We generalize the classical study of Alexander polynomials of smooth or PL locally-flat knots to PL knots that are not necessarily locally-flat. We introduce three families of generalized Alexander polynomials and study their properties. For knots with point singularities, we obtain a classification of these polynomial…
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48 results for “polynomial classification”
Complete classification of knotoids up to seven crossings.
problem Classifying spherical knotoids up to a certain number of crossings.
method Enumerating diagrams, simplifying them, distinguishing equivalence classes using various invariants.
result Conjecture that classification up to seven crossings is complete.
We give the classification, up to homeomorphisms, of reduced complex polynomials with 2 variables with one critical value.
This paper describes a novel method to approximate the polynomial coefficients of regression functions, with particular interest on multi-dimensional classification. The derivation is simple, and offers a fast, robust classification technique that is resistant to over-fitting.
New upper bound on Jones polynomial for fibered positive links.
problem Classifying positive and non-positive knots of crossing number ≤ 12.
method Proved a new upper bound on the maximum degree of Jones polynomial for fibered positive knots.
result Maximum degree of Jones polynomial for fibered positive knots is at most four times the minimum degree.
Paper revisits graph-CNNs using Laplace-Beltrami spectral filters and polynomials.
problem Improving spectral graph convolutional neural networks (graph-CNNs).
method Developed Laplace-Beltrami CNN (LB-CNN) by replacing graph Laplacian with LB operator and approximating spectral filters using Chebyshev, Laguerre, and Hermite polynomials.
result Classification accuracy of LB-CNN is not dependent on the type of polynomials or operators.
FPC tackles big data challenges with polynomial kernels and ADMM.
problem Efficiently classify massive data with scalability and storage challenges.
method Polynomial feature mapping and ADMM for non-smooth convex optimization.
result FPC significantly reduces computational burden and storage memory without sacrificing generalization ability.
The vanishing ideal is a set of polynomials that takes zero value on the given data points. Originally proposed in computer algebra, the vanishing ideal has been recently exploited for extracting the nonlinear structures of data in many applications. To avoid overfitting to noisy data, the polynomials are often designe…
New approach for classification using trigonometric polynomial kernels from signal processing.
problem Classifying objects in arbitrary compact metric spaces.
method Localized trigonometric polynomial kernels for separating different probability measures.
result Minimal number of queried labels for perfect classification.
In this short note we show the existence of an epimorphism between groups of -bridge knots by means of an elementary argument using the Riley polynomial. As a corollary, we give a classification of -bridge knots by Riley polynomials.
Study bi-Lipschitz equivalence of mixed polynomials under specific conditions.
problem Classify bi-Lipschitz equivalence of mixed polynomials with inner non-degeneracy.
method Defined metric links and introduced new data to determine bi-Lipschitz equivalence.
result Neither Newton boundary nor C-face diagram is an invariant for bi-Lipschitz equivalence.
We generalized the periodic links to \emph{transitive} links in a -manifold . We find a complete classification theorem of transitive links in a -dimensional sphere . We study these links from several different aspects including polynomial invariants using the relation between link polynomials of…
The paper classifies pretzel links with 2 components and gives conditions for those with 3 or more.
problem Classifying pretzel links based on their self delta-equivalence.
method Using Conway polynomials to determine self delta-equivalence for links with 2 or more components.
result Necessary and sufficient conditions for self delta-equivalence of pretzel links with 3 or more components.
Classifies small links in an unmarked solid torus.
problem Classifying knots and links in an unmarked solid torus.
method Invariants and Dehn twists to detect and transform links.
result Classification of all non-split links up to 6 crossings.
A method for classifying points with minimal queries using Hermite polynomials.
problem Classifying points from an unknown probability measure with minimal label queries.
method Convex combination of conditional probabilities, Hermite polynomial kernel for hierarchical support estimation.
result The method achieves high -score for classification in hyper-spectral images and MNIST.
Study on knot classification using 3-braid closures and ribbon surfaces.
problem Classifying smoothly slice knots from 3-braid closures.
method Construct ribbon surfaces and use twisted Alexander polynomial.
result Classification of knots up to 20 crossings.
