Two new algorithms select matrix rows and columns to preserve distances.
problem Preserving distances in large matrix visualizations.
method Selects rows and columns to preserve distances.
result Preserves distances as closely as possible.
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48 results for “Distance Preservation”
Smooth maps preserve distances on specific revolution surfaces.
problem Existence of smooth maps on revolution surfaces.
method Proving existence of maps preserving distances on meridians and parallels.
result Smooth maps exist from revolution surfaces to Euclidean plane.
This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.
problem Hexagonal grid firing patterns in grid cells.
method Learning a distance-preserving position embedding in neural space using a recurrent neural network.
result The conformal isometric embedding of 2D physical space into neural space explains hexagonal grid firing patterns.
This study improves graph coarsening methods by preserving graph spectrum and distances.
problem Solving large-scale graph problems by working on a smaller graph.
method Developed a geometric approach using Gromov--Wasserstein distance to minimize the difference between graph distances and their coarsened versions.
result Minimizing the difference between graph distances and their coarsened versions can be achieved using the weighted kernel -means method.
The article explains Rao distances and conformal mappings for 3D objects.
problem Calculating distances and preserving angles in 3D objects.
method Proposed constructions of distances and angle-preserving mappings.
result Application to virtual tourism and line integrals in complex planes.
Any generalized distance-squared mapping of equidimensional case has singularities, and their singularity types are wrapped into mystery in higher dimensional cases. Any generalized distance-squared mapping of equidimensional case is not injective. Nevertheless, in this paper, it is shown that the non-singular property…
New curvature concept preserves graph distances under operations.
problem Preserving graph distances under graph operations.
method Characterization of distance matrix and its null space.
result Linear system may not have a solution.
Maps preserving Carathéodory distance between symmetric domains are rigid.
problem Rigidity of maps preserving Carathéodory distance between bounded symmetric domains.
method Large-scale geometry of Carathéodory distance, horocompactification, Gromov product.
result Maps preserving Carathéodory distance are rigid and either holomorphic or antiholomorphic.
New method preserves distances in time series data.
problem Preserving distances in time series data under interpolation.
method Developed lines-preserving terminal embeddings.
result First dimension-free coresets for Fréchet distance clustering.
We introduce an asymmetric distance function, which we call the `left Hausdorff distance function', on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity res…
Paper proposes dp-VAE for preserving spatial context in gene expression data.
problem Inaccessibility of spatial context in single-cell gene expression data.
method Generic representation learning and transfer learning framework with a distance-preserving regularizer.
result dp-VAE effectively reconstructs and imputes spatial context from gene expression data.
Can convolutional ResNets approximately preserve input distances? A frequency analysis perspectivecs.LG
ResNets can approximate input distances under certain conditions, but existing theory is flawed.
problem Theoretical justification for regularizing ResNets to preserve input distances is flawed.
method Frequency analysis perspective to explain effectiveness of regularization schemes.
result Regularization schemes enforce a lower Lipschitz bound on low-frequency projections of images.
Landmark-Based Node Representations for Shortest Path Distance Approximations in Random Graphsstat.ML
Landmark-based node embeddings approximate shortest path distances in random graphs.
problem Capturing global graph distances in node representations.
method Landmark-based node embeddings using shortest path distances from a subset of reference nodes (landmarks).
result Random graphs require lower dimensions in landmark-based embeddings compared to worst-case graphs.
Establishes a link between heat diffusion and manifold distances in data.
problem No theoretical link between diffusion-based manifold learning and geodesic distances.
method Formulates heat geodesic embeddings based on Riemannian geometry.
result Method outperforms state-of-the-art in preserving manifold distances and cluster structure.
Proposes Isometric Graph Neural Networks to preserve graph distances.
problem Lack of faithful distance representation in graph neural networks.
method Introduces a new technique to modify GNNs' input space and loss function.
result Significant improvement in reflecting graph distances, as measured by KT.
TTRP method preserves distances in high-dimensional data with reduced storage and speed.
problem Preserving distances in high-dimensional datasets efficiently and accurately.
method Tensor train random projection (TTRP) using TT-ranks of one.
result TTRP is an expected isometric projection with bounded variance.
