Approximate vanishing ideal is a concept from computer algebra that studies the algebraic varieties behind perturbed data points. To capture the nonlinear structure of perturbed points, the introduction of approximation to exact vanishing ideals plays a critical role. However, such an approximation also gives rise to a…
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48 results for “basis construction”
New basis confirms Thurston's conjecture and reveals knot configurations.
problem Understanding cluster algebras and their bases from surfaces.
method Topological construction of band basis and comparison with Kazhdan-Lusztig type basis.
result Common triangular basis matches band basis in quantum cluster algebras.
Gradient boosts monomial-order-free basis construction algorithms.
problem Lack of theoretical properties in monomial-order-free basis construction algorithms.
method Exploits gradient to sidestep spurious vanishing, achieve consistent output, and remove redundant bases.
result Proposes methods that equip monomial-order-free algorithms with theoretical properties.
Springer varieties appear in both geometric representation theory and knot theory. Motivated by knot theory and categorification Khovanov provides a topological construction of Springer varieties. We extend Khovanov's construction to all two-row Springer varieties. Using the combinatorial and diagrammatic …
Optimizes basis for density-based atomic representations to enhance compactness and accuracy.
problem Improving the efficiency and accuracy of machine learning models for atomic properties.
method An unsupervised approach to determine the optimal basis set for atom density representations using splines.
result Optimal basis sets that encode structural information more compactly and accurately.
We construct a canonical basis of two-cycles, on a surface, in which the intersection form takes the canonical form . The basic elements are realized by formal sums of smooth submanifolds.
A new kernel improves statistical surrogates for stochastic manifolds with diverse data.
problem Handling statistical surrogates for stochastic manifolds with heterogeneous data.
method A transient anisotropic kernel is introduced to improve statistical surrogates for stochastic manifolds with heterogeneous data.
result The transient anisotropic kernel provides a better representation of statistical dependencies in the learned probability measure.
The study explores various localized bases and their duals for scattered data approximation.
problem Scattered data approximation using radial basis functions.
method Examines different localized bases including Lagrange, Newton, and multiresolution versions, and their duals.
result Localized orthogonal bases, such as the Newton basis, offer symmetric preconditioners and are feasible for scattered data approximation.
Constructs manifolds from quantum codes with novel geometric properties.
problem Creating manifolds with specific geometric constraints.
method Reverse engineering manifolds from quantum code chain complexes.
result First examples of power law systolic freedom.
We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to : the {\em reduced web} basis associated to Kuperberg's combinatorial description of the spider category; and the {\em left cell basis} for the left cell construction of Kazhdan and Lusztig. I…
Derives representations invariant under crystallographic groups for functions.
problem Representing and learning functions invariant under crystallographic groups.
method Derives linear and nonlinear representations of functions invariant under crystallographic groups.
result Derives orthonormal crystallographically invariant basis functions and embedding maps.
Nonnegative matrix factorization (NMF) is a widely used linear dimensionality reduction technique for nonnegative data. NMF requires that each data point is approximated by a convex combination of basis elements. Archetypal analysis (AA), also referred to as convex NMF, is a well-known NMF variant imposing that the bas…
We construct a new inductive basis of the Birman-Murakami-Wenzl algebra. Using it, we provide a new proof of the existence of the Markov trace on the BMW algebras affording the two-variable Kauffman polynomial. We prove also that all the transverse Markov traces on the BMW algebras are determined by the self-linking nu…
Recently there has been renewed interest in the mapping-class group of a compact surface of genus and also in its finite order elements. A finite order element of the mapping-class group will be a conformal automorphisms on some Riemann surface of genus . Here we give the details of the proof that there is…
-non-degenerate spaces are spacetimes that can be characterized uniquely by their scalar curvature invariants. The ultimate goal of the current work is to construct a basis for the scalar polynomial curvature invariants in three dimensional Lorentzian spacetimes. In particular, we seek a minimal set of alg…
Optimizing Basis Function Selection in Constructive Wavelet Neural Networks and Its Applicationscs.LG
A new WNN framework selects wavelet bases for efficient learning.
problem Challenges in constructing accurate wavelet bases and high computational costs in WNN.
method Introduces a constructive WNN that selects initial bases and trains functions by introducing new bases for predefined accuracy while reducing computational costs.
result Significantly improves computational efficiency through a frequency estimator and wavelet-basis increase mechanism.
Kernel methods are widespread in machine learning; however, they are limited by the quadratic complexity of the construction, application, and storage of kernel matrices. Low-rank matrix approximation algorithms are widely used to address this problem and reduce the arithmetic and storage cost. However, we observed tha…
Radial-basis-function networks are traditionally defined for sets of vector-based observations. In this short paper, we reformulate such networks so that they can be applied to adjacency-matrix representations of weighted, directed graphs that represent the relationships between object pairs. We re-state the sum-of-squ…
We compare two important bases of an irreducible representation of the symmetric group: the web basis and the Specht basis. The web basis has its roots in the Temperley-Lieb algebra and knot-theoretic considerations. The Specht basis is a classic algebraic and combinatorial construction of symmetric group representatio…
The paper constructs a Saito basis for a specific class of divisors and applies it to logarithmic Poisson geometry.
problem Investigating a class of non-quasi-homogeneous free divisors and their logarithmic vector fields.
method Explicitly constructing a Saito basis for the module of logarithmic vector fields and applying it to logarithmic Poisson geometry.
result The construction of the Saito basis and the Lie-Rinehart algebra structure on the sheaf of logarithmic 1-forms.
In view of the result of Kontsevich, now often called ``the fundamental theorem of Vassiliev theory'', identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the ``4T-relation'' (and in the unframed case, th…
This paper optimizes PCE for efficient surrogate modeling in engineering.
problem Efficiently selecting polynomial regressors for surrogate modeling in computationally expensive models.
method Three state-of-the-art basis-adaptive sparse PCE methods are compared and analyzed.
result Automