Estimates manifold distances using graph Laplacian, proving consistency.
problem Estimating distances in compact Riemannian manifolds.
method Graph Laplacian estimates of the Laplace-Beltrami operator, bounding errors.
result Proof of consistency for manifold distances.
Centered plug-in estimators reduce bias in Wasserstein distance estimation.
problem Conservative bias in plug-in estimators of Wasserstein distances.
method Centering procedure based on linear combinations to reduce bias.
result Centered plug-in estimators provide informative upper and lower bounds on Wasserstein distances.
This work robustifies Wasserstein distance estimation with MoM estimators for outlier-polluted data.
problem Estimating Wasserstein distance between two distributions with outliers.
method Introducing MoM-based robust estimators for Wasserstein distance.
result Consistent MoM-based estimators for Wasserstein distance with convergence rates.
Private minimum Hellinger distance estimators maintain robustness and efficiency while ensuring privacy.
problem Ensuring privacy in robust statistical estimation.
method Derive private minimum Hellinger distance estimators satisfying Hellinger differential privacy.
result Private minimum Hellinger distance estimators retain robustness and efficiency under privacy constraints.
Paper proposes a method to estimate total variation distance for synthetic data fidelity.
problem Assessing the fidelity of synthetic data generated by AI.
method Discriminative approach to estimate total variation distance between two distributions.
result Estimation of total variation distance reduces to quantifying Bayes risk in classification.
We address noisy Euclidean distances in high dimensions, estimating noise levels and correcting distances.
problem Distorted pairwise Euclidean distances due to heteroskedastic noise.
method Developed a hyperparameter-free approach to jointly estimate noise magnitudes and correct distances.
result Our method provides accurate noise magnitude estimates and corrected distances in high-dimensional settings.
An important class of distance metrics proposed for training generative adversarial networks (GANs) is the integral probability metric (IPM), in which the neural net distance captures the practical GAN training via two neural networks. This paper investigates the minimax estimation problem of the neural net distance ba…
A method for fast estimation of Wasserstein distances using sliced Wasserstein distances.
problem Efficiently computing Wasserstein distances for multiple pairs of distributions.
method Regression on sliced Wasserstein distances to predict true Wasserstein distances.
result The proposed method provides a better approximation of Wasserstein distance than state-of-the-art models, especially in low-data regimes.
Wasserstein distances are increasingly used in a wide variety of applications in machine learning. Sliced Wasserstein distances form an important subclass which may be estimated efficiently through one-dimensional sorting operations. In this paper, we propose a new variant of sliced Wasserstein distance, study the use …
Robust test for distributions under Hellinger distance, simpler than optimal tests.
problem Testing and estimating distributions robustly under Hellinger distance.
method Simple robust hypothesis test with optimal sample complexity, robust to Hellinger distance perturbations.
result Empirically demonstrated robustness and power of the test on canonical distributions.
Although recovering an Euclidean distance matrix from noisy observations is a common problem in practice, how well this could be done remains largely unknown. To fill in this void, we study a simple distance matrix estimate based upon the so-called regularized kernel estimate. We show that such an estimate can be chara…
Study robust distribution estimation with Wasserstein distance, achieving optimal risk.
problem Robust distribution estimation under adversarial corruption.
method Combining partial OT and minimum distance estimation, proving structural properties and deriving a novel dual form.
result Achieves minimax-optimal robust estimation risk in many settings.
Differentially private data structures for estimating distances between strings.
problem Estimating distances between query strings and database strings while ensuring privacy.
method Proposes differentially private data structures for Hamming and edit distances using randomized response technique.
result Efficient data structures that provide accurate distance estimates with strong privacy guarantees.
New method estimates SW distance using CDFs for scalable data parallelism.
problem Estimating SW distance efficiently for large datasets.
method Estimators based on CDFs of projected measures, avoiding sorting.
result Efficient estimation for large datasets and federated learning.
