arXiv research

FairACE improves fairness in GNNs by balancing node performance across degree groups.

The degree-$d$ Chow parameters of a Boolean function $f: \{-1,1\}^n \to \mathbb{R}$ are its degree at most $d$ Fourier coefficients. It is well-known that degree-$d$ Chow parameters uniquely characterize degree-$d$ polynomial threshold functions (PTFs) within the space of all bounded functions. In this paper, we prove …

Extends graph degree theorem to simplicial closure of Auter space.

This paper finds all prime alternating knots with minimal warping degree two.

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…

Study Kazdan-Warner equations on graphs using Brouwer degree theory.

For harmonic maps of degree 2, a similar quantitative stability estimate does not hold uniformly.

The study finds lower bounds for the warping degree of a knot projection.

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…

New formula recovers degree of colored Jones polynomials for pretzel knots.

Christoffel function characterizes the corruption a bounded-degree certificate cannot remove in robust halfspace learning.

Research examines curves of degree 8 with specific singularities.

Low-degree method fails to predict robust subspace recovery problem.

The study classifies graphs with specific curvature and maximum degree.

The paper corrects for node degree in spectral clustering using random walk Laplacian.

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.

The paper calculates the slicing degree of knots using advanced homology theories.

Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.

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…

The study explores mapping degree sets and their properties for manifolds.

In [3] Borzellino and Brunsden started to develop an elementary differential topology theory for orbifolds. In this paper we carry on their project by defining a mapping degree for proper maps between orbifolds, which counts preimages of regular values with appropriate weights. We show that the mapping degree satisfies…

A new simple proof for surface map degree inequality.

Simply-connected surfaces of general type for n≥5.

Survey on using low-degree polynomials to assess statistical tasks complexity.

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 $M\times N$ admits a self-map of degree $d$, when neither $D(M)$ nor $D(N)$ contains $d$. We find sufficient conditions so that $D(M\times N)$ contains e…

We calculate Euclidean distance degrees for common manifold optimization types.

Upper bounds on map degrees for various 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…

Maps between surfaces have degree constraints based on their Euler characteristics.

In this paper we study rational real algebraic knots in $\R P^3$. We show that two real algebraic knots of degree $\leq5$ 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…

Study on knots formed by gluing ellipses, defining gluing degree.

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.

Study local sensitivity of HDD and CDD 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…

New $\varepsilon$-harmonic maps of low degree are rigid under certain energy bounds.

New findings on computational limits for estimating hidden structures.

We study the degree of polynomial representations of knots. We obtain the lexicographic degree for two-bridge torus knots and generalized twist knots. The proof uses the braid theoretical method developed by Orevkov to study real plane curves, combined with previous results from [KP10] and [BKP14]. We also give a sharp…

The Schwarzian derivative helps classify minimal surfaces by their degree.

In this paper, using exclusively homotopy theoretical methods, we study degrees of maps between $(n-2)$-connected $(2n-1)$-dimensional Poincar\' e complexes which have torsion free integral homology. Necessary and sufficient algebraic conditions for the existence of map degrees between such Poincar\' e complexes are es…

We study the degree of polynomial representations of knots. We give the lexicographic degree of all two-bridge knots with 11 or fewer crossings. First, we estimate the total degree of a lexicographic parametrisation of such a knot. This allows us to transform this problem into a study of real algebraic trigonal plane c…

Paper solves whether zero sets are mapping degree sets.

We develop the theory of equivariant harmonic self-maps of compact cohomogeneity one manifolds and construct new harmonic self-maps of the compact Lie groups SO(4L+2), L >= 1, with degree -3, of SO(8), SO(14) and SO(26) with degree -5 each, of SO(10) with degree -7, and of SO(14) with degree -11 by exhibiting linear so…

Proves rigidity for maps between manifolds using degree theory and current developments.

The degree of certain holomorphic 2-spheres is bounded.