The Cartesian subgroup in graph products of groups is studied with bounds and algorithms.
problem Understanding the structure of Cartesian subgroups in graph products of groups.
method Theory of polyhedral products, lower and upper bounds, algorithm for small presentations.
result Bounds on the number of relations and deficiency in presentations of Cartesian groups.
Paper learns Cartesian product graphs with Laplacian constraints.
problem Learning Cartesian product graphs from Laplacian constraints.
method Penalized maximum likelihood estimation (MLE) and efficient algorithm.
result Statistical consistency for Cartesian product Laplacian estimation.
New graph Fourier transform distinguishes directions in multi-dimensional signals.
problem Existing graph Fourier transform fails to distinguish directions in multi-dimensional signals.
method Algebraic properties of Cartesian products rearrange 1-D spectra into multi-dimensional frequency domain.
result Solves multi-valuedness of spectra and enables directional frequency analysis.
COMBO optimizes Bayesian Optimization for combinatorial search spaces.
problem Optimizing objectives on combinatorial search spaces with high-order interactions.
method COMBO uses a combinatorial graph and ARD diffusion kernel with Horseshoe prior for efficient modeling and variable selection.
result COMBO outperforms state-of-the-art methods consistently across various benchmarks.
The study bounds the effective diameter of graphs with positive Ollivier curvature.
problem Bounding the effective diameter of graphs with positive Ollivier curvature.
method Introducing reflective graphs and proving discrete Bonnet Myers theorem.
result The effective diameter bound is attained only for specific graphs.
Study on balanced Hermitian structures on Lie algebras twisted by representations.
problem Conditions for balanced and locally conformally balanced Hermitian structures on Lie algebras.
method Analysis of Hermitian structures on twisted cartesian products of Lie algebras.
result Classification of six-dimensional balanced Hermitian twisted cartesian products Lie algebras.
New bounds for average graph distance using curvature and centrality.
problem Finding bounds for average graph distance.
method Using weighted average Ollivier curvature with edge betweenness centrality.
result Equality in bounds achieved for specific reflective graphs.
Decomposes 3-manifolds into simpler parts, but higher dimensions are more complex.
problem Understanding how 3-manifolds can be broken down into smaller parts.
method Examined Cartesian products of nondecomposable manifolds, focusing on 3-manifolds and higher dimensions.
result For 3-manifolds, similar results to Borsuk's for surfaces hold, but higher dimensions are more complex.
We study the Ollivier-Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most 3 linear parts, with at most 2 linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartes…
The paper explores curvatures on graphs and their implications for Ricci flatness.
problem Comparing and understanding different curvature notions on graphs and their implications for Ricci flatness.
method Analyzing Ollivier Ricci curvature and Bakry-Émery curvature on combinatorial graphs, investigating graph products, and proving curvature properties.
result Non-negativity of Ollivier Ricci curvature implies non-negativity of Bakry-Émery curvature under specific conditions.
Examines how irreducibility and rigidity affect digital images.
problem Understanding interactions between irreducibility and rigidity in digital images.
method Analyzes Cartesian products, wedges, and cold and freezing sets.
result Interactions between irreducibility and rigidity in digital images.
New curvature tensor and matrices for connection graphs derived from Bakry-Émery curvature.
problem Deriving Buser-type bounds on eigenvalues of connection Laplacians.
method Reformulation of Bakry-Émery curvature through curvature matrices and tensor representations.
result Extension of curvature matrices to connection graphs, addressing eigenfunction challenges.
We show that the Cartesian product of three hereditarily infinite dimensional compact metric spaces is never hereditarily infinite dimensional. It is quite surprising that the proof of this fact (and this is the only proof known to the author) essentially relies on algebraic topology.
We study the Bakry-Émery curvature function KG,x:(0,∞]→R of a vertex x in a locally finite graph G systematically. Here KG,x(N) is defined as the optimal curvature lower bound K in the Bakry-Émery curvature-dimension inequality $CD(\mathcal{K},\ma…
We show that non-domination results for targets that are not dominated by products are stable under Cartesian products.
