Notes a flaw in a proof about embedding graphs.
problem A flaw in proving Sachs' conjecture about graph embeddings.
method Analyzing Stanfield's proof for gaps.
result Identifies a significant error in the proof.
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arXiv research
A locally-built, LLM-digested index of recent arXiv papers in quant finance, geometry/topology, and statistical ML — keyword search served straight from SQLite on this machine.
168,742 papers · 148 categories
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48 results for “linear linkless embedding”
A maximally linkless graph is a graph that can be embedded in without any links, but cannot be embedded in such a way if any other edge is added to the graph. Recently, a family of maximally linkless graphs was found with edges. We improve upon this by demonstrating a new family of maximally lin…
Graphs embeddable on torus and linklessly in 3D can be embedded linklessly in standard torus.
problem Embedding linklessly in a standard torus for graphs embeddable on torus and in 3D.
method Analyzing graphs of order 9 and below, showing linkless embedding in standard torus.
result For graphs of order 9 and below, linkless embedding in standard torus is possible.
We announce results about flat (linkless) embeddings of graphs in 3-space. A piecewise-linear embedding of a graph in 3-space is called {\it flat} if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have shown: (i) An embedding is flat if and only if the fundamental group of the compleme…
Classifies intrinsically linked tournaments by their score sequences.
problem Classifying intrinsically linked tournaments using their score sequences.
method Examining the score sequences of tournaments and identifying linkless sequences.
result The vast majority of score sequences for 8-vertex tournaments are linkless.
New bounds on maximal linkless graphs with improved edge-to-vertex ratios.
problem Finding maximal linklessly embeddable graphs with improved edge-to-vertex ratios.
method Constructing families of graphs and proving necessary and sufficient conditions for clique sums.
result Improved edge-to-vertex ratios for maximal linklessly embeddable graphs.
New graphs found that can be drawn without crossing links.
problem Finding graphs that can be drawn without links crossing.
method Provided specific examples for each .
result Infinite family of graphs linklessly embeddable and Tutte-4-connected.
A simpler proof for apex graphs in McCarty and Thomas' conjecture.
problem Proving a conjecture about apex graphs and their linklessly embeddable properties.
method Shorter and simpler proof for the apex case.
result A shorter and simpler proof for the apex case of the conjecture.
We consider intrinsic linking and knotting in the context of directed graphs. We construct an example of a directed graph that contains a consistently oriented knotted cycle in every embedding. We also construct examples of intrinsically 3-linked and 4-linked directed graphs. We introduce two operations, consistent edg…
Constructions stemming from non-separating planar graphs and their Colin de Verdière invariantmath.CO
New constructions from non-separating planar graphs improve understanding of graph linkability and knotability.
problem Understanding linkability and knotability of graph complements.
method Using maximal non-separating planar graphs to construct examples of maximal linkless and knotless graphs, and analyzing their Colin de Verdière invariant.
result The Colin de Verdière invariant of the complement of a maximal non-separating planar graph satisfies μ(cG) ≤ n-4, and equality holds.
Study finds maximal linklessly embeddable graphs up to 11 vertices and their complements.
problem Characterizing linklessly embeddable graphs and their complements.
method Comprehensive search and verification of graphs up to 11 vertices.
result For graphs of order 11, either the graph or its complement is intrinsically linked.
Study embeddability of 2-complexes in 4-space, proving Heawood family's excluded minors.
problem Whether a 2-dimensional CW complex embeds in .
method Operations preserving embeddability, constructions of non-preserving transformations, study of 4-flat graphs.
result Prove 78 graphs of Heawood family are excluded minors for 4-flat graphs.
Classifies linear embeddings of grassmannians and ind-grassmannians.
problem Understanding linear embeddings of grassmannians and ind-grassmannians.
method Classification through isomorphism of Picard groups and direct limits.
result Most linear embeddings of grassmannians are equivariant.
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.
This paper examines linear embeddings for high-dimensional Bayesian optimization, identifying and addressing issues to improve performance.
problem Scaling Bayesian optimization to high-dimensional spaces while maintaining sample efficiency.
method Study and empirical evaluation of linear embeddings for BO, addressing design choices and their impact on performance.
result Properly addressing issues in linear embeddings significantly improves their efficacy in BO.
