arXiv research

Polynomially parametrize interesting knotted surfaces.

We show that every rational knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_a(t), y = T_b(t), z = C(t)$ where $T_k(t)$ are the Chebyshev polynomials, $a=3$ and $b+ °C = 3N.$ We show that every rational knot also admits a polynomial parametrization with $a=4$. If $C (t)= T_c(t)$ is a Chebyshev p…

New algorithm reduces dynamic regret for noisy gradient feedback with piecewise polynomial comparators.

Weierstrass representation is a classical parameterization of minimal surfaces. However, two functions should be specified to construct the parametric form in Weierestrass representation. In this paper, we propose an explicit parametric form for a class of parametric polynomial minimal surfaces of arbitrary degree. It …

We propose an algorithm which allows to derive the generalized Alexander polynomial invariants of knots and links with the help of the q,p-numbers, appearing in bosonic two-parameter quantum algebra. These polynomials turn into HOMFLY ones by applying special parametrization. The Jones polynomials can be also obtained …

Paper connects AJ conjecture and colored Jones polynomial potential function.

In this paper we develop the theory of parametric polynomial regression in Riemannian manifolds and Lie groups. We show application of Riemannian polynomial regression to shape analysis in Kendall shape space. Results are presented, showing the power of polynomial regression on the classic rat skull growth data of Book…

Gradient descent learns over-param neural nets better than NTK.

Polynomial chaos surrogates handle intrinsic noise in stochastic models.

We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras $\mathfrak{cga}_\ell(d,{\mathbb C})$ with $d=1$ for any integer value $\ell \in \mathbb{N}$. The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type, and id…

We prove that the coefficients of the colored Jones polynomial of alternating links stabilize under increasing the number of twists in the twist regions of the link diagram. This gives us an infinite family of $q$-power series derived from the colored Jones polynomial parametrized by the color and the twist regions of …

A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich $(g+1)$-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformatio…

Study algebraic relations of Vassiliev invariants for families of knots.

Model inference for dynamical systems aims to estimate the future behaviour of a system from observations. Purely model-free statistical methods, such as Artificial Neural Networks, tend to perform poorly for such tasks. They are therefore not well suited to many questions from applications, for example in Bayesian fil…

If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily altern…

In many applications (in particular information systems, such as pattern recognition, machine learning, cheminformatics, bioinformatics to name but a few) the assessment of uncertainty is essential - i.e., the estimation of the underlying probability distribution function. More often than not, the form of this function…

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 paper is devoted to quadratic Poisson structures compatible with the canonical linear Poisson structures on trivial 1-dimensional central extensions of semisimple Lie algebras. In particular, we develop the general theory of such structures and study related families of functions in involution. We also show that th…

We show that every two-bridge knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_3(t), y = T_b(t), z =C(t)$ where $T_k(t)$ are the Chebyshev polynomials and $b+°C = 3N$. If $C (t)= T_c(t)$ is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots …

Study on colored Jones polynomial of figure-eight knot for complex parameters.

New method detects projective equivalences and symmetries in rational 3D curves.

Proof of isotopy invariance in Khovanov link homology.

This paper investigates gradient recovery schemes for data defined on discretized manifolds. The proposed method, parametric polynomial preserving recovery (PPPR), does not require the tangent spaces of the exact manifolds, and they have been assumed for some significant gradient recovery methods in the literature. Ano…

This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la arXiv:1506.00339, (ii) evaluating Racah/mixing matrices for various numbers of strands in var…

A Chebyshev curve C(a,b,c,φ) has a parametrization of the form x(t)=Ta(t); y(t)=T_b(t) ; z(t)= Tc(t + φ), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ\in \RR. When C(a,b,c,φ) has no double points, it defines a polynomial knot. We determine all possible knots when a, b and c are given.

In this note we explore a connection between finite covers of surfaces and the Teichmüller polynomial of a fibered face of a hyperbolic 3--manifold. We consider the action of a homological pseudo-Anosov homeomorphism $ψ$ on the homology groups of a class of finite abelian covers of a surface $Σ_{g,n}$. Eigenspaces of t…

Adaptive gradient methods like AdaGrad are widely used in optimizing neural networks. Yet, existing convergence guarantees for adaptive gradient methods require either convexity or smoothness, and, in the smooth setting, only guarantee convergence to a stationary point. We propose an adaptive gradient method and show t…

Polynomial-time algorithm learns causal graphs without parametric assumptions.

Study the expressivity and training complexity of polynomial neural networks.

Using the recently proposed differential hierarchy (Z-expansion) technique, we obtain a general expression for the HOMFLY polynomials in two arbitrary symmetric representations of link families, including Whitehead and Borromean links. Among other things, this allows us to check and confirm the recent conjecture of arX…

We present a class of models that, via a simple construction, enables exact, incremental, non-parametric, polynomial-time, Bayesian inference of conditional measures. The approach relies upon creating a sequence of covers on the conditioning variable and maintaining a different model for each set within a cover. Infere…

Paper improves SDR estimation speed and conditions.

ParamBoost uses gradient boosting to create interpretable non-linear models with constraints.

New results show neural networks generalize well due to polynomial regression, not just overparametrization.

In this paper, we treat the problem of evaluating the asymptotic error in a numerical integration scheme as one with inherent uncertainty. Adding to the growing field of probabilistic numerics, we show that Gaussian process regression (GPR) can be embedded into a numerical integration scheme to allow for (i) robust sel…

Paper introduces a new, tractable measure of model complexity.

We obtain defining equations of the smooth equivariant compactification of the Grassmannian of the complex associative $3$-planes in $\C^7$, which is the parametrizing variety of all quaternionic subalgebras of the algebra of complex octonions $\OO\cong \C^8$. By studying the torus fixed points, we compute the Poincaré…

Using nonparametric methods has been increasingly explored in Bayesian hierarchical modeling as a way to increase model flexibility. Although the field shows a lot of promise, inference in many models, including Hierachical Dirichlet Processes (HDP), remain prohibitively slow. One promising path forward is to exploit t…

Active learning recovers choice model from noisy data.

Develops a nonparametric method to estimate isotropic covariance functions efficiently.

A famous result of Bennequin states that for any braid representative of the unknot the Bennequin number is negative. We will extend this result to all n-trivial closed n-braids. This is a class of infinitely many knots closed under taking mirror images. Our proof relies on a non-standard parametrization of the Homfly …

Estimates hybrid dynamical systems with polynomial expansions and Markovian switching.

Estimates classifier errors without ground truth using algebraic geometry.

The paper proposes a method to learn the structure of continuous-action games with non-parametric utilities using a limited number of samples.

The paper proposes a new method for probabilistic load forecasting using Bernstein-Polynomial Normalizing Flows.

We investigate solutions of the elliptic sinh-Gordon equation of spectral genus g<3. These solutions are parametrized by complex matrix-valued polynomials called potentials. On the space of these potentials there act two commuting flows. The orbits of these flows are called Polynomial Killing fields and are double peri…

We utilize copulas to constitute a unified framework for constructing and optimizing variational proposals in hierarchical Bayesian models. For models with continuous and non-Gaussian hidden variables, we propose a semiparametric and automated variational Gaussian copula approach, in which the parametric Gaussian copul…

Machine learning uses invariant theory to restrict function classes.