Statistical Query Hardness of Multiclass Linear Classification with Random Classification Noisecs.LG
Study shows SQ hardness for multiclass linear classification with random noise.
problem Complexity of multiclass linear classification with random noise.
method Proves super-polynomial SQ lower bounds for MLC with RCN.
result Super-polynomial SQ lower bounds for MLC with RCN.
BPR matches NN accuracy in crop classification while being more transparent.
problem Lack of auditability and alignment with domain knowledge in neural networks for high-dimensional climate data.
method Bagged polynomial regression with random projections (BPR), averaging many low-degree polynomial models.
result BPR matches neural networks in accuracy but is more transparent.
We show that every rational knot of crossing number admits a polynomial parametrization where are the Chebyshev polynomials, and We show that every rational knot also admits a polynomial parametrization with . If is a Chebyshev p…
Popular graph neural networks implement convolution operations on graphs based on polynomial spectral filters. In this paper, we propose a novel graph convolutional layer inspired by the auto-regressive moving average (ARMA) filter that, compared to polynomial ones, provides a more flexible frequency response, is more …
The paper characterizes biharmonic maps between spheres using polynomial functions.
problem Characterizing biharmonic maps between spheres using polynomial functions.
method Proved a characterization formula and constructed biharmonic maps.
result Classification of all proper biharmonic quadratic forms from spheres.
Formula for arborescent link tails using theta functions.
problem Understanding tails of colored Jones polynomials for arborescent links.
method Explicit formula using theta and false theta functions.
result Numerical evidence for modularity of tails for alternating knots.
Polynomial term structure modelsq-fin.MF
In this article, we explore a class of tractable interest rate models that have the property that the price of a zero-coupon bond can be expressed as a polynomial of a state diffusion process. Our results include a classification of all such time-homogeneous single-factor models in the spirit of Filipovic's maximal deg…
Kernel discriminant analysis uses nonlinear embeddings to improve classification.
problem Limited effectiveness of linear discriminant analysis in capturing nonlinear features.
method Study of nonlinear embeddings in kernel discriminant analysis using polynomial and Gaussian kernels, solving generalized eigenvalue problems.
result Polynomial and Gaussian discriminants capture class differences through population moments and randomized projections.
Classifies matrices in the quaternionic hyperbolic unitary group.
problem Understanding the structure of matrices in the quaternionic hyperbolic unitary group.
method Used complex representation and characteristic polynomial to study matrices.
result Computed the characteristic polynomial and studied its sign.
We give a concise proof of a classification of lens spaces up to orientation-preserving homeomorphisms. The chief ingredient in our proof is a study of the Alexander polynomial of ` symmetric' links in .
Classifies 3D non-degenerate left-symmetric algebras.
problem Classifying left-symmetric algebras in 3D.
method Using Nijenhuis geometry and algebraic independence of coefficients in characteristic polynomial.
result Classification of differentially non-degenerate LSA in dimension 3.
New quantum invariants for planar knotoids improve knot classification.
problem Classifying and distinguishing planar knotoids with up to five crossings.
method Define biframed planar knotoids and construct new invariants.
result Improved classification of planar knotoids with up to five crossings.
The study classifies Heintze groups up to isometry and quasi-isometry in low dimensions.
problem Classifying Heintze groups up to isometry and quasi-isometry in low dimensions.
method Analyzing quasi-isometries and isometries of Heintze groups, applying existing tools to groups of dimension 4 and 5.
result Complete classification of simply connected solvable groups in dimension 4 and groups of polynomial growth in dimension 5 up to isometry.
Classifies uncolored bonded knots with up to 7 singularity points.
problem Classifying uncolored bonded knots with up to 7 singularity points.
method Generation of planar graphs, conversion into bonded knot diagrams, use of Yamada polynomial, and brute-force Reidemeister moves.
result Systematic classification of uncolored bonded knots with singularity number at most seven.
Classifies connected shelves up to order six.
problem Classifying finite right-distributive binary algebraic structures called shelves.
method Symbolic computations with Python to classify shelves up to isomorphism, exploring group structure, and defining shelf polynomials.
result Classified all connected shelves with order less than six up to isomorphism.
We show that every two-bridge knot of crossing number admits a polynomial parametrization where are the Chebyshev polynomials and . If is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots …
Study slice-regular polynomial functions via twistor space group actions.
problem Characterize slice-regular functions and their polynomial subclasses.
method Employ the twistor construction and group actions of .
result Characterize slice-regular functions with planar twistor lifts and normal classes of polynomials.
