New cohomology theory for planar graphs with perfect matchings.
problem Understanding cohomology of planar trivalent graphs with perfect matchings.
method Introducing a cohomology theory and defining new polynomials.
result 2-factor polynomial can indicate 4-face colorability.
The paper refines 2-factor homology to a stable homotopy type for planar trivalent graphs with perfect matchings.
problem Developing a stable homotopy type for planar trivalent graphs with perfect matchings.
method Defining a cover functor from the 2-factor flow category to the cube flow category, realizing the 2-factor spectrum, and showing it's an invariant.
result The stable homotopy type of the 2-factor spectrum is an invariant of planar trivalent graphs with perfect matchings.
FMI uses matching to mimic interventions for causal feature learning.
problem Challenges in causal discovery from observational data.
method Feature Matching Intervention (FMI) using matching to emulate perfect interventions.
result FMI outperforms in identifying causal features from observational data.
The paper introduces a quantum state system to count perfect matchings in graphs.
problem Counting perfect matchings in graphs using quantum state systems.
method Topological quantum field theory (TQFT) and spectral sequences.
result The filtered n-color vertex homology for n=2 is generated by perfect matchings. New link polynomials linked to cluster theory.
problem Connecting link polynomials to cluster theory.
method Introducing new link polynomials and their expansion over perfect matchings.
result Bracket polynomials of certain links can be realized as specializations of cluster variables.
OmniMatch algorithm perfectly matches graphs without edge correlation.
problem Graph matching in the absence of edge correlation.
method OmniMatch algorithm for seeded multiple graph matching.
result OmniMatch aligns O(sα) unseeded vertices across multiple networks efficiently and perfectly. We give an algorithmic computation for the height of Kauffman's clock lattice obtained from a knot diagram with two adjacent regions starred and without crossing information specified. We show that this lattice is more familiarly the graph of perfect matchings of a bipartite graph obtained from the knot diagram by over…
This paper presents an algorithm to construct a weighted adjacency matrix of a plane bipartite graph obtained from a pretzel knot diagram. The determinant of this matrix after evaluation is shown to be the Jones polynomial of the pretzel knot by way of perfect matchings (or dimers) of this graph. The weights are Tutte'…
The paper characterizes discrete Morse functions on knot diagrams and generalizes a clock theorem.
problem Characterizing discrete Morse functions on knot diagrams and generalizing a clock theorem.
method Using matchings on the Tait graph, the paper constructs discrete Morse functions and counts them with a formula involving the graph Laplacian. It also proves a bijection between these functions and certain rooted spanning forests.
result The paper provides a closed formula for counting discrete Morse functions and generalizes a clock theorem.
A simple method for counterfactual inference from observational data.
problem Learning representations for counterfactual inference from observational data.
method Augmenting samples with their propensity-matched nearest neighbours.
result PM outperforms state-of-the-art methods in inferring counterfactual outcomes.
Algorithm uses RBM to solve matching problems on weighted graphs.
problem Perfect matching problem on bipartite weighted graphs.
method Iterative RBM algorithm to maximize energy function and assignment.
result Algorithm successfully solves real-world matching problems.
Graph matching with feature vectors is solved using a two-layer graph neural network.
problem Graph matching in the presence of sparse binary features.
method Two-layer graph neural network with graph structure.
result Graph neural network can recover correct mapping with high probability under certain conditions.
In recent work the author investigates perfect matchings of a bipartite graph obtained from a knot diagram and demonstrates that these correspond to discrete Morse functions on a 2-complex for the 2-sphere. This relationship is expounded below for the opposite audience: those who may be unfamiliar with knots.
Proves curvature of conference graphs and finds local matchings.
problem Proving precise values of curvature in conference graphs.
method Combining parameter relations and combinatorial approach.
result Existence of local perfect matchings in broader classes of graphs.
GAN-based semi-supervised learning improves classifier generalization.
problem Improving classifier performance with limited labeled data.
method Theoretical analysis of GAN-SSL, proving equivalence of discriminator optimization and supervised learning, and exploring conditions for perfect discriminator.
result GAN-SSL theoretically outputs a perfect discriminator on both labeled and unlabeled data.
The paper connects knot theory and cluster algebras via dimer face polynomials.
problem Understanding the relationship between knot theory and cluster algebras.
method Analyzing dimer face polynomials and their connections to Alexander polynomials and cluster algebras.
result Dimer face polynomials are multivariate generalizations of Alexander polynomials and F-polynomials in cluster algebras. We give polynomial-time algorithms for the exact computation of lowest-energy (ground) states, worst margin violators, log partition functions, and marginal edge probabilities in certain binary undirected graphical models. Our approach provides an interesting alternative to the well-known graph cut paradigm in that it …
New algorithms for efficient inference and sampling in complex Ising models.
problem Efficiently computing partition functions and sampling configurations for complex Ising models.
method Equivalent linear transition to perfect matching counting and sampling on an expanded dual graph.
result Polynomial-time inference and sampling algorithms for K33-free topologies. Holographic principle matches deformed Liouville theory action.
problem Matching deformed Liouville theory action in holography.
method Developed a holographic scheme involving bending energy.
result Perfect match between deformed theory actions on field and gravity sides.
This paper tackles matching two complete graphs with correlated edge weights in geometric models.
problem Matching two complete graphs with edge weights correlated through latent geometries.
method Derives an approximate maximum likelihood estimator for recovering hidden vertex correspondence.
result The estimator provably achieves perfect recovery under certain noise conditions.
Paper tackles graph matching with partially correct seeds, improving performance guarantees.
problem Graph matching with partially correct seeds.
method Proposes algorithms for matching vertices based on 1-hop and 2-hop neighborhoods, analyzing their performance guarantees.
result New 2-hop algorithm requires fewer correct seeds than the 1-hop algorithm, especially for sparse graphs.
Study reconstructs hidden perfect matchings in random graphs with specific edge weights.
problem Reconstructing hidden perfect matchings in random weighted bipartite graphs.
method Analyzes the maximum likelihood estimator for matching reconstruction under different probability distributions of edge weights.
result Sharp threshold and infinite-order phase transition in reconstruction error for different probability distributions.
Analysis of Vlasov plasma dynamics using matched pair Lie-Poisson formulation.
problem Understanding the dynamics of Vlasov plasma and its kinetic moments.
method Hamiltonian (Lie-Poisson) analysis and matched pair decomposition.
result Observation of mutual interactions between subdynamics in Vlasov plasma.
For a perfect Lie algebra h we classify all Lie algebras containing h as a subalgebra of codimension 1. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product h⋉(k∗×AutLie(h)). In the non-…
The paper proposes multicalibration to improve matching in graphs with imperfect predictors.
problem Finding the best matching in graphs with imperfect predictors.
method Introduces multicalibration as a fairness notion to ensure unbiasedness on protected sets of contexts.
result Constructing a multicalibrated predictor that outperforms standard optimal rules in matching algorithms.
The paper explores triangulations on spheres and tori, extending clock theorems.
problem Investigating triangulations of spheres and tori with colored triangles.
method Analyzing matchings between white and black triangles, focusing on their lattices and state transitions.
result Clock theorems extend to spheres but not to tori, with different lattice structures.
We describe a new optimization scheme for finding high-quality correlation clusterings in planar graphs that uses weighted perfect matching as a subroutine. Our method provides lower-bounds on the energy of the optimal correlation clustering that are typically fast to compute and tight in practice. We demonstrate our a…
Matching correlated VAR time series databases by recovering matching permutations.
problem Matching perturbed and permuted correlated VAR time series.
method Probabilistic framework modeling, maximum likelihood estimator (MLE), linear assignment, convex relaxations.
result Recovery guarantees for perfect or partial recovery of matching permutations, thresholds for σ. The paper confirms a conjecture linking link bipyramid volume and Mahler measure.
problem Link bipyramid volume and Mahler measure relationship for alternating links.
method Using isoradial graphs and spanning trees on lattices, the authors confirm the conjecture for two examples and calculate five more.
result The conjecture is confirmed for specific examples of alternating links.
The Penrose-Kauffman polynomial connects knot theory to graph coloring.
problem Understanding the Penrose-Kauffman polynomial for cubic graphs.
method Using knot theory, the polynomial is shown equivalent to 3-coloring link diagrams.
result The Four Color Theorem is linked to 3-coloring link diagrams.
We introduce a novel approach for training adversarial models by replacing the discriminator score with a bi-modal Gaussian distribution over the real/fake indicator variables. In order to do this, we train the Gaussian classifier to match the target bi-modal distribution implicitly through meta-adversarial training. W…
The study examines perfect fluid spacetimes and their properties.
problem Characterizing properties of perfect fluid spacetimes with concircular vector fields.
method Analyzing the conformal curvature tensor, state equation, and solitons in perfect fluid spacetimes.
result Perfect fluid spacetimes with concircular vector fields have specific properties related to the state equation and solitons.
In his seminal 1951 paper "Extreme forms" Coxeter \cite{cox51} observed that for n≥9 one can add vectors to the perfect lattice $\sfA_9$ so that the resulting perfect lattice, called $\sfA_9^2$ by Coxeter, has exactly the same set of minimal vectors. An inhomogeneous analog of the notion of perfect lattice is tha…
The study examines properties of perfect fluid spacetimes in Einstein's theory.
problem Analyzing curvature properties of perfect fluid spacetimes.
method Assuming perfect fluid as the source, the paper investigates solutions to Einstein's field equations.
result Properties of perfect fluid spacetimes are explored in the context of Einstein's theory.
Algorithm decides if pseudo-Anosov flows have perfect fits.
problem Determining if pseudo-Anosov flows have specific asymptotic properties.
method Algorithm based on box decompositions and universal cover analysis.
result Algorithmic decision on pseudo-Anosov flows' perfect fit status.
Paper proves a rigidity result for static perfect fluids.
problem Proving a rigidity result for static perfect fluids.
method Robinson's divergence formula and boundary conditions.
result Rigidity result for static perfect fluids.
Paper introduces ρ-Perfect to estimate model-human correlation in subjective datasets.
problem Inherent noise in subjective ratings limits model-human correlation quantification.
method Defines ρ-Perfect as highest achievable correlation between perfect predictor and human ratings. Estimates based on heteroscedastic noise scenarios. result Demonstrates ρ-Perfect can distinguish model limitations from data quality issues. The notion of a locally continuously perfect group is introduced and studied. This notion generalizes locally smoothly perfect groups introduced by Haller and Teichmann. Next, we prove that the path connected identity component of the group of all homeomorphisms of a manifold is locally continuously perfect. The case o…
Joint matching over a collection of objects aims at aggregating information from a large collection of similar instances (e.g. images, graphs, shapes) to improve maps between pairs of them. Given multiple matches computed between a few object pairs in isolation, the goal is to recover an entire collection of maps that …
Paper proves almost all Gaussian graphical models are perfect.
problem Determining when Gaussian graphical models are perfect.
method Direct approach to Gaussian graphical models, extending Lněnička and Matúš's construction.
result Almost all Gaussian graphical models are perfect.
New conditions for GRW space-times to be perfect-fluid space-times.
problem Conditions for GRW space-times to be perfect-fluid.
method Gray's decomposition of the gradient of the Ricci tensor, determining Ricci tensor forms in invariant subspaces.
result For most GRW space-times, the Ricci tensor is Einstein or perfect fluid.
The paper examines uniform perfectness of diffeomorphism groups on open manifolds.
problem Uniform perfectness of diffeomorphism groups on open manifolds.
method Study of uniform perfectness, boundedness, and simplicity of diffeomorphism groups of compact and open manifolds.
result Obtained upper bounds of diameters for commutator length, balls, and conjugation-generated norm.
Study on static perfect fluid space-time geometry and boundary estimates.
problem Investigate the geometry and boundary properties of static perfect fluid space-time.
method Used generalized Reilly's formula to establish geometric inequalities and boundary estimates.
result Obtained new boundary estimates involving the Brown-York mass and first eigenvalue of the Jacobi operator.
We introduce the concept of hereditarily non uniformly perfect sets, compact sets for which no compact subset is uniformly perfect, and compare them with the following: Hausdorff dimension zero sets, logarithmic capacity zero sets, Lebesgue 2-dimensional measure zero sets, and porous sets. In particular, we give an exa…
Method uses neural networks to solve combinatorial problems.
problem Solving combinatorial problems on raw input data.
method Integrates blackbox combinatorial solvers into neural networks.
result Efficient backward pass through blackbox solvers implemented.
Uniformly perfect Morse boundaries characterize geometric properties of groups.
problem Characterizing geometric properties of groups using Morse boundaries.
method Introducing and geometrically characterizing uniformly perfect Morse boundaries for proper geodesic metric spaces.
result The Morse boundary of any finitely generated, non-elementary group is uniformly perfect if it is nonempty.
Study mapping class groups of infinite type surfaces with noncompact boundaries.
problem Classify pure mapping class groups of infinite type surfaces.
method Developed a method to cut surfaces into simpler ones and combined recent results.
result Complete classification of perfect and uniformly perfect pure mapping class groups.
The property of perfectness plays an important role in the theory of Bayesian networks. First, the existence of perfect distributions for arbitrary sets of variables and directed acyclic graphs implies that various methods for reading independence from the structure of the graph (e.g., Pearl, 1988; Lauritzen, Dawid, La…