CDOT optimizes transport between domains preserving both feature and geometric structure.
problem Optimizing transport between heterogeneous domains with preserved feature and geometric structure.
method CDOT uses operator-based regularization to align distance structures, proving pseudometric properties.
result CDOT improves robustness to local geometric variations and is provably convex.
Logit distance bounds representational similarity of models.
problem Approximating linear similarity when distributions are close.
method Defined a logit distance and proved its relationship to representational dissimilarity.
result Logit distance bounds representational similarity, providing nontrivial control in practice.
A fast binary embedding method preserves Euclidean distances in high-dimensional data.
problem Preserving Euclidean distances in high-dimensional datasets.
method Stable noise-shaping quantization of with a sparse Gaussian random matrix, followed by a linear transformation.
result Euclidean distances are approximated by the norm on binary sequences, leading to accurate binary codes.
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.
Proposes a new graph kernel framework using regularized Wasserstein distances.
problem Learning optimal transport distances for graph kernels.
method Introduces Regularized Wasserstein (RW) discrepancy with two regularization terms.
result Empirically validated method outperforms state-of-the-art methods.
In this work we study the properties of deep neural networks (DNN) with random weights. We formally prove that these networks perform a distance-preserving embedding of the data. Based on this we then draw conclusions on the size of the training data and the networks' structure. A longer version of this paper with more…
We present Graph Random Neural Features (GRNF), a novel embedding method from graph-structured data to real vectors based on a family of graph neural networks. The embedding naturally deals with graph isomorphism and preserves the metric structure of the graph domain, in probability. In addition to being an explicit em…
These lectures were a part of the geometry course held during the Fall 2011 Mathematics Advanced Study Semesters (MASS) Program at Penn State (\url{http://www.math.psu.edu/mass/}). The lectures are meant to be accessible to advanced undergraduate and early graduate students in mathematics. We have placed a great emphas…
Unified framework recovers exact input from SOM activation patterns.
problem Generating high-dimensional data from Self-Organizing Maps (SOMs).
method Inverting SOM activation patterns to recover input, using linear system and Tikhonov regularization.
result MUSIC framework produces coherent semantic transitions and maintains high classifier confidence.
We show that if the Hempel distance of a Heegaard splitting is larger than three then the mapping class group of the Heegaard splitting is isomorphic to a subgroup of the mapping class group of the ambient 3-manifold. This implies that given two handlebody sets in the curve complex for a surface that are distance at le…
In a recently published paper [1], it is shown that deep neural networks (DNNs) with random Gaussian weights preserve the metric structure of the data, with the property that the distance shrinks more when the angle between the two data points is smaller. We agree that the random projection setup considered in [1] pres…
This work investigates the properties of Gaussian-smoothed sliced divergences for comparing distributions.
problem Comparing probability distributions while preserving privacy.
method Investigates the theoretical properties of Gaussian-smoothed sliced Wasserstein distance and generalized versions.
result Gaussian smoothed sliced Wasserstein distance converges with a rate of \(O(n^{-1/2})\).
Model tracks structural changes in Brownian particle configurations on a sphere.
problem Tracking structural changes in Brownian particle configurations on a sphere.
method Introduces Frustrated Distance Matrix (FDM) model for dynamic distance matrices on S^2.
result Preserves static BBS template with dynamics as redistributed spectral mass.
Distance metric learning (DML) has been studied extensively in the past decades for its superior performance with distance-based algorithms. Most of the existing methods propose to learn a distance metric with pairwise or triplet constraints. However, the number of constraints is quadratic or even cubic in the number o…
dtSNE preserves local densities in low-dimensional embeddings.
problem Local density differences are not accurately preserved in tSNE and UMAP.
method dtSNE, which approximately conserves local densities.
result dtSNE provides more accurate local density depictions.
Quantitative Alexandrov theorem and asymptotic behavior of the volume preserving mean curvature flowmath.AP
New theorem shows shapes close to balls, flow converges to balls in 2D and 3D.
problem Understanding the asymptotic behavior of volume-preserving mean curvature flow.
method Proved a new quantitative Alexandrov theorem and used it to show flow convergence.
result Weak solutions of volume-preserving mean curvature flow converge to disjoint balls in R^2 and R^3.