Tropical division approximates polynomial division for neural networks.
problem Approximating polynomial division in max-plus semiring.
method Approximating Newton Polytope of dividend by divisor, then applying to neural networks.
result Minimizes a two-layer fully connected network for binary classification.
Piecewise regression mixture for simultaneous functional data clustering and optimal segmentationstat.ME
This paper introduces a novel mixture model-based approach for simultaneous clustering and optimal segmentation of functional data which are curves presenting regime changes. The proposed model consists in a finite mixture of piecewise polynomial regression models. Each piecewise polynomial regression model is associat…
We continue the study of the genus of knot diagrams, deriving a new description of generators using Hirasawa's algorithm. This description leads to good estimates on the maximal number of crossings of generators and allows us to complete their classification for knots of genus 4. As applications of the genus 4 classifi…
Algorithm classifies surface homeomorphisms with polynomial time complexity.
problem Classifying surface homeomorphisms with polynomial time complexity.
method Algorithm to compute curve distances and decide Nielsen-Thurston types.
result Polynomial time classification of surface homeomorphisms.
Polynomial invariants classify molecular chains based on their contact arrangements.
problem No established invariants for molecular chains with both hard and soft contacts.
method Developed polynomial invariants for circuit topology of molecular chains.
result Polynomial invariants efficiently classify chains with various contact types.
Factorization machines and polynomial networks are supervised polynomial models based on an efficient low-rank decomposition. We extend these models to the multi-output setting, i.e., for learning vector-valued functions, with application to multi-class or multi-task problems. We cast this as the problem of learning a …
Machine learning identifies boundaries of real solutions in polynomial systems.
problem Locating boundaries in parameter space for real solutions of polynomial systems.
method Supervised machine learning approach using nearest neighbor and deep learning approximations.
result Efficiently approximates the real discriminant locus for multidimensional parameter spaces.
The study classifies rational isoparametric functions on Damek-Ricci spaces.
problem Classifying isoparametric functions on Damek-Ricci spaces.
method Using polynomial functions divided by for classification.
result New isoparametric functions discovered and their properties studied.
Strongly polynomial algorithm for approximate Forster transforms and halfspace learning.
problem Computing approximate Forster transforms and halfspace learning.
method Strongly polynomial time algorithm for approximate Forster transforms and halfspace learning.
result First strongly polynomial time algorithm for distribution-free PAC learning of halfspaces.
The study classifies geometrically finite polynomials on the boundary of Blaschke products.
problem Understanding the boundaries of hyperbolic components of Blaschke products.
method Combinatorial classification and construction of self-bumps.
result The closure of the main hyperbolic component is not a topological manifold with boundary for .
Neural networks approximate and estimate binary classifiers with polynomial input dependence.
problem Approximating and estimating binary classification functions with neural networks in high dimensions.
method ReLU neural networks, empirical risk minimization, Barron class.
result Approximation and estimation rates are independent of input dimension, overcoming curse of dimensionality.
The study eliminates infinite families of knots with nontrivial Alexander polynomials and improves unknotting number data.
problem Identifying knots with nontrivial Alexander polynomials and improving knot classification.
method Elimination of infinite families of knots and use of determinants to improve unknotting number data.
result Elimination of infinite families of knots with nontrivial Alexander polynomials and improvement of unknotting number data.
Study of panhandle polynomials of torus links with geometric applications.
problem Characterizing the HOMFLY-PT polynomial of torus knots and links.
method Utilizing quantum group representations and the Rosso-Jones formula.
result Established panhandle-like structure of HOMFLY-PT polynomials for torus knots and links.
We generalize a classification result for self-shrinkers of the mean curvature flow with nonnegative mean curvature, which was obtained by T. Colding and W. Minicozzi, replacing the assumption on polynomial volume growth with a weighted condition on the norm of the second fundamental form. Our approach adopt the …
The paper reformulates an invariant and calculates it for lens spaces.
problem Classifying lens spaces using an invariant.
method Reformulation of an invariant and calculation for lens spaces.
result The invariant is calculated for lens spaces.