FairACE improves fairness in GNNs by balancing node performance across degree groups.
problem Degree biases in GNNs lead to unequal prediction performance among nodes with varying degrees.
method Integrates asymmetric contrastive learning with adversarial training to balance performance between high-degree and low-degree nodes.
result Significantly improves degree fairness metrics while maintaining competitive accuracy.
Develops degree theory for orbifolds, a generalization of differential topology.
problem No suitable problem statement as the abstract focuses on the development of theory.
method Defined a mapping degree for proper maps between orbifolds, satisfying invariance properties.
result The mapping degree counts preimages of regular values with appropriate weights.
Extends graph degree theorem to simplicial closure of Auter space.
problem Connectivity of graphs in Auter space.
method Defines degree for simplicial closure, extends Hatcher-Vogtmann theorem.
result Simplicial closure of Auter space is (d-1)-connected for degree d.
This paper finds all prime alternating knots with minimal warping degree two.
problem Finding knots with minimal warping degree.
method Examined all prime alternating knots and determined those with minimal warping degree two.
result All prime alternating knots with minimal warping degree two were identified.
Paper shows how to identify and reconstruct degree-d PTFs robustly from their Fourier coefficients.
problem Identifying and reconstructing degree-d polynomial threshold functions (PTFs) from their Fourier coefficients.
method Proves a robust version of the theorem that degree-d Chow parameters uniquely characterize degree-d PTFs, and uses this to develop efficient algorithms.
result Boolean degree-d PTFs are robustly identifiable from their degree-d Chow parameters.
The stochastic block model is a powerful tool for inferring community structure from network topology. However, it predicts a Poisson degree distribution within each community, while most real-world networks have a heavy-tailed degree distribution. The degree-corrected block model can accommodate arbitrary degree distr…
For harmonic maps of degree 2, a similar quantitative stability estimate does not hold uniformly.
problem Investigate the quantitative stability of harmonic maps of degree 2.
method Prove a local quantitative stability result for harmonic maps of degree 2, showing dependence on the given harmonic map.
result A uniformly quantitative stability estimate does not hold for degree 2 harmonic maps.
Study Kazdan-Warner equations on graphs using Brouwer degree theory.
problem Proving existence of solutions to Kazdan-Warner equations on finite graphs.
method Degree theory approach to uniformly bound and compute Brouwer degree.
result New proofs of existence results for Kazdan-Warner equations.
In Stochastic blockmodels, which are among the most prominent statistical models for cluster analysis of complex networks, clusters are defined as groups of nodes with statistically similar link probabilities within and between groups. A recent extension by Karrer and Newman incorporates a node degree correction to mod…
The study finds lower bounds for the warping degree of a knot projection.
problem Determining the warping degree of a knot projection.
method Examining the maximal number of regions sharing no crossings for a fixed crossing in a knot projection.
result Lower bounds for the warping degree of a knot projection are provided.
New formula recovers degree of colored Jones polynomials for pretzel knots.
problem Determining the degree of colored Jones polynomials for specific knots.
method Alternate expansion of the colored Jones polynomial for pretzel links, focusing on 3-tangle knots.
result Determined the degrees of the colored Jones polynomials for a new family of 3-tangle pretzel knots.
Christoffel function characterizes the corruption a bounded-degree certificate cannot remove in robust halfspace learning.
problem Robust halfspace learning under malicious noise
method Sum-of-Squares degree of outlier-removal certificate
result Christoffel function bounds the corruption a bounded-degree certificate cannot remove
Research examines curves of degree 8 with specific singularities.
problem Existence of curves with prescribed singularities.
method Algebraic and symplectic approaches.
result Characterization of curves with specific singularities.
Low-degree method fails to predict robust subspace recovery problem.
problem Predicting computational tractability of robust subspace recovery problem.
method Low-degree polynomial framework, anti-concentration properties.
result Low-degree method fails to predict computational tractability of robust subspace recovery problem even up to high degree.
The study classifies graphs with specific curvature and maximum degree.
problem Graphs with nonnegative Ricci curvature and maximum degree constraints.
method Classification of graphs with Lin-Lu-Yau-Ollivier Ricci curvature, maximum degree ≤ 3, and diameter ≥ 6.
result Classification of graphs meeting the specified criteria.