New tools for estimating and inferring Wasserstein distance in topic models.
problem Estimating and inferring the Wasserstein distance between mixing measures in topic models.
method New canonical interpretation and tools for inference on Wasserstein distance in topic models.
result First minimax lower bounds and fully data-driven inferential tools for the Wasserstein distance in topic models.
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. Study on estimating distances between covariance operators and Gaussian processes.
problem Estimating distances between covariance operators and Gaussian processes.
method Riemannian distances, concentration results for Hilbert space-valued random variables, RKHS covariance and cross-covariance operators.
result Both distances converge in the Hilbert-Schmidt norm and can be consistently and efficiently estimated.
Paper tackles robust Euclidean distance estimation with sparse outliers.
problem Estimating point positions from corrupted distance measurements.
method Proposes a novel algorithm using Nyström method and robust PCA.
result Achieves accurate recovery with minimal anchors and sparse outliers.
Identifying statistical dependence between the features and the label is a fundamental problem in supervised learning. This paper presents a framework for estimating dependence between numerical features and a categorical label using generalized Gini distance, an energy distance in reproducing kernel Hilbert spaces (RK…
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.
The paper proves inequalities linking Wasserstein distances and eigenfunctions in RCD(K,∞) spaces.
problem Estimating Wasserstein distances and their bounds in RCD(K,∞) spaces.
method Similar techniques used to prove inequalities involving p-Wasserstein distances and Laplace eigenfunctions. result Proves a conjectured lower bound on p-Wasserstein distance between positive and negative parts of Laplace eigenfunctions. Optimally estimate distances on surfaces using reconstructed meshes.
problem Estimating intrinsic distances on smooth submanifolds.
method Reconstruction of the surface using a tangential Delaunay complex, and Isomap variant.
result Minimax optimality achieved for distance estimation.
Two algorithms estimate Wasserstein distance matrices from few entries for manifold learning.
problem Estimating Wasserstein distance matrices from limited data for manifold learning.
method Proposes two algorithms: matrix completion and Nyström completion for square Wasserstein matrices.
result Nyström completion can outperform matrix completion with a fixed sample budget and improve classification stability.
Error estimates found between SGD with momentum and Langevin diffusion.
problem Quantifying the difference between SGD with momentum and Langevin diffusion.
method Established error estimates using 1-Wasserstein and total variation distances.
result Quantitative error estimates between SGD with momentum and underdamped Langevin diffusion.
New method beats volumetric barrier for manifold recovery.
problem Reconstructing latent geometry from noisy distances.
method Orthogonal Ring Distance Estimation Routine (ORDER).
result Achieves pointwise distance estimation of order n−2/(d+5). Private density estimation in Wasserstein distance for geographic populations.
problem Private estimation of population density distributions.
method Differentially private algorithms for Wasserstein distance, instance-optimal.
result Uniformly achievable instance-optimal rates in both 1D and 2D.
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, i…
Diffusion-weighted MR imaging (DWI) is the only method we currently have to measure connections between different parts of the human brain in vivo. To elucidate the structure of these connections, algorithms for tracking bundles of axonal fibers through the subcortical white matter rely on local estimates of the fiber …
We provide a simple method and relevant theoretical analysis for efficiently estimating higher-order lp distances. While the analysis mainly focuses on l4, our methodology extends naturally to p = 6,8,10..., (i.e., when p is even). Distance-based methods are popular in machine learning. In large-scale applications, sto…
The paper explores selecting the parameter α for Fermat distance to balance geometry and noise.
problem Choosing the optimal parameter α for Fermat distance to navigate geometry and noise.
method Theoretical and simulation studies to determine the best α value.
result An optimal α value is identified to balance geometry and noise.
We generalize Mallows model to learn distance metrics from data.
problem Learning optimal distance metrics from noisy ranking data.
method Propose Lα distances and develop FPTAS for sampling and MLE. result Strong consistency of estimators for various α and β. Sharp estimates for Finsler metrics in convex domains.
problem Estimating distances in Finsler metrics near convex points.
method Sharp estimates for intrinsic distances of Finsler metrics.
result Characterization of k-quasi hyperbolic metric in convex geometry. Many statistical and machine learning approaches rely on pairwise distances between data points. The choice of distance metric has a fundamental impact on performance of these procedures, raising questions about how to appropriately calculate distances. When data points are real-valued vectors, by far the most common c…
Study on computing and estimating calibration distance, showing hardness and efficiency.
problem Computing and estimating calibration distance under different assumptions.
method Efficient algorithm for exact computation, polynomial-time approximation scheme; sample-based estimation for upper bounds.
result The problem becomes NP-hard when assumptions are removed, but efficient algorithms exist under certain conditions.
This work improves scalability of Wasserstein distances in high dimensions.
problem Scalability issues in computing Wasserstein distances in high dimensions.
method Empirical convergence rates, robustness to data contamination, and computational methods.
result Established fast rates and robust estimation risks for sliced Wasserstein distances.
Estimating entropy and mutual information consistently is important for many machine learning applications. The Kozachenko-Leonenko (KL) estimator (Kozachenko & Leonenko, 1987) is a widely used nonparametric estimator for the entropy of multivariate continuous random variables, as well as the basis of the mutual inform…
This paper provides performance guarantees for neural estimation of statistical distances.
problem Developing performance guarantees for neural estimation of statistical distances.
method Non-asymptotic error bounds using function approximation theorems and empirical process theory.
result Established a fundamental tradeoff between approximation and estimation errors in neural estimation of statistical distances.
Method recovers particle orientations from cryo-EM projections.
problem Unknown orientations in cryo-EM images.
method Two-step process: estimating distances and recovering orientations.
result Accurate orientation recovery from noisy projections.
We estimate the distance in the curve graph of a surface S of finite type using Teichmueller geodesics and assuming to be able to detect curves of distance at least three.
Transforms distance-based outlier scores into interpretable probabilistic estimates.
problem Difficult interpretation of distance-based outlier scores.
method Generic transformation of scores into probabilistic estimates using distance probability distributions.
result Probabilistic transformation improves interpretability without impacting detection performance.
Distance-based hierarchical clustering (HC) methods are widely used in unsupervised data analysis but few authors take account of uncertainty in the distance data. We incorporate a statistical model of the uncertainty through corruption or noise in the pairwise distances and investigate the problem of estimating the HC…
For localization and mapping of indoor environments through WiFi signals, locations are often represented as likelihoods of the received signal strength indicator. In this work we compare various measures of distance between such likelihoods in combination with different methods for estimation and representation. In pa…
New method estimates shape distance in neural representations with limited data.
problem Measuring geometric similarity between high-dimensional network representations.
method Method-of-moments estimator with tunable bias-variance tradeoff.
result New estimator achieves lower bias than standard methods in high-dimensional settings.
Robustly aligns datasets with partial GW distance to handle contamination.
problem Aligning contaminated datasets using Gromov-Wasserstein distances.
method Proposes a partial GW distance estimator to minimize distortion from outliers.
result The partial GW distance estimator is minimax optimal and near-optimal in finite samples.
In this note, we prove a uniform distance distortion estimate for Ricci flows with uniformly bounded scalar curvature, independent of the lower bound of the initial μ-entropy. Our basic principle tells that once correctly renormalized, the metric-measure quantities obey similar estimates as in the non-collapsing case…
Gromov-Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov-Hausdorff distance is equivalent to solving an NP-Hard optimization problem, deeming the notion impractical for applications. In this paper we pro…
Distance metric learning is an important component for many tasks, such as statistical classification and content-based image retrieval. Existing approaches for learning distance metrics from pairwise constraints typically suffer from two major problems. First, most algorithms only offer point estimation of the distanc…
The paper establishes distance estimates for manifolds with lower scalar curvature bounds.
problem Distance estimates on manifolds with lower scalar curvature bounds.
method Introduced a definition of relative index via a deformed Dirac operator trick and proved index coincidence with Callias operators.
result Proved short neck inequality and quantitative shielding result with positive scalar curvature.