Classifies graphs with Bonnet-Myers sharp curvature and connects it to Bakry-Émery curvature.
problem Classifying graphs with specific curvature properties.
method Introducing Bonnet-Myers and Lichnerowicz sharpness in Ollivier Ricci curvature, combinatorial reformulation, and relating to Bakry-Émery curvature.
result Classification of graphs including hypercubes, cocktail party graphs, and Johnson graphs J(2n,n). We study properties of Cartesian products of digital images, using a variety of adjacencies that have appeared in the literature.
New proof shows how to fill a square with Fibonacci curve.
problem How to fill a square with a space-filling curve.
method Used Cartesian product of Fibonacci substitution with itself.
result Different proof of space-filling curve construction.
The paper reformulates Bakry-Émery curvature on graphs using eigenvalues.
problem Analyzing curvature on weighted graphs.
method Reformulating curvature as the smallest eigenvalue of a rank one perturbation of the curvature matrix.
result The curvature function is analytic, strictly monotone increasing, and concave until a threshold, after which it is constant.
Efficiently infers sparse networks from count data with reduced memory usage.
problem Sparse network inference for count data with high dimensions and dependencies.
method Improved Bigraphical Lasso using eigenvalue decomposition of Cartesian product graph.
result Reduced computational complexity from O(n2p2) to O(n2+p2). Characterizes graphs with Lin-Lu-Yau curvature at least one and explores bone-idle graphs.
problem Characterizing graphs with specific curvature properties.
method Study of Ollivier-Ricci curvature and Lin-Lu-Yau curvature, exploration of regular graphs, and exact formula derivation.
result Characterizes edges that are bone-idle in regular graphs and provides a complete characterization of 4-regular bone-idle graphs.
New method detects non-product domains using squeezing function.
problem Detecting non-product bounded pseudoconvex domains.
method New application of squeezing function and optimal estimates.
result Identifies new family of holomorphic homogeneous regular domains.
Modified proof constructs holomorphic quilts on closed surfaces.
problem Compare Lagrangian Floer theory with quilted Lagrangian Floer theory.
method Modified proof of holomorphic quilts from Wehrheim and Woodward.
result Supports Bottman and Wehrheim's conjecture on isomorphism.
Method segments graphs to estimate network models using power graph fused lasso.
problem Estimating non-parametric network models from noisy data.
method Power graph fused lasso (PGFL) for graph segmentation.
result PGFL achieves optimal error rate for graphon estimation under subGaussian noise.
We prove that if a compact nilmanifold Γ\G is endowed with a Vaisman structure, then G is isomorphic to the Cartesian product of the Heisenberg group with R.
Flow preserves curvature sharpness on weighted graphs.
problem Curvature flow on weighted graphs.
method Adapting Bakry-Émery calculus for Markovian preservation and analyzing limits.
result Flow limits to curvature sharp weighted graphs.
We propose a novel adaptive learning algorithm based on iterative orthogonal projections in the Cartesian product of multiple reproducing kernel Hilbert spaces (RKHSs). The task is estimating/tracking nonlinear functions which are supposed to contain multiple components such as (i) linear and nonlinear components, (ii)…
In this paper, we establish the rigidity result for local holomorphic volume preserving maps from an irreducible Hermitian manifold of compact type into its Cartesian products.
For any n>1 we give an explicit example of an n-axially symmetric Cartesian current in B^3 x S^2 with non-trivial vertical part and non-constant graph part minimizing the relaxed Dirichlet energy among the n-axially symmetric Cartesian currents with the same boundary. This stands in sharp contrast with a results of Har…
We make an estimation of the value of the Gromov norm of the Cartesian product of two surfaces. Our method uses a connection between these norms and the minimal size of triangulations of the products of two polygons. This allows us to prove that the Gromov norm of this product is between 32 and 52 when both factors hav…
A free action of the direct product of two copies of the symmetric group on 3 elements on the cartesian product of two copies of the 3-sphere is constructed. This nonlinear action is constructed using surgery. The action provides a counterexample to a conjecture of Lewis made in 1968.
4D manifolds without positive scalar curvature but products do.
problem Existence of positive scalar curvature on manifolds and products.
method Counterexamples and product constructions.
result Found 4D manifolds without positive scalar curvature but products do.
