The article discusses how to create a special type of triangle mesh for surfaces in 3D space.
problem Creating a special type of triangle mesh for surfaces in 3D space.
method Using sufficient conditions and the diagonal switch algorithm to find an embedded Delaunay triangulation.
result The diagonal switch algorithm can find an embedded Delaunay triangulation for a point cloud on an embedded surface in R3. Embedding principle explains loss landscape of deep neural networks.
problem Understanding the structure of loss landscapes in deep neural networks.
method Proposed an embedding principle that critical points of narrower DNNs can be embedded to critical points of wider DNNs.
result Wide DNNs are often attracted by highly-degenerate critical points embedded from narrower DNNs.
Paper explores duality in DPPs using embedding structure analysis.
problem Understanding the geometric structure of determinantal point processes.
method Analyzes the exponential family embedding of DPPs and uses the e-embedding curvature tensor.
result Discovers the duality between marginal and L-ensemble kernels.
Study on neural networks in overparameterized cases, focusing on flat minima and saddle points.
problem Understanding the landscape of training error in neural networks with overparameterization.
method Three methods of embedding a network into a wider one with more hidden units, analyzing the embedded point's properties.
result Smooth and ReLU activation networks have different partially flat landscapes around the embedded point.
Condition for embedding metric spaces into curved manifolds.
problem Embedding conditions for metric spaces in curved manifolds.
method If-and-only-if condition on five-point metric spaces.
result Five-point metric spaces admit embeddings into nonnegatively curved Riemannian manifolds.
A new invariant captures geometric features of circle embeddings.
problem Capturing geometric features of circle embeddings invariantly.
method Chordal distance transform and persistent homology.
result Persistent homology of chordal distance transform is invariant.
The paper studies spheres with small diameter in 3D manifolds concentrating at scalar curvature critical points.
problem Understanding the behavior of Willmore spheres with small diameter in 3D manifolds.
method Analyzes spheres under bounded Willmore energy and small diameter constraints, focusing on scalar curvature critical points.
result Embedded Willmore spheres concentrate at critical points of scalar curvature under small diameter and bounded energy conditions.
Improves manifold learning by identifying and fixing stuck points.
problem Local minima in nonlinear embedding methods lead to poor data structure visualization.
method Introduces a method to temporarily allow pressured points to use an extra dimension in the embedding space.
result Significantly improves the objective function value of existing methods even after getting stuck in a poor local minimum.
Generative model for 3D point clouds using invertible flows.
problem Generating realistic 3D point clouds.
method Invertible flow-based models for point cloud generation with parameter sharing and embedding vectors.
result The model generates high-quality 3D point clouds with good similarity.
Let F be a closed orientable surface. We give an explicit formula for the number mod 2 of quadruple points occurring in any generic regular homotopy between any two regularly homotopic embeddings e,e':F -> R^3. The formula is in terms of homological data extracted from the two embeddings.
Projections of knotted spheres in high dimensions show complex singularities.
problem Understanding the singularities of projections of knotted spheres in high-dimensional spaces.
method Construction of specific knotted spheres and analysis of their projections.
result Projections of knotted spheres can have double points and connected embedded double point sets.
Probabilistic McShane's identity measures paths through a point.
problem Understanding the probabilistic nature of McShane's identity.
method Interpreting McShane's identity as a measure on path spaces.
result A probabilistic measure on path spaces.
We embed KKT points in neural networks of different sizes.
problem Classifying data using homogeneous neural networks.
method Introducing KKT point embedding principle and proving it for different network types.
result KKT points of a smaller network can be mapped to those of a larger network via linear transformations.
DSNE visualizes data velocity in lower dimensions.
problem Understanding movement patterns in high-dimensional data.
method DSNE is a variation of Stochastic Neighbor Embedding that learns velocity embeddings using Euclidean distances on a unit sphere.
result DSNE enables visualization of data movement in lower dimensions.
Every point on an asymptotically flat 3D space has a minimal plane nearby.
problem Finding minimal surfaces in asymptotically flat 3D spaces.
method Proving the existence of minimal planes for every point in the manifold.
result Every point in an asymptotically flat 3D space has a complete properly embedded minimal plane.
In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang-Yau quasi-local mass for a surface in spacetime introduced in [7] and [8] is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the …
Robustly computes intrinsic coordinates on point clouds using resampling and averaging.
problem Computing intrinsic coordinates on noisy or outlier-prone point clouds.
method Subsample data, vary hyperparameters, cluster candidate embeddings, identify representative embeddings, and average them using Procrustes analysis.
result Robust to noise and outliers, validated on synthetic and real data.
Reformulates Z2n-manifolds using functor of points.
problem Understanding Z2n-graded manifolds in a categorical framework. method Uses functor of points to embed Z2n-manifolds into contravariant functors of Z2n-points. result Equivalence of Z2n-manifolds and locally trivial functors. Paper proposes a novel method to test differences in spatial point patterns.
problem Detecting differences in the first-order structures of spatial point patterns.
method Kernel mean embedding with approximate version tailored for spatial point processes, reducing comparison to Euclidean space t-tests.
result The proposed method is powerful and well-calibrated, demonstrated on real-world data.
Improves visualization of high-dimensional data by correcting misleading artifacts in neighbor embedding methods.
problem Misleading visual artifacts in t-SNE and UMAP due to lack of data-independent manifold learning interpretations.
method LOO-map framework that extends embedding maps to the entire input space, identifying and correcting map discontinuities.
result Developed point-wise diagnostic scores to detect unreliable embedding points and improve hyperparameter selection.
Interactive machine learning with weak supervision and pre-trained embeddings.
problem Training machine learning models with limited labeled data.
method Use pre-trained embeddings to define a distance function and extend source votes to nearby points.
result Significantly outperforms traditional weakly-supervised and fully-supervised methods.
IDK improves anomaly detection for points and groups without explicit learning.
problem Anomaly detection for points and groups using kernel methods.
method Isolation Distributional Kernel (IDK) addresses data independence and intractable dimensionality issues.
result IDK outperforms existing methods for both point and group anomaly detection.
New lower bounds on embedding dimensions for neural network architectures.
problem Ensuring neural networks can handle symmetries like permutations in high dimensions.
method Novel technique to prove lower bounds on embedding dimensions.
result Proves new lower bounds on embedding dimensions for Deep Sets and Janossy pooling.
A manifold is T-embedded into an affine space if its tangent spaces at distinct points are disjoint. We prove that an n-dimensional disc cannot be T-embedded into 2n-dimensional space.
The study embeds graphs on translation surfaces, proving essential-systolic embeddings and estimating surface genera.
problem Embedding graphs on translation surfaces with specific properties.
method Proving essential-systolic embeddings and estimating surface genera.
result Finite graphs admit essential-systolic embeddings on translation surfaces with estimated genera.
We prove that every proper n-dimensional length metric space admits an "approximate isometric embedding" into Lorentzian space R3n+6,1. By an "approximate isometric embedding" we mean an embedding which preserves the energy functional on a prescribed set of geodesics connecting a dense set of points.
Generative LLE modifies LLE to generate stochastic embeddings.
problem Nonlinear dimensionality reduction and manifold learning.
method Generative LLE modifies LLE by using stochastic linear reconstruction.
result Generative LLE can generate various LLE embeddings stochastically.
Study embeddings of free group products into automorphism groups.
problem Embeddings of direct products of free groups into automorphism groups.
method Proved existence of canonical fixed points in the boundary of Outer space.
result Complete description of embeddings of direct products of free groups.
A new clustering method improves recovery guarantees by re-embedding data.
problem Improving recovery guarantees in clustering algorithms.
method Chaining four techniques: leapfrog distances, multidimensional scaling, spectral methods, and sum-of-norms clustering.
result Re-embedding data improves recovery guarantees of clustering.
We show that if P is an embedded least area (area minimizing) plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded.
New invariant metrics preserved under deformed Markov embeddings.
problem Preserving invariance in probability measure spaces under deformed embeddings.
method Deforming Markov embeddings while maintaining sufficiency, proving existence and uniqueness of invariant families.
result Existence and uniqueness of invariant families of tensor fields under deformed embeddings.
The square-peg problem is solved using configuration spaces and multijet transversality.
problem Proving that every simple closed curve in the plane has an odd number of inscribed squares.
method Using the multijet transversality theorem and configuration spaces, we find a dense set of smooth embeddings for which the configuration space of points is transverse to any submanifold.
result A dense family of smoothly embedded circles in the plane and in Rn have an odd number of inscribed square-like quadrilaterals. We embed objects as elliptical distributions using the Wasserstein metric.
problem Embedding complex objects as vectors in low dimensional spaces.
method Embedding objects as elliptical probability distributions with the 2-Wasserstein metric.
result Wasserstein elliptical embeddings provide more intuitive and numerically stable tools than Gaussian embeddings.
PERCEPT detects changes in high-dimensional data streams using topological data analysis.
problem Detecting changes in high-dimensional data streams, especially when embedded in a low-dimensional space.
method Leverages topological data analysis to learn embedded topology as a point cloud via persistence diagrams, then applies non-parametric monitoring for detecting changes.
result Demonstrates efficient detection of online changes from high-dimensional data streams.
Small neural networks embed arbitrary metric spaces into Gaussian mixtures.
problem Embedding arbitrary metric spaces into a fixed space with low distortion.
method Probabilistic transformers of small depth and width.
result Embeddings with low metric distortion for various metric spaces.
This work models uncertainty in instance embeddings using hedging.
problem Uncertainty in ambiguous inputs is not well represented by traditional embeddings.
method Hedged instance embedding (HIB) models embeddings as random variables and trains under variational information bottleneck.
result Improved performance in image matching and classification tasks, more structured embedding space, and per-exemplar uncertainty measure.
DINOSAUR improves retrieval by accounting for embedding uncertainty in recommender systems.
problem Retrieval bias towards popular items due to noisy embeddings.
method Samples multiple embeddings per item and queries with sampled embeddings to account for uncertainty.
result Improves coverage of long-tail niche content without sacrificing recall.
The paper addresses data uncertainty in graph embedding by modeling data points as Gaussian distributions.
problem Data uncertainty in machine learning pipelines leads to misleading embeddings and lower accuracy.
method The paper proposes modeling data uncertainty using Gaussian distributions and reformulates graph embedding techniques.
result The proposed methods improve the accuracy of graph embedding by accounting for data uncertainty.
In this paper we prove that an embedded and simply connected constant mean curvature surface with curvature large at a point contains a multi-valued graph around that point on the scale of ∣A∣2, where ∣A∣2 is the norm squared of the second fundamental form. This generalizes Colding and Minicozzi's result for mini…
Combines foundation models with weak supervision to improve NLP and video tasks.
problem Leveraging weak supervision with foundation models without labeled data.
method Liger, a combination of foundation model embeddings and weak supervision techniques.
result Liger outperforms existing weak supervision methods by 14.1 points on benchmark NLP and video tasks.
Disk and sphere graphs embed quasi-isometrically into Euclidean spaces.
problem Embedding disk and sphere graphs into Euclidean spaces.
method Disk and sphere graphs defined based on handlebodies and marked points.
result Quasi-isometric embeddings of disk and sphere graphs into Euclidean spaces.
Explains SNE, t-SNE, and their variants for manifold learning.
problem Dimensionality reduction and manifold learning.
method Probabilistic approach using Gaussian and Student-t distributions.
result Out-of-sample extension and acceleration methods for t-SNE.
The paper proves local isometric embeddings for singular metrics near a point.
problem Existence of local isometric embeddings for singular Riemannian metrics.
method Ramified local isometric embeddings using Leray's ramified Cauchy-Kovalevskaya Theorem.
result Existence of local analytic isometric embeddings into Euclidean space.
A graph embedded in the 3-sphere is called irreducible if it is non-splittable and for any 2-sphere embedded in the 3-sphere that intersects the graph at one point the graph is contained in one of the 3-balls bounded by the 2-sphere. We show that irreducibility is preserved under certain deformations of embedded graphs…
Let X be a smooth subvariety of CP^N. We study a flow, called balancing flow, on the space of projectively equivalent embeddings of X, which attempts to deform the given embedding into a balanced one. If L->X is an ample line bundle, considering embeddings via H^0(L^k) gives a sequence of balancing flows. We prove that…
The paper proves the existence of minimal planes in specific 3-manifolds.
problem Existence of minimal planes in asymptotically flat 3-manifolds.
method Improving a previous result, the authors prove the existence of minimal planes under specific conditions.
result Properly embedded minimal planes can be found in asymptotically flat 3-manifolds.
Develops a new theory of width for embedded circles in Riemannian manifolds.
problem Defining and understanding the width of embedded circles in Riemannian manifolds.
method Morse-Lusternik-Schnirelmann theory applied to geodesics and minimising configurations.
result Classifies configurations of minimising geodesics intersecting embedded circles.
Study on inflection points of plane curve shadows with fixed embedded shapes.
problem Minimum number of inflection points in plane curves with fixed embedded shadows.
method Finite coorientation problem on building polygons, dynamic programming, universal lower bound, tree-necklace shadows.
result Exact formula for minimum number of normalized inflections for tree-like shadows.