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.
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.
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.
Study on limits of recovering sparse variables from phaseless measurements.
problem Support recovery in phase retrieval model with noisy phaseless measurements.
method Information-theoretic analysis, considering discrete and Gaussian models, Gaussian measurement matrices.
result Sharp thresholds with near-matching constant factors for sparsity and signal-to-noise ratio in various scaling regimes.
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.
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 . 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.
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.
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.
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.
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.
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.
Improved spectral gap for MwG with adaptive RWM proposals.
problem Improving mixing efficiency of MwG for log-concave distributions.
method Using adaptive RWM proposals tuned to match conditional variances of log-concave target distributions.
result Established a spectral gap lower bound of order O ( 1 / κ d ) \mathcal{O}(1/κd) O ( 1/ κ d ) for MwG. We propose a novel and flexible rank-breaking-then-composite-marginal-likelihood (RBCML) framework for learning random utility models (RUMs), which include the Plackett-Luce model. We characterize conditions for the objective function of RBCML to be strictly log-concave by proving that strict log-concavity is preserved…
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.
Graphical models for structured domains are powerful tools, but the computational complexities of combinatorial prediction spaces can force restrictions on models, or require approximate inference in order to be tractable. Instead of working in a combinatorial space, we use hinge-loss Markov random fields (HL-MRFs), an…
New sampling method improves efficiency for diffusion models.
problem Efficient sampling from arbitrary smooth distributions in polynomial time.
method Randomized midpoint method for log-concave sampling.
result Achieves best known dimension dependence ( O ~ ( d 5 / 12 ) \widetilde O(d^{5/12}) O ( d 5/12 ) ) for total variation distance. 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.
Enhances LMC for log-concave sampling, reducing computational cost.
problem High computational cost of LMC for high-dimensional problems.
method Random coordinate descent (RCD) combined with variance reduction techniques (SAGA, SVRG).
result Achieves computational cost reduction compared to classical LMC, same number of iterations as LMC.
The unadjusted Langevin algorithm converges faster for some variables in high dimensions.
problem Sampling probability distributions in high-dimensional settings.
method Analysis of the unadjusted Langevin algorithm for strongly log-concave distributions.
result The delocalization of bias effect allows for faster convergence for a small number of variables.
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.
New algorithm speeds up sampling from log-concave distributions.
problem Sampling from log-concave distributions efficiently.
method Markov chain Monte Carlo (MCMC) based on underdamped Langevin diffusion (ULD).
result Significantly faster than previous methods, achieving ε·D error in O(κ^7/6/ε^1/3 + κ/ε^2/3) steps.
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.
Poisson Midpoint Method improves Langevin Dynamics for diffusion models.
problem Slow convergence of LMC in diffusion models requiring many small steps.
method Poisson Midpoint Method approximates LMC with larger steps, proving quadratic speed up.
result Poisson Midpoint Method maintains quality of DDPM with fewer calls.
New method reduces variance in random coordinate descent for Langevin Monte Carlo.
problem Efficient sampling from log-concave distributions in high dimensions.
method Introduces RCAD, a variance reduction technique for RCD-LMC.
result RCAD-O-LMC and RCAD-U-LMC converge within the same number of iterations as classical LMC methods, saving computational cost.
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 algorithm speeds up sampling from log-concave distributions over polytopes.
problem Sampling from log-concave distributions constrained to polytopes efficiently.
method Improved Markov chain with efficient linear solvers and randomized estimators.
result Per-step complexity is nearly optimal, with reduced arithmetic operations.
This paper shows how to approximate any log-concave distribution using well-conditioned affine coupling flows.
problem Understanding the representational power of affine coupling flows for log-concave distributions.
method Leveraging connections between affine coupling architectures, Langevin dynamics, and Hénon maps to prove log-concave approximation.
result Any log-concave distribution can be approximated using well-conditioned affine-coupling flows.
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.
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.
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.
New algorithm speeds up sampling from complex Bayesian mixture models.
problem Sampling from non-log-concave, multi-modal posterior distributions in Bayesian Gaussian mixtures.
method Introduced Reflected Metropolis-Hastings Random Walk (RMRW) algorithm.
result Proved mixing time bound for RMRW in symmetric two-component Gaussian mixtures.
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.
Deep models can't generate heavy-tailed samples well.
problem Understanding the limitations of deep generative models in generating samples with heavy tails.
method Unified framework using concentration of measure and convex geometry, Gromov-Levy inequality.
result Deep generative models are not universal generators and can only produce concentrated samples with light tails.
We propose a computationally efficient random walk on a convex body which rapidly mixes and closely tracks a time-varying log-concave distribution. We develop general theoretical guarantees on the required number of steps; this number can be calculated on the fly according to the distance from and the shape of the next…
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.
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…
Least Squares EM converges globally for log-concave mixtures.
problem Location estimation in mixtures of two log-concave densities.
method Least Squares EM algorithm applied to log-concave mixtures.
result Least Squares EM converges globally to the true location parameter.
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.
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.
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.
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.
We construct a compact symplectic manifold with a Hamiltonian circle action for which the Duistermaat-Heckman function is not log-concave.
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.