In this note we derive enumerative formulas for several types of labelled acyclic directed graphs by slight modifications of the familiar recursive formula for simple acyclic digraphs. These considerations are motivated by, and based upon, recent combinatorial results in geometric topology obtained by S.Choi, who estab…
Scalable Gaussian processes with latent Kronecker structure for large datasets.
problem Limited scalability of Gaussian processes for large datasets.
method Leveraging latent Kronecker structure, projecting kernel matrix onto latent Kronecker product, using iterative linear system solvers and pathwise conditioning.
result Outperforms state-of-the-art sparse and variational GPs on real-world datasets with up to five million examples.
Defines products for fibered corners manifolds, generalizing resolutions.
problem Resolving fibered corners manifolds.
method Introduces a category of fibered corners manifolds with products and transverse fiber products, defining the 'ordered product' for wedge metrics.
result The 'ordered product' is a natural product for wedge metrics.
We study homogenous Weyl connections with non-positive sectional curvatures. The Cartesian product S1×M carries canonical families of Weyl connections with such a property, for any Riemmanian manifold M. We prove that if a homogenous Weyl connection on a manifold, modeled on a unimodular Lie group, …
Let B be a fiber bundle with compact fiber F over a compact Riemannian n-manifold M. There is a natural Riemannian metric on the total space B consistent with the metric on M. With respect to that metric, the volume of a rectifiable section s:M--> B is the mass of the image s(M) as a rectifiable n-current in B. Theorem…
Constructs special Kähler structures on Lie groups.
problem Creating special Kähler structures on Lie groups.
method Introducing twisted cartesian product and double extension process.
result Characterizes left invariant flat special Kähler structures.
Digital trees have approximate fixed point property, and conditions for products are explored.
problem Conditions for the approximate fixed point property in digital tree products.
method Analyzes digital trees and their products, explores conditions for the AFPP.
result Conditions are found for the AFPP in digital tree products.
EiGLasso speeds up sparse Kronecker-sum covariance estimation.
problem Sparse Kronecker-sum inverse covariance estimation challenges in scalability and parameter identification.
method Newton's method combined with eigendecomposition of sample and feature graphs, approximating Hessian for speed.
result Two to three orders-of-magnitude speed-up on simulated and real-world data.
New limits of minimal surface systems have surprising large interior parts.
problem Minimal surface system limits with large interior vertical and non-minimal portions.
method Construction of limits with smallest possible dimension and codimension.
result Limits of minimal surface systems can have surprising large interior parts.
New Calabi-Yau metrics constructed with detailed geometry at infinity.
problem Constructing complete Calabi-Yau metrics with specific properties.
method Weighted blow-up and Hölder spaces for Laplacian analysis.
result Examples of Calabi-Yau metrics with conical singularities and non-uniqueness of tangent cones.
CAN approximates explicit feature interactions for CTR prediction.
problem Learning explicit feature interactions from sparse features.
method Co-Action Network approximates explicit pairwise feature interactions without introducing too many additional parameters.
result CAN outperforms state-of-the-art CTR models and the cartesian product method.
A simplified proof for embedding higher-dimensional complexes into manifolds.
problem Embedding higher-dimensional complexes into manifolds with constraints.
method A short and accessible proof for the Patak-Tancer theorem.
result A simplified proof for the Heawood inequality in higher dimensions.
Generalized are the investigated in other works of the author transports along paths in fibre bundles to transports along arbitrary maps in them. Their structure and some properties are studied. Special attention is paid to the linear case and the case when the map's domain is a Cartesian product of two sets. Also cons…
Study torsion obstructions to positive scalar curvature on manifolds.
problem Obstructions to positive scalar curvature metrics on manifolds.
method Analysis of torsion classes in integral homology.
result Construction of manifolds without positive scalar curvature metrics but whose Cartesian products do.
Paper improves Weyl Law for product manifolds.
problem Improving the error term in the Weyl Law for product manifolds.
method Using Duistermaat and Guillemin's result, reducing the problem to lattice point distribution in Euclidean space.
result Quantified bound for Cartesian product of spheres, showing o(λd−1) error. Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star, Lie derivative, pullback and interior product. Partitions of unity exist in thi…