An embedding of a graph into is said to be linear, if any edge of the graph is sent to be a line segment. And we say that an embedding of a graph into is free, if is a free group. It was known that for any complete graph its linear embedding is always free.…
In order to model entanglements of polymers in a confined region, we consider the linking numbers and writhes of cycles in random linear embeddings of complete graphs in a cube. Our main results are that for a random linear embedding of in a cube, the mean sum of squared linking numbers and the mean sum of square…
Survey of Locally Linear Embedding and its variants.
problem Representing high-dimensional data in a lower-dimensional space while preserving local structure.
method Explains various LLE and variant methods, including kernel LLE, inverse LLE, feature fusion, out-of-sample embedding, incremental LLE, landmark LLE, supervised LLE, robust LLE, fusion with other methods, and weighted LLE.
result Comprehensive overview of LLE and its variants.
Theoretical Connection between Locally Linear Embedding, Factor Analysis, and Probabilistic PCAstat.ML
This work connects LLE, factor analysis, and probabilistic PCA through a stochastic perspective.
problem Exploring the theoretical connection between LLE, factor analysis, and probabilistic PCA.
method Solving the stochastic linear reconstruction of LLE using expectation maximization.
result LLE, factor analysis, and probabilistic PCA are shown to be connected through a stochastic perspective.
Non-linear Hopf manifolds can be embedded into linear ones and admit LCK metrics.
problem Understanding non-linear Hopf manifolds and their properties.
method Holomorphic embeddings and LCK metrics.
result Non-linear Hopf manifolds admit LCK metrics.
Improved recommendation systems using multi-layer embeddings reduce model size while maintaining accuracy.
problem Improving model accuracy in recommendation systems while minimizing model size.
method Introducing a multi-layer embedding training (MLET) architecture that trains embeddings via a sequence of linear layers.
result Substantial advantages in model accuracy and memory footprint are achieved with reduced embedding dimensions.
In 1983 Conway and Gordon proved that any embedding of the complete graph into contains at least one nontrivial knot as its Hamiltonian cycle. After their work knots (also links) are considered as intrinsic properties of abstract graphs, and numerous subsequent works have been continued until recen…
This paper finds a linear relationship between t-SNE perplexity and data set size.
problem Choosing the right perplexity for t-SNE embeddings.
method Analyzed the relationship between perplexity and data set size.
result Embeddings remain structurally consistent when perplexity is adjusted accordingly.
We use the theory of oriented matroids to show that any linear embedding of , the complete graph on nine vertices, contains a non-split link with three components.
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.
A new embedding method for high-dimensional data.
problem Handling large sample sizes in high-dimensional spaces.
method Partitioning space into simplices and embedding into barycentric coordinates.
result Linear classifier in rich feature space yields highly non-linear decision boundaries.
Supervised (linear) embedding models like Wsabie and PSI have proven successful at ranking, recommendation and annotation tasks. However, despite being scalable to large datasets they do not take full advantage of the extra data due to their linear nature, and typically underfit. We propose a new class of models which …
Linear representations help embed manifolds into matrix spaces.
problem Embedding manifolds into matrix spaces with effective bounds.
method Defining linear representations of -manifolds as maps into matrix spaces, encoding -actions as matrix products.
result Explicit bounds for Mostow-Palais -equivariant embeddings of -manifolds into -modules , showing for compact .
Random complexes can be embedded linearly if certain conditions on parameters are met.
problem Embedding random simplicial complexes linearly in Euclidean space.
method Established strict inequalities on parameters for linear embedding into R^(2d).
result Necessary and sufficient conditions for linear embedding of random complexes.
Linear autoregressive models serve as basic representations of discrete time stochastic processes. Different attempts have been made to provide non-linear versions of the basic autoregressive process, including different versions based on kernel methods. Motivated by the powerful framework of Hilbert space embeddings o…
A new method simplifies HLLE for better robustness.
problem Improving robustness of Hessian locally linear embedding.
method Replacing Hessian with arbitrary weights and modifying manifold dimension.
result Achieved a new LLE-type method called tangential LLE.
We classify all rotational surfaces in Euclidean space whose principal curvatures and satisfy the linear relation , where and are two constants. We give a variational characterization of these surfaces in terms of its generating curve. As a consequence of our classification, we find clos…
FREDE efficiently embeds graphs using linear space and guarantees quality.
problem Efficiently embedding graphs with quality guarantees and linear space complexity.
method FREDE combines matrix sketching with a nonlinear transform of PageRank similarities to achieve linear space and quality guarantees.
result FREDE provides column-covariance approximation guarantees that are nearly as good as SVD, even with limited node similarities.
