Note on the computational complexity of Gromov-Wasserstein distance.
problem Computational difficulty of Gromov-Wasserstein distance.
method Analysis of the optimization problem structure and providing explicit examples.
result Gromov-Wasserstein distance optimization problem is non-convex quadratic.
A new supervised tree-Wasserstein distance improves document classification.
problem Measuring document similarity efficiently and accurately.
method Rewriting Wasserstein distance on tree metric, using contrastive loss for optimization.
result The Supervised Tree-Wasserstein (STW) distance improves document classification accuracy.
New slicing methods speed up Gaussian mixture Wasserstein distance computations.
problem High computational cost of the mixture Wasserstein distance.
method Slicing-based approximations to reduce computational complexity.
result Significant reduction in computational complexity while preserving key properties.
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.
New formula and algorithm for computing distances on complex Riemann surfaces.
problem Computing distances on higher-genus Riemann surfaces is challenging due to infinite terms in the formula.
method Derived a computable distance formula and developed an efficient algorithm.
result Reduced distance computation from an infimum to a minimum over a finite set of terms.
Matrix profile has been recently proposed as a promising technique to the problem of all-pairs-similarity search on time series. Efficient algorithms have been proposed for computing it, e.g., STAMP, STOMP and SCRIMP++. All these algorithms use the z-normalized Euclidean distance to measure the distance between subsequ…
Researchers compute the full spectrum of Laplace operator on distance spheres in symmetric spaces.
problem Computing the full Laplace spectrum on distance spheres in symmetric spaces.
method Lie-theoretic methods to explicitly compute the spectrum.
result Unified formula for the full spectrum of Laplace operator on distance spheres in symmetric spaces of rank one.
Optimal transportation distances are a fundamental family of parameterized distances for histograms. Despite their appealing theoretical properties, excellent performance in retrieval tasks and intuitive formulation, their computation involves the resolution of a linear program whose cost is prohibitive whenever the hi…
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.
We investigate the use of Minimax distances to extract in a nonparametric way the features that capture the unknown underlying patterns and structures in the data. We develop a general-purpose and computationally efficient framework to employ Minimax distances with many machine learning methods that perform on numerica…
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…
New metric learning approach for tree data reduces computation cost.
problem Efficiently computing distances between ordered labeled trees.
method Introduced pq-grams and a differentiable weighted pq-gram distance, combined with LMNN for optimization.
result Significantly reduces computation time for tree classification problems.
Bounds on geodesic distances on Stiefel manifold derived from new metrics.
problem Improving geodesic computation algorithms and understanding Stiefel manifold.
method New geometric insights and Lipschitz constants for geodesic distances.
result Explicit bounds on geodesic distances and conditions for attaining bounds.
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.
The area distance to a convex plane curve is an important concept in computer vision. In this paper we describe a strong link between area distances and improper affine spheres. This link makes possible a better understanding of both theories. The concepts of the theory of affine spheres lead to a new definition of an …
Improved computational efficiency for estimating Wasserstein distance.
problem Inefficient computation of Wasserstein distance for large samples.
method Developed Sample-Sketch-Solve paradigm using grid sketches.
result Approximates Wasserstein distance within ε error in ε^(-max(2, (d+1+o(1))/(1+α))) time.
A new algorithm computes elastic shape distances between curves efficiently.
problem Computing elastic shape distances between curves in high dimensions.
method Dynamic Programming for optimal diffeomorphisms and Kabsch-Umeyama algorithm for optimal rotation matrices.
result Efficient computation of elastic shape distances with improved efficiency for closed curves.
Tomova, along with results of Bachman and Schleimer, showed that any high distance knot has a stair-step bridge spectrum. In this paper, we compute the bridge spectra and distance of generalized Montesinos knots. In particular, we produce the first example of a class of knots which attain the stair-step bridge spectra …
This paper approximates 1-Wasserstein distance using tree-based embedding.
problem Computational inefficiency of estimating 1-Wasserstein distance.
method L1-regularized approach to learn tree weights, using shortest path distance as a linear model.
result Tree-Wasserstein distance (TWD) approximates 1-Wasserstein distance efficiently.
The medoid of a set of n points is the point in the set that minimizes the sum of distances to other points. It can be determined exactly in O(n^2) time by computing the distances between all pairs of points. Previous works show that one can significantly reduce the number of distance computations needed by adaptively …
Large scale agglomerative clustering is hindered by computational burdens. We propose a novel scheme where exact inter-instance distance calculation is replaced by the Hamming distance between Kernelized Locality-Sensitive Hashing (KLSH) hashed values. This results in a method that drastically decreases computation tim…
The paper uses distance covariance to improve fairness in machine learning models.
problem Improving fairness in machine learning models.
method Using conditional and distance covariance statistics to assess independence and add a penalty for fairness.
result The method effectively reduces the fairness gap in machine learning models.
Introduces new Wasserstein distances for more intrinsic metrics.
problem Improve metric for comparing distributions.
method Introduces RWp distances, designs algorithms for computation. result New distances are more intrinsic and computable.
This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, …
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.
Unified framework for hyperbolic embeddings from mixed data types.
problem Computing hyperbolic embeddings from noisy metric and non-metric data.
method Semidefinite programming and spectral factorization methods.
result Efficient computation of hyperbolic embeddings from arbitrary data.
Paper connects surface shape analysis and unbalanced optimal transport.
problem Computing the SRNF shape distance on piecewise linear surfaces.
method Characterizes SRNF shape distance as WFR distance pullback, proposes new algorithm for WFR distance computation.
result Direct computation of SRNF shape distance on piecewise linear surfaces.
A new Wasserstein distance method for comparing incomparable distributions.
problem Comparing distributions that are not supported on the same metric space.
method Distributional slicing, embeddings, and closed-form computation of Wasserstein distance.
result HWD preserves properties like rotation-invariance and can be efficiently learned.
New method improves Wasserstein distance for large-scale data.
problem High computational cost of Wasserstein distance for large-scale machine learning.
method Augmented Sliced Wasserstein Distances (ASWDs) using neural network mappings.
result ASWDs significantly outperform other Wasserstein variants in synthetic and real-world problems.
Paper presents efficient computation of robust Wasserstein distance using Riemannian optimization.
problem Intractability of optimizing Projection Robust Wasserstein (PRW) distance due to non-convexity and non-smoothness.
method Riemannian optimization to efficiently compute PRW/Wasserstein Projection Pursuit (WPP) distance.
result The original formulation of PRW/WPP can be efficiently computed in practice, providing better behavior than its convex relaxation.
LOT Wassmap speeds up Wasserstein space manifold learning.
problem Finding low-dimensional structures in Wasserstein space datasets.
method Linearized optimal transport and approximation schemes.
result LOT Wassmap provides accurate embeddings with computational efficiency.
Study infinite Euclidean distance discriminants of algebraic varieties.
problem Understanding the structure of data points with infinitely many critical points in Euclidean distance correspondence.
method Developed computer code to compute discriminants and proved properties of fibers.
result Infinite Euclidean distance discriminants contain all data points with infinitely many critical points for the nearest-point problem.
Ball k-means reduces point-centroid distance computations for faster k-means clustering.
problem Efficiently finding k-means clusters in large datasets.
method Uses a ball to describe clusters, dividing them into stable and active areas, and adjusting points within annulus areas.
result Significantly reduces point-centroid distance computations, making k-means faster and more efficient.
There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into ac…
Researchers developed a differentially private method for computing Wasserstein distances.
problem Computing divergences between distributions while preserving privacy.
method They focused on the Sliced Wasserstein Distance and added Gaussian perturbations to make it differentially private.
result They introduced a new differentially private distance, the Smoothed Sliced Wasserstein Distance, which performs well in generative models and domain adaptation.
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.
GT is a new method for denoising and enhancing datasets using Gaussian density estimates.
problem Improving latent structures in datasets.
method GT is an iterative method that generates a new distance function by computing the ℓ2-Wasserstein distance between Gaussian density estimates. result GT is stable under perturbations and asymptotically ellipsoidal neighborhoods in the continuous case.
A new distance measure balances projection exploration and informativeness.
problem Inefficient and incomplete projection sampling in existing sliced-Wasserstein distances.
method Proposes Distributional Sliced-Wasserstein (DSW) that optimally balances projection exploration and informativeness.
result DSW generalizes Max-SW and can be computed efficiently.
New distances for comparing multivariate normal distributions.
problem Comparing multivariate normal distributions efficiently and accurately.
method Approximated Fisher-Rao distance and pullback SPD cone distances.
result Efficient computation of distances between normal distributions.
In this work, we present a method to compute the Kantorovich-Wasserstein distance of order one between a pair of two-dimensional histograms. Recent works in Computer Vision and Machine Learning have shown the benefits of measuring Wasserstein distances of order one between histograms with n bins, by solving a classic…
The Sinkhorn "distance", a variant of the Wasserstein distance with entropic regularization, is an increasingly popular tool in machine learning and statistical inference. However, the time and memory requirements of standard algorithms for computing this distance grow quadratically with the size of the data, making th…
Optimal transport (\OT) theory defines a powerful set of tools to compare probability distributions. \OT~suffers however from a few drawbacks, computational and statistical, which have encouraged the proposal of several regularized variants of OT in the recent literature, one of the most notable being the \textit{slice…
A scalable version of MADD improves big-data classification speed.
problem High computational complexity of MADD in big data.
method Selecting a representative set and using Random Fourier Features.
result Achieves similar performance to MADD but at a fraction of the computing time.
A new sliced IGW distance for Gromov-Wasserstein alignment.
problem Scalability issues in Gromov-Wasserstein alignment for high-dimensional problems.
method Proposed a sliced IGW distance with rotational invariance.
result Natural rotational invariance of the sliced IGW distance.
Computing optimal transport distances such as the earth mover's distance is a fundamental problem in machine learning, statistics, and computer vision. Despite the recent introduction of several algorithms with good empirical performance, it is unknown whether general optimal transport distances can be approximated in …
A new distance metric compares probability distributions using kernel covariance operators.
problem Comparing probability distributions in machine learning tasks.
method Introduces a novel distance metric based on Schatten norm of kernel covariance operators.
result The new distance metric is more discriminative and robust to hyperparameters.
Physics: Similar long-distance properties can mask vastly different short-distance metrics.
problem Classifying homogeneous metrics on group manifolds by long-distance properties.
method Apply universality concept to geometry, focusing on metrics on Lie groups.
result Many metrics on low-dimensional Lie groups have similar long-distance properties despite differing short-distance properties.
A new method efficiently approximates Gromov-Wasserstein distance.
problem High computational complexity of Gromov-Wasserstein distance.
method Importance sparsification method to construct a sparse coupling matrix.
result Efficient approximation of GW distance with reduced complexity.