Regular neighborhoods of singular submanifolds are isotopic to bundle morphisms.
problem Isotoping regular neighborhoods of singular submanifolds to bundle morphisms.
method Leaf preserving isotopy and homogeneity assumptions on foliations.
result Every leaf preserving diffeomorphism of a regular neighborhood is isotopic to a bundle morphism.
Proposes TTNPE for tensor data embedding with improved trade-offs.
problem Embedding multi-dimensional tensor data into low dimensions.
method Tensor Train Neighborhood Preserving Embedding (TTNPE) with novel optimization approaches.
result Improves classification, computation, and dimensionality reduction trade-offs.
Classifies stable hypersurfaces in real projective spaces and confirms the isoperimetric conjecture.
problem Volume preserving stability and isoperimetric problem in real projective spaces.
method Classification of stable hypersurfaces and analysis of antipodal invariant hypersurfaces.
result Solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces.
The paper studies curvature conditions on manifolds with boundary.
problem Curvature preservation on manifolds with smooth boundaries.
method Constructing a family of metrics that agree with given metrics on the boundary and interior.
result Deforming metrics to ones with totally geodesic boundary while preserving curvature conditions.
Urban2Vec combines street view imagery and POIs for better urban neighborhood embeddings.
problem Lack of comprehensive representation of urban neighborhoods using heterogeneous data.
method Unsupervised multi-modal framework using CNN for visual features and bag-of-words for POI data.
result Urban2Vec achieves better performance than baseline models and comparable to fully-supervised methods.
LPL optimizes embeddings to align local neighborhoods, improving cross-lingual word alignment.
problem Aligning embeddings across different datasets and languages.
method Locality Preserving Loss (LPL) optimizes model to project embeddings while maintaining local neighborhoods and aligning them.
result LPL-based alignment leads to better and consistent accuracy, especially in small training set settings.
Linearizability of singular foliations is preserved under a specific equivalence relation.
problem Preserving properties of singular foliations under equivalence relations.
method Characterization of tubular neighborhood embeddings using Euler-like vector fields.
result Linearizability along a leaf is a Morita invariant.
In this paper, we develop Leray-Serre-type spectral sequences to compute the intersection homology of the regular neighborhood and deleted regular neighborhood of the bottom stratum of a stratified PL-pseudomanifold. The E^2 terms of the spectral sequences are given by the homology of the bottom stratum with a local co…
The local linear embedding algorithm (LLE) is a non-linear dimension-reducing technique, widely used due to its computational simplicity and intuitive approach. LLE first linearly reconstructs each input point from its nearest neighbors and then preserves these neighborhood relations in the low-dimensional embedding. W…
Prediction tasks over nodes and edges in networks require careful effort in engineering features used by learning algorithms. Recent research in the broader field of representation learning has led to significant progress in automating prediction by learning the features themselves. However, present feature learning ap…
Combines OT and PCA for DR, preserving clusters.
problem Analyzing high-dimensional data with global dependencies.
method Optimal transport (OT) for minimizing reconstruction error, combined with PCA.
result Effective preservation of high-dimensional clusters in embeddings.
Bagging reduces variance in LID estimation by preserving local distribution of NN distances.
problem High estimation variance from limited data in small neighborhoods.
method Subbagging to preserve local distribution of NN distances, combined with ensemble size.
result Bagging significantly reduces variance and MSE in LID estimation.
A scalable framework preserves personalized higher-order network proximities.
problem Lack of expressive methods to preserve personalized higher-order network proximities.
method Incorporates random walk into a sound objective to preserve arbitrary higher-order proximities and introduces random walk with restart for personalized-weighted preservation.
result Consistently and substantially outperforms state-of-the-art methods on real-world networks.
We establish that over a C^{2,1} manifold the exponential map of any Lipschitz connection or spray determines a local Lipeomophism and that, furthermore, reversible convex normal neighborhoods do exist. To that end we use the method of Picard-Lindelof approximation to prove the strong differentiability of the exponenti…
If π:M→B is a Riemannian Submersion and M has positive sectional curvature, O'Neill's Horizontal Curvature Equation shows that B must also have positive curvature. We show there are Riemannian submersions from compact manifolds with positive Ricci curvature to manifolds that have small neighborhoods of…
Caratheodory's axiom limits arbitrage in resource-limited systems.
problem Non-arbitrage constraints in resource-limited financial systems.
method Preserving Caratheodory's axiom in resource-limited systems.
result Exponential family is the necessary geometric structure for both thermodynamics and finance.