The paper corrects for node degree in spectral clustering using random walk Laplacian.
problem Node degree heterogeneity in spectral clustering.
method Graph spectral embedding using the random walk Laplacian.
result The embedding provides uniformly consistent estimates of degree-corrected latent positions.
We define and study the statistical models in exponential family form whose sufficient statistics are the degree distributions and the bi-degree distributions of undirected labelled simple graphs. Graphs that are constrained by the joint degree distributions are called dK-graphs in the computer science literature and…
GCNs favor high-degree nodes, leading to biased performance; a new method mitigates this.
problem Degree-related biases in GCNs, especially for low-degree nodes.
method Developed a novel SL-DSGC that reduces model and data biases.
result SL-DSGC improves GCN accuracy significantly for low-degree nodes.
The paper calculates the slicing degree of knots using advanced homology theories.
problem Determining the minimum slicing degree of knots.
method Rasmussen's s-invariant, knot Floer homology, and singular instanton homology.
result Computed slicing degrees for many small knots and some families of torus knots.
A new model corrects SBM's bias for power-law degree networks.
problem SBM's incapability to handle power-law degree distributions.
method Introducing degree decay variables to encode varying degree distributions.
result PLD-SBM approximately preserves the scale-free feature in real networks and corrects SBM's bias.
The problem of finding all minimal surfaces presented in parametric form as polynomials of certain degree is discussed by many authors. It is known that the classical Enneper surface is (up to position in space and homothety) the only polynomial minimal surface of degree 3 in isothermal parameters. In higher degrees th…
Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.
problem Understanding harmonic maps between singular spaces.
method Analyzing homogeneous harmonic maps between simplicial cones and their degrees.
result Degrees of homogeneous harmonic maps are related to eigenvalues of discrete graph Laplacians.
The study explores mapping degree sets and their properties for manifolds.
problem Understanding the structure and properties of mapping degree sets for manifolds.
method Analyzes the properties of mapping degree sets and their relationships with self-mapping degree sets.
result Not every multiplicative set containing 0,1 is a self-mapping degree set.
A new simple proof for surface map degree inequality.
problem Degree of maps between closed surfaces.
method Elementary proof without additional techniques.
result A new proof of the inequality χ(M) ≤ d·χ(N).
Survey on using low-degree polynomials to assess statistical tasks complexity.
problem Understanding the complexity of statistical tasks using polynomial functions.
method Applying low-degree polynomials to measure the complexity of statistical tasks, including detection, recovery, and estimation.
result Low-degree polynomials provide a framework to predict and explain statistical-computational tradeoffs.
Simply-connected surfaces of general type for n≥5.
problem Topological structures of Galois covers of surfaces of minimal degree.
method Investigation of Galois covers of surfaces of minimal degree in complex projective space.
result Galois covers of surfaces of minimal degree are simply-connected for n≥5.
For ordinary knots in R3, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant …
Every closed oriented manifold M is associated with a set of integers D(M), the set of self-mapping degrees of M. In this paper we investigate whether a product M×N admits a self-map of degree d, when neither D(M) nor D(N) contains d. We find sufficient conditions so that D(M×N) contains e…
We calculate Euclidean distance degrees for common manifold optimization types.
problem Optimizing on manifold structures.
method Closed-form expressions for stationary points of Euclidean distance function.
result Closed-form expressions for all stationary points on manifold optimization.
Upper bounds on map degrees for various manifold types.
problem Understanding the maximum degree of maps between different types of manifolds.
method Analyzing Lipschitz maps and dividing manifolds into topological types.
result New upper bounds on map degrees for different manifold types.
In this paper, we explore degrees of freedom in deep sigmoidal neural networks. We show that the degrees of freedom in these models is related to the expected optimism, which is the expected difference between test error and training error. We provide an efficient Monte-Carlo method to estimate the degrees of freedom f…
In this paper, we give the sharp estimates for the degree of symmetry and the semi-simple degree of symmetry of certain four dimensional fiber bundles by virtue of the rigidity theorem of harmonic maps due to Schoen and Yau. As a corollary of this estimate, we compute the degree of symmetry and the semi-simple degree o…
14 homogeneous convex foliations of degree 5 found on complex projective plane.
problem Classifying homogeneous convex foliations of degree 5 on the complex projective plane.
method Analyzing properties of specific foliations and using automorphisms of PC2 to establish results. result Every reduced convex foliation of degree 5 is linearly conjugated to one of two specific foliations.