In this article we study the induced geodesic distance of fractional order Sobolev metrics on the groups of (volume preserving) diffeomorphisms and symplectomorphisms. The interest in these geometries is fueled by the observation that they allow for a geometric interpretation for prominent partial differential equation…
Maps preserving mass and injective on boundary are isometries.
problem Stability of mass-preserving maps in integral current spaces.
method Proving rigidity of mass-preserving 1-Lipschitz maps.
result Maps preserving mass and injective on boundary are isometries.
Energy distance measures feature heterogeneity in federated learning.
problem Heterogeneity across data sources hinders model aggregation in federated learning.
method Introduced Taylor approximations of energy distance for efficient computation.
result Taylor approximations accurately capture feature discrepancies, improving convergence.
Null distance encodes causal structure in spacetimes.
problem Encoding causal structure in Lorentzian manifolds.
method Using null distance defined by Sormani and Vega, and proving causal structure is encoded by null distance.
result Lorentzian isometry between spacetimes with bijective map preserving null distance and cosmological time function.
Investigates projections onto explicit subspaces and their variance effects.
problem Understanding the variance preservation in explicit subspace projections.
method Investigates projections onto explicit subspaces of varying dimensionality and analyzes the variance effects.
result Developed new bounds for Euclidean distances and inner products.
The paper proves stability of manifolds with boundary under volume and distance constraints.
problem Stability of manifolds with boundary under volume and distance constraints.
method Volume preserving intrinsic flat convergence of metrics with boundary constraints.
result The stability of manifolds with boundary under volume and distance constraints is proven.
Mercat preserves angles to create accurate low-dimensional embeddings.
problem Reconstructing global relationships in low-dimensional embeddings.
method Reconstructing angles between data points to preserve both local and global structures.
result Mercat yields good reconstruction across various experiments and metrics.
Learning a distance function or metric on a given data manifold is of great importance in machine learning and pattern recognition. Many of the previous works first embed the manifold to Euclidean space and then learn the distance function. However, such a scheme might not faithfully preserve the distance function if t…
Training neural networks under a strict Lipschitz constraint is useful for provable adversarial robustness, generalization bounds, interpretable gradients, and Wasserstein distance estimation. By the composition property of Lipschitz functions, it suffices to ensure that each individual affine transformation or nonline…
Distance-based tests, also called "energy statistics", are leading methods for two-sample and independence tests from the statistics community. Kernel-based tests, developed from "kernel mean embeddings", are leading methods for two-sample and independence tests from the machine learning community. A fixed-point transf…
A method for learning embeddings from multi-view data using Gromov-Wasserstein.
problem Challenges in learning low-dimensional representations from multi-view relational data with differing geometries.
method Bary-GWMDS and Mean-GWMDS-C, Gromov-Wasserstein-based methods operating on distance matrices.
result Stable and geometrically meaningful embeddings learned from synthetic and real-world datasets.
Person re-identification (Re-ID) aims at matching images of the same person across disjoint camera views, which is a challenging problem in multimedia analysis, multimedia editing and content-based media retrieval communities. The major challenge lies in how to preserve similarity of the same person across video footag…
Polarimetric Synthetic Aperture Radar (PolSAR) images are establishing as an important source of information in remote sensing applications. The most complete format this type of imaging produces consists of complex-valued Hermitian matrices in every image coordinate and, as such, their visualization is challenging. Th…
The study proves properties of intersections of horospheres in harmonic spaces.
problem Properties of intersections of horospheres in harmonic spaces.
method Constructing volume preserving mappings using Busemann functions.
result Upper bound of the volume of intersection of horospheres is independent of Busemann function differences.
Expands newsvendor model with moment constraints using Wasserstein distance.
problem Optimizing order quantity under distributional ambiguity.
method Formulates infinite dimensional primal problem, derives finite dimensional dual problem using problem of moments duality.
result Distributional ambiguity affects optimal order quantity and profits/costs.
We give positive answers for questions by Berestovskii. Namely, we prove that every bijection of locally compact geodesically complete and connected at infinity CAT(0)-space onto itself preserving some fixed distance or satellite relations is an isometry of this space. The proof of this theorem is based on another …