We propose a new notion called \emph{infinity-harmonic maps}between Riemannain manifolds. These are natural generalizations of the well known notion of infinity harmonic functions and are also the limiting case of p% -harmonic maps as p→∞. Infinity harmoncity appears in many familiar contexts. For example,…
This research improves classification performance by learning a distance metric from balanced data.
problem Data imbalance in learning methods.
method Extracts a low-dimensional manifold, learns local neighborhood relationships, and optimizes distance metric.
result The proposed method outperforms other approaches, especially in imbalanced datasets.
Spectral graph sparsification preserves geometry of GNN embeddings.
problem Maintaining geometric properties of graph neural network embeddings during sparsification.
method Proving spectral sparsification preserves squared pairwise distances, class means, and covariance structure in embedding space.
result Spectral sparsification preserves the geometry of learned embeddings in GNNs.
SDSPCAAN combines supervised and local data structures for better dimensionality reduction.
problem Preserving both global and local data structures for noisy high-dimensional data.
method Supervised discriminative sparse PCA with adaptive neighbors (SDSPCAAN).
result SDSPCAAN improves classification accuracy on high-dimensional datasets.
Local Lorentzian theorem preserves metrics or makes them flat.
problem Analyzing conformal vector fields on Lorentzian manifolds.
method Proves local isometry or conformal flatness using global arguments.
result Optimal improvement of conformal vector field normal forms.
WSFN overcomes saddle points for non-convex functionals in Wasserstein space.
problem Minimizing non-convex functionals over the Wasserstein space with saddle point avoidance.
method WSFN is a second-order method that preconditions the Wasserstein gradient to avoid saddle points.
result WSFN escapes saddle regions and reaches a global minimizer in polynomial time.
Finding relationships between multiple views of data is essential both for exploratory analysis and as pre-processing for predictive tasks. A prominent approach is to apply variants of Canonical Correlation Analysis (CCA), a classical method seeking correlated components between views. The basic CCA is restricted to ma…
RECS improves graph embeddings by preserving network structure and stability.
problem Stable and accurate graph embeddings for multi-graph problems.
method RECS uses connection subgraphs and analogy to graphs with electrical circuits to learn stable node representations.
result RECS outperforms state-of-the-art algorithms by up to 36.85% on multi-label classification problems.
Develops an online Gaussian process method that maintains convergence guarantees without sample complexity issues.
problem The computational intractability of Gaussian processes with streaming data.
method Parsimonious Online Gaussian Processes (POG) that maintains asymptotic consistency with bounded memory.
result POG preserves convergence guarantees to the population posterior with finite memory, even for constant error radius.
Meta-Neighborhoods adapts predictions based on input neighborhoods.
problem Adaptive prediction based on input neighborhoods for AI.
method Semi-parametric method with induced neighborhoods and meta-learning.
result Meta-Neighborhoods more accurately represents predictive distributions.
We study "warped Berger" solutions $\big(\mc S^1\times\mc S^3,G(t)\big)$ of Ricci flow: generalized warped products with the metric induced on each fiber {s}×SU(2) a left-invariant Berger metric. We prove that this structure is preserved by the flow, that these solutions develop finite-time neckpinch …
We derive spectral sequences for the intersection homology of stratified fibrations and approximate tubular neighborhoods in manifold stratified spaces. These neighborhoods include regular neighborhoods in PL stratified spaces.
NNK algorithm improves neighborhood and graph construction for machine learning.
problem Ad hoc selection of k and ε parameters in kNN and ε-neighborhood methods.
method NNK algorithm for better sparse signal approximation.
result NNK leads to superior performance in local neighborhood and graph-based machine learning tasks.
Smooth curves from polygonal chains with vertex preservation and explicit curvature control.
problem Preserving vertices while smoothing polygonal chains to C∞ curves. method Directional mollification operator for polygonal chains.
result Smooth curves that intersect original vertices and maintain explicit curvature bounds.