Maps between surfaces have degree constraints based on their Euler characteristics.
problem Constraints on the degree of maps between surfaces based on their Euler characteristics.
method Used the Kneser-Edmonds factorization theorem and provided a simple proof.
result Maps between surfaces have degree constraints based on their Euler characteristics.
In this paper we study rational real algebraic knots in RP3. We show that two real algebraic knots of degree ≤5 are rigidly isotopic if and only if their degrees and encomplexed writhes are equal. We also show that any irreducible smooth knot which admits a plane projection with less than or equal to four cro…
We show any set of degrees can be realized by manifolds.
problem Realizing any set of mapping degrees between manifolds.
method Using aspherical manifolds as building blocks.
result Any finite set containing zero is a mapping degree set between manifolds.
New degree theory for perturbed self-adjoint operators.
problem Equivariant gradient perturbations of unbounded self-adjoint operators.
method Equivariant gradient degree for perturbations of self-adjoint operators with discrete spectrum.
result Definition and application of new degree theory.
New findings show mapping degree sets can be realized as sums of arithmetic progressions.
problem Realizing mapping degree sets as sums of arithmetic progressions.
method Using multiplication by finite subsets of Z and sets from additive submonoids of Z.
result Every set of the described type occurs as a mapping degree set.
Study on knots formed by gluing ellipses, defining gluing degree.
problem Properties of glued knots.
method Defined gluing degree to relate to knot properties.
result Classified all knots up to gluing degree 6.
Classifies Killing forms of arbitrary degree on specific nilpotent Lie groups.
problem Classifying Killing forms of arbitrary degree on specific Lie groups.
method Analyzing left-invariant Killing forms on simply connected 2-step nilpotent Lie groups with left-invariant metrics.
result Classified Killing forms when center is at most 2-dimensional.
In this paper, it is shown that every orientable closed 3-manifold maps with nonzero degree onto at most finitely many homeomorphically distinct irreducible non-geometric orientable closed 3-manifolds. Moreover, given any nonzero integer, as a mapping degree up to sign, every orientable closed 3-manifold maps with that…
Polynomial neural networks explore thresholds for maximum expressiveness.
problem Understanding the limits of polynomial neural networks' expressiveness.
method Introducing activation degree threshold to measure network expressiveness and proving its existence and upper bounds.
result Polynomial neural networks with equi-width architectures achieve the maximum expressiveness.
Study local sensitivity of HDD and CDD temperature derivatives prices.
problem Understanding how temperature derivatives prices change with small temperature changes.
method Analyzes sensitivity of HDD and CDD futures and options prices to temperature perturbations using a CAR process.
result Identifies the order of the CAR process and its impact on temperature derivatives prices.
The G-degree of colored graphs is a key concept in the approach to Quantum Gravity via tensor models. The present paper studies the properties of the G-degree for the large class of graphs representing singular manifolds (including closed PL manifolds). In particular, the complete topological classification up to G-deg…
Study shows surfaces with many twistor lines can't be odd-degree.
problem Characterizing algebraic surfaces with infinitely many twistor lines.
method Utilized quaternionic slice regularity and normalization map theory.
result Constructive existence of even-degree surfaces with infinitely many twistor lines.
New ε-harmonic maps of low degree are rigid under certain energy bounds.
problem Understanding the rigidity of ε-harmonic maps of low degree. method Analysis of ε-harmonic maps and their critical points. result Non-trivial ε-harmonic maps of degree zero exist with energy above 8π. New findings on computational limits for estimating hidden structures.
problem Estimating hidden structures in noisy data.
method Use of low-degree polynomials as a restricted model of computation.
result Established low-degree hardness of recovery problems for easy detection problems.
The Schwarzian derivative helps classify minimal surfaces by their degree.
problem Classifying minimal surfaces based on their geometric properties.
method Using the Schwarzian derivative, constructing sequences of meromorphic differentials.
result Minimal surfaces can be approximated by sequences of increasing degree.