Linearly Free Graphsmath.GT
The paper explores linearly free graphs and their embeddings into 3D space.
problem Understanding the conditions under which a graph's embedding into 3D space is free.
method Developed a sufficient condition for a linear embedding to be free and applied it to specific graph cases.
result Established sufficient conditions for a graph to be linearly free and provided examples and counterexamples.
The Local Isometric Embedding of Two-Metrics of Low Differentiability in Euclidean Three Spacemath.DG
Proves local isometric embedding of low-differentiability metrics in 3D space.
problem Isometric embedding of metrics of low differentiability in Euclidean 3-space.
method Simplified notation, geodesic and level parameters, solutions of initial value problems for first order non-linear PDEs, classical linear algebraic systems.
result Local isometric embedding exists for metrics of C1 differentiability.
LLE produces unwanted results without regularization, which can be prevented with regularization.
problem LLE's inherent unwanted results without regularization.
method Mathematical proof and numerical examples of regularization effectiveness.
result Regularization prevents unwanted results in LLE.
Entangled embedded periodic nets and crystal frameworks are defined, along with their dimension type, homogeneity type, adjacency depth and periodic isotopy type. We obtain periodic isotopy classifications for various families of embedded nets with small quotient graphs. We enumerate the 25 periodic isotopy classes of …
EGORSE optimizes high-dimensional problems using random and supervised embeddings.
problem Efficiently solving computationally expensive high-dimensional optimization problems.
method EGORSE combines random and supervised linear embeddings for adaptive optimization.
result EGORSE outperforms state-of-the-art methods in high-dimensional optimization.
We study piecewise linear co-dimension two embeddings of closed oriented manifolds in Euclidean space, and show that any such embedding can always be isotoped to be a closed braid as long as the ambient dimension is at most five, extending results of Alexander (in ambient dimension three), and Viro and independently Ka…
This paper introduces an acceleration structure for hyperbolic embeddings.
problem Efficiently embedding and visualizing high-dimensional data in hyperbolic spaces.
method Building upon a polar quadtree, the paper introduces a new acceleration structure for hyperbolic embeddings.
result The new method computes embeddings in significantly less time compared to existing methods.
The paper strengthens a theorem on crossings under linear perturbations with Hausdorff measure estimates.
problem Understanding multiple-point crossings under linear perturbations.
method Establishes a transversality theorem with Hausdorff measure estimates for exceptional parameter sets.
result Explicit upper bounds on the Hausdorff dimension of the exceptional set.
Word embeddings generated by neural network methods such as word2vec (W2V) are well known to exhibit seemingly linear behaviour, e.g. the embeddings of analogy "woman is to queen as man is to king" approximately describe a parallelogram. This property is particularly intriguing since the embeddings are not trained to a…
Improved graph embedding through refined linear transformation and community recovery.
problem Identifying meaningful latent communities in graph data.
method Refined graph encoder embedding via linear transformation, self-training, and latent community recovery.
result Improved vertex embedding and better decision boundaries for vertex classification.
Local Linear embedding (LLE) is a popular dimension reduction method. In this paper, we first show LLE with nonnegative constraint is equivalent to the widely used Laplacian embedding. We further propose to iterate the two steps in LLE repeatedly to improve the results. Thirdly, we relax the kNN constraint of LLE and p…
Study estimates gaps in semigroup products, proving embedding properties.
problem Estimating singular value gaps in semigroup products.
method Lower estimates for singular value gaps of free products of semigroups in ping-pong position.
result Groups generated by semigroups in ping-pong position are quasi-isometrically embedded.
We consider an embedding of a -dimensional CW complex into the -sphere, and construct it's dual graph. Then we obtain a homogeneous system of linear equations from the -dimensional CW complex in the first homology group of the complement of the dual graph. By checking that the homogeneous system of linear equa…
Identifying coordinate transformations that make strongly nonlinear dynamics approximately linear is a central challenge in modern dynamical systems. These transformations have the potential to enable prediction, estimation, and control of nonlinear systems using standard linear theory. The Koopman operator has emerged…
Most of existing manifold learning methods rely on Mean Squared Error (MSE) or norm. However, for the problem of image quality assessment, these are not promising measure. In this paper, we introduce the concept of an image structure manifold which captures image structure features and discriminates image dist…