Enhances nearest neighbor classifier performance with local distance metric learning.
problem Inconsistent data distribution across feature space.
method Local Mahalanobis Distance Learning (LMDL) considers neighborhood influence and learns multiple distance metrics for prototypes.
result LMDL improves nearest neighbor classifier performance on various datasets.
Bi-contact surgery operations can be applied to Anosov flows.
problem Characterizing Anosov flows and their properties.
method Metric and contact geometric characterizations, Liouville geometry, Reeb dynamics.
result Bi-contact surgery operations can be applied to Anosov flows.
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.
The paper uses EVT to improve tail risk measures under ambiguity sets.
problem Misspecification of tail risk measures leads to inflated risk estimates.
method Applies Extreme Value Theory to derive worst-case tail risk under ambiguity sets.
result Proposes a tail-calibrated ambiguity design that preserves nominal tail asymptotic scaling.
Study geodesics entering a fixed cusp neighborhood multiple times.
problem Understanding geodesics entering a specific cusp neighborhood multiple times.
method Investigate reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
result Characterized the class of reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
New study shows mean estimation algorithms can't beat sub-Gaussian rate in general.
problem Improving mean estimation beyond worst-case scenarios.
method Constructing counterexamples and introducing neighborhood optimality.
result No reasonable estimator can achieve better than sub-Gaussian error rate for any distribution.
Moves connect multibranched surfaces with same neighborhoods.
problem Connecting multibranched surfaces with identical neighborhoods.
method Introduces moves to connect multibranched surfaces.
result Any two multibranched surfaces can be connected in finitely many steps.
Proposes a parametric t-SNE without perplexity tuning.
problem Non-parametric t-SNE's perplexity parameter limits DR quality.
method Multi-scale parametric t-SNE with deep neural network.
result Produces reliable embeddings with competitive neighborhood preservation.
Skeleta and other pure subsets of manifold stratified spaces are shown to have neighborhoods which are teardrops of stratified approximate fibrations (under dimension and compactness assumptions). In general, the stratified approximate fibrations cannot be replaced by bundles, and the teardrops cannot be replaced by ma…
In this article we study Whitney (B) regular stratified spaces with the action of a compact Lie group G which preserves the strata. We prove an equivariant submersion theorem and use it to show that such a G-stratified space carries a system of G-equivariant control data. As an application, we show that if $A \su…
MixHop learns complex neighborhood relationships in graphs.
problem Existing graph neural networks cannot learn certain neighborhood mixing relationships.
method MixHop repeatedly mixes feature representations of neighbors at various distances.
result MixHop outperforms on challenging baselines and visualizes neighborhood information prioritization.
New graph kernels for evolving graphs with ordered neighborhoods.
problem Graphs with evolving edges over time.
method Combining convolutional subgraph kernels and string kernels, new scalable algorithms for generating graph feature maps.
result Neighborhood ordering yields more informative features.
New framework distinguishes knots via neighborhood invariants.
problem Distinguishing knots and knotoids.
method Study of knotoid spectra and neighborhood invariants.
result Neighborhood invariants can distinguish knots of higher Gordian distance.
This paper tackles selection bias in recommender systems by considering the neighborhood effect.
problem Selection bias in recommender systems due to filtering and user selection.
method Formalizes neighborhood effect as interference problem, introduces treatment representation, and proposes ideal loss.
result Proposed methods achieve unbiased learning when both selection bias and neighborhood effect are present.
Many prediction problems can be phrased as inferences over local neighborhoods of graphs. The graph represents the interaction between entities, and the neighborhood of each entity contains information that allows the inferences or predictions. We present an approach for applying machine learning directly to such graph…
Maximally hyperbolic solutions contain future neighborhoods of intersecting hypersurfaces.
problem Maximally globally hyperbolic solutions of higher-dimensional vacuum Einstein equations.
method Analyzing intersections of characteristic hypersurfaces.
result Contains a future neighborhood of intersecting hypersurfaces.
Compact leaves with amenable groups are stable under small perturbations.
problem Stability of compact leaves with amenable fundamental groups under small perturbations.
method Proving Thurston's conjecture for foliations close to the original foliation.
result Compact leaves with amenable groups are stable under small perturbations.
The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism f of a closed manifold is isotopic to a map realizing the Nielsen number of f, which is a lower bound for the number of fixed points among all maps homotopic to f. The main theorem of this paper proves this conjecture for all orientation pre…