability measures for data analysis.
problem Data distributional analysis in high-dimensional probability.
method Unified stochastic localization under Eldan's α-scheme, coupled probability measures via shared Brownian motion.
result Eldan's α-distance as a scalable surrogate for Wasserstein distance.
For a given 1-Lipschitz map u:Rn→Rm we define a partition, up to a set of Lebesgue measure zero, of Rn into maximal closed convex sets such that restriction of u is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave meas…
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.
The paper analyzes tensor recovery from symmetric rank-one measurements using information theory.
problem Recovering tensors with low symmetric rank from symmetric rank-one measurements.
method Covering numbers argument, Carbery-Wright inequality, orthogonal polynomials, Fano's inequality.
result Near-optimal sample complexity bounds for log-concave distributions.
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 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 studies randomized approximations of Tukey's depth for log-concave isotropic data.
problem The challenge of approximating Tukey's depth in high dimensions.
method The study examines randomized algorithms for approximating Tukey's depth for log-concave isotropic data.
result Randomized algorithms correctly approximate maximal depth and close to zero depths but not intermediate depths.
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. Stein's method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds are readily available for a diversity of univariate targets, few multivariate targ…
Discrete time analogues of ergodic stochastic differential equations (SDEs) are one of the most popular and flexible tools for sampling high-dimensional probability measures. Non-asymptotic analysis in the L2 Wasserstein distance of sampling algorithms based on Euler discretisations of SDEs has been recently develop…
This paper tackles denoising of complex measures using optimal transport and curvature analysis.
problem Denoising of complex, possibly non-log-concave measures.
method Score function and optimal transport theory to revert Langevin diffusion chains.
result The difficulty of denoising depends on the curvature complexity of the initial measure at specific SNR scales.
We discuss a certain Riemannian metric, related to the toric Kahler-Einstein equation, that is associated in a linearly-invariant manner with a given log-concave measure in R^n. We use this metric in order to bound the second derivatives of the solution to the toric Kahler-Einstein equation, and in order to obtain spec…
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.
Study improves sampling from complex distributions using annealed Langevin Monte Carlo.
problem Sampling from non-log-concave and multimodal distributions.
method Annealed Langevin Monte Carlo algorithm with theoretical guarantees.
result Oracle complexity of O(dβ²A²/ε⁶) for achieving ε² accuracy in Kullback-Leibler divergence.
The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, our method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian…
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 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.
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 tackles sampling from Gibbs measures with constrained support, providing a sampling guarantee.
problem Sampling from Gibbs measures with constrained support, especially in the pre-asymptotic regime.
method Analyzing the spectral gap of Langevin dynamics to provide a non-asymptotic sampling guarantee.
result The low-temperature Gibbs distribution concentrates on a neighborhood of its mode in the pre-asymptotic regime.
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.
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…
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.
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.
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.
Let M be a compact, connected symplectic 2n-dimensional manifold on which an(n-2)-dimensional torus T acts effectively and Hamiltonianly. Under the assumption that there is an effective complementary 2-torus acting on M with symplectic orbits, we show that the Duistermaat-Heckman measure of the T-action is log-concave.…
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.