Paper calculates Gromov-Hausdorff distance between simplexes and 2-distance spaces.
problem Calculating Gromov-Hausdorff distance between simplexes and 2-distance spaces.
method Formulas derived for clique covering number and chromatic number of graphs.
result Complete solution to generalized Borsuk problem for 2-distance spaces.
End-to-end algorithm for W-2 distance using neural networks.
problem Training optimal transport mappings for W-2 distance.
method Input convex neural networks and cycle-consistency regularization.
result Algorithm scales well to high dimensions without bias.
Algorithm classifies point clouds using deep set linearized optimal transport.
problem Classifying point clouds efficiently and accurately.
method Deep Set Linearized Optimal Transport, ICNNs, and a discriminator network.
result Efficiently distinguishes between various classes of point clouds.
The paper provides convergence guarantees for ODE-based generative models using transformers.
problem Theoretical guarantees for ODE-based generative models.
method A pre-trained autoencoder maps inputs to a latent space, and a transformer predicts the velocity field.
result The distribution of samples generated via estimated ODE flow converges to the target distribution in Wasserstein-2 distance.
To optimize a neural network one often thinks of optimizing its parameters, but it is ultimately a matter of optimizing the function that maps inputs to outputs. Since a change in the parameters might serve as a poor proxy for the change in the function, it is of some concern that primacy is given to parameters but tha…
This study analyzes the quadratic Wasserstein metric's effects on inverse data matching.
problem Analyzing the quadratic Wasserstein metric's impact on inverse data matching.
method Characterizes and numerically analyzes the smoothing effect and convexity improvement of W2 distance. result The W2 distance improves convexity and reduces resolution for reconstructed objects at a given noise level. SGHMC improves sampling and optimization under local conditions.
problem Nonconvex optimization and sampling under local conditions.
method Nonasymptotic analysis of SGHMC convergence.
result SGHMC provides high-precision results uniformly in iterations.
A simpler proof shows L2-metric completion is CAT(0).
problem Completing Riemannian metrics space.
method Easier proof of existing result by Brian Clarke.
result Completion of Riemannian metrics is CAT(0).
LOT embeds distributions for linear separability and classification.
problem Distribution discrimination in various scientific fields.
method Linear Optimal Transport (LOT) embedding into L2 space. result LOT embeds distributions into linearly separable spaces for certain transformations and perturbations.
Improved efficiency in HMC samplers reduces dissipative behavior.
problem Reducing dissipative behavior in HMC samplers.
method Variable integration time and partial velocity refreshment.
result Efficiency improved by a √κ factor in Wasserstein-2 distance.
KIPLMC methods improve statistical inference in latent variable models.
problem Statistical inference in latent variable models.
method Joint diffusion process in parameter and latent variable spaces, with two explicit discretizations.
result KIPLMC methods achieve accelerated convergence rates in Wasserstein-2 distance.
We study the problem of sampling from a probability distribution π on $\rset^d$ which has a density \wrt\ the Lebesgue measure known up to a normalization factor $x \mapsto \rme^{-U(x)} / \int_{\rset^d} \rme^{-U(y)} \rmd y$. We analyze a sampling method based on the Euler discretization of the Langevin stochastic dif…
Study finds new knot distances and chirally cosmetic bands using grid diagrams.
problem Computing distances and identifying chirally cosmetic bands between knots.
method Using grid diagrams to explore non-coherent band attachments between knots.
result Discovery of 33 new H(2)-distance one pairs for knots up to 8 crossings.
Efficiently reconstructs jump-diffusion processes from data using neural networks.
problem Reconstructing jump-diffusion processes from data.
method Temporally decoupled squared Wasserstein distance method using parameterized neural networks.
result Enhanced reconstruction of jump-diffusion processes from data.
We show in this note that the Sobolev Discrepancy introduced in Mroueh et al in the context of generative adversarial networks, is actually the weighted negative Sobolev norm ∣∣.∣∣H˙−1(νq), that is known to linearize the Wasserstein W2 distance and plays a fundamental role in the dynamic formulation of…
New algorithm tames non-linear growth in stochastic optimization.
problem Computational challenges in E-step of EM framework.
method Employing interacting particle systems and taming techniques to create tIPLA.
result Non-asymptotic convergence error estimates in Wasserstein-2 distance for tIPLA.
DDPMs are robust to noisy score estimates and achieve optimal convergence rates in Wasserstein-2 distance.
problem Evaluating the quality of DDPMs in Wasserstein distance with noisy score estimates.
method Established finite-sample guarantees in Wasserstein-2 distance for DDPMs, considering noisy score estimates.
result Optimal convergence rates in Wasserstein-2 distance for DDPMs, matching Gaussian case.
Corrects local error estimates for UBU integrator in SDEs, improving complexity guarantees.
problem Improper local error estimates in UBU integrator for SDEs.
method Reconciles theory with practice by correcting local error estimates.
result Stronger assumptions needed for O(d1/4ε−1/2) steps in Wasserstein-2 distance. Improved reliability of machine learning predictions using variational auto-encoders.
problem Individual unreliability of machine learning models.
method Modified variational auto-encoders to identify a low-dimensional space for reliable classification.
result Improved reliability of predictions and robust identification of adversarial samples.
New limits found for training deep learning models efficiently.
problem Optimizing the training speed of deep learning models without sacrificing accuracy.
method Applied stochastic thermodynamics to set speed limits for neural network training.
result Training neural networks is optimal within certain scaling assumptions.
Develops wavelet-based neural network approximation theory.
problem Analyzing neural network approximation capabilities over various activation functions.
method Wavelet frame theory on spaces of homogeneous type, sufficient conditions for approximation, error estimates.
result Derives sufficient conditions for neural networks to approximate any functions in a given space, including non-smooth activations.
A new tamed stochastic gradient Hamiltonian Monte Carlo algorithm for superlinearly growing stochastic gradients.
problem Sampling and stochastic optimization problems with superlinearly growing stochastic gradients.
method Tamed Stochastic Gradient Hamiltonian Monte Carlo (tSGHMC) algorithm.
result Established a non-asymptotic error bound in Wasserstein-2 distance with a convergence rate of 1/4. Improved outlier detection in hierarchical Gaussian Processes using Wasserstein-2 kernels.
problem Outlier detection limitations in stacked Gaussian Processes.
method Proposed a hybrid kernel combining Euclidean and Wasserstein-2 distances, emphasizing variance in Wasserstein-2 computations.
result Improved performance and enhanced out-of-distribution detection on various datasets.
Proves stability of cone-volume measure with nearly constant density.
problem Stability of cone-volume measure with near constant density.
method Proves stability of cone-volume measure with near constant density.
result Homothetic copy of the body is close to the unit ball in the L2-distance. The Johnson-Lindenstrauss Lemma allows for the projection of n points in p−dimensional Euclidean space onto a k−dimensional Euclidean space, with k≥3ε2−2ε324lnn, so that the pairwise distances are preserved within a factor of 1±ε. Here, working directly with the distributions of the …
Imagine that measurements are made at times t0 and t1 of the trajectory of a physical system whose governing laws are given approximately by a class A of so-called {\em prior vector fields}. Because the physical laws are not known precisely, it might be that the measurements are not realised by the integ…
We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its C2-distance fro…
Generative AutoEncoders require a chosen probability distribution in latent space, usually multivariate Gaussian. The original Variational AutoEncoder (VAE) uses randomness in encoder - causing problematic distortion, and overlaps in latent space for distinct inputs. It turned out unnecessary: we can instead use determ…
New analysis for learning and applying preconditioners in MCMC improves efficiency.
problem Improving efficiency of MCMC algorithms by modifying them with preconditioners.
method Analyzes and compares computational costs of MCMC schemes with and without preconditioners.
result Establishes non-asymptotic guarantees for MCMC algorithms that learn and use preconditioners.
The study provides guarantees for diffusion-based models under log-concave data, offering best-known convergence rates.
problem Theoretical guarantees for convergence of diffusion-based generative models under log-concave data distributions.
method Assumption of strongly log-concave data distributions, Lipschitz continuous functions for score estimation, and novel auxiliary process.
result Best known upper bounds for Wasserstein-2 distance between Gaussian distribution and sampling algorithm.
CNFs learn distributions from samples with error bounds.
problem Learning probability distributions from finite samples.
method Continuous normalizing flows with linear interpolation and flow matching objective function.
result Non-asymptotic error bounds for distribution estimator in Wasserstein-2 distance.
Proposes variance reduction techniques for sliced Wasserstein distance estimation.
problem Intractability of estimating sliced Wasserstein distances.
method Uses control variates based on Gaussian approximations of projected measures.
result Significant reduction in variance of SW distance estimators.
Alternative proof of weak solutions to mean curvature flow using minimizing movements.
problem Existence of weak solutions to mean curvature flow and volume preserving mean curvature flow.
method Proposes a new existence proof using a minimizing movements scheme and a novel proxy for distance.
result Unconditional convergence towards a De Giorgi solution for the minimizing movements scheme.
In this paper we develop a new perspective on generalization of neural networks by proposing and investigating the concept of a neural network stiffness. We measure how stiff a network is by looking at how a small gradient step in the network's parameters on one example affects the loss on another example. Higher stiff…
The paper provides bounds for regression schemes using nonstationary training samples.
problem Developing confidence intervals for nonparametric regression with nonstationary data.
method The approach involves Rademacher and Vapnik-Chervonenkis theories to analyze the cost and optimality of regression schemes.
result The paper establishes nonasymptotic bounds for regression schemes and optimality in L2-distance. New method improves sampling efficiency in complex stochastic systems.
problem Sampling efficiency in nonconvex stochastic gradient cases.
method Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo.
result Quantitative Gaussian concentration bounds and convergence rates established.
Generative Distribution Embeddings learn multiscale representations of distributions.
problem Learning representations of entire distributions for multiscale reasoning.
method Introducing GDE framework that lifts autoencoders to the space of distributions, using conditional generative models and distributional invariance.
result GDEs learn predictive sufficient statistics embedded in Wasserstein space, recovering distances and trajectories for Gaussian and Gaussian mixture distributions.
This research converts visual information into audio for users to perceive.
problem Brevity in conveying visual information through spoken language.
method Pretrained image embedding network, GAN for metric space mapping, human subject testing.
result Users can accurately classify audio sonifications of faces.
New method learns discrete graph diffusion via free-energy gradient flows.
problem Challenges in translating continuous diffusion models to discrete spaces.
method Proposes a novel computational approach using a specific metric on the simplex.
result Recover the underlying functional for various graph classes.
Study shows optimal rates for independence testing via U-statistic permutation tests.
problem Developing a valid test of independence for pairs with additional smoothness constraints.
method Defining a measure of dependence, using a permutation test based on a basis expansion and U-statistic estimator.
result Proves minimax optimality of the test in separation rates for certain cases.
Study nondegenerate singularities in mean curvature flow.
problem Understanding the behavior of nondegenerate cylindrical singularities.
method New L2-distance monotonicity formula and discrete almost monotonicity. result Topology change agrees with level sets change near a critical point of a Morse function.
The paper proves stability of eigenvalue inequalities on surfaces.
problem Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces.
method Employing eigenvalues of measures and Sobolev space W−1,2, the paper proves stability estimates for the first and second nonzero Laplace eigenvalues on surfaces. result Metrics almost maximizing the normalized eigenvalue are W−1,2-close to a maximal metric. New sampling method on Lie groups converges quickly.
problem Sampling on non-Euclidean Lie groups.
method Kinetic Langevin dynamics with noise added.
result Exponential convergence rate proved under W2 distance. The paper develops manifold learning in Wasserstein space for probability measures.
problem Learning latent manifold structure in Wasserstein space of probability measures.
method Introduces submanifolds in Wasserstein space, learns latent structure from samples and distances, recovers tangent spaces via spectral analysis.
result The latent manifold structure can be learned from samples and pairwise extrinsic Wasserstein distances.
Many interesting machine learning problems are best posed by considering instances that are distributions, or sample sets drawn from distributions. Previous work devoted to machine learning tasks with distributional inputs has done so through pairwise kernel evaluations between pdfs (or sample sets). While such an appr…
The paper extends statistical estimation techniques under differential privacy.
problem Establishing sample complexity bounds for estimation tasks under differential privacy.
method Proposes analogues of Le Cam's method, Fano's inequality, and Assouad's lemma under central differential privacy.
result Optimal sample complexity bounds for discrete distribution estimation under total variation and ℓ2 distances. Why does training deep neural networks using stochastic gradient descent (SGD) result in a generalization error that does not worsen with the number of parameters in the network? To answer this question, we advocate a notion of effective model capacity that is dependent on {\em a given random initialization of the netw…
New algorithm samples superlinearly growing log-gradient distributions.
problem Sampling from distributions with superlinearly growing log-gradient.
method Proposes a novel taming Langevin-based scheme called sTULA.
result Derives non-asymptotic convergence bounds in KL, TV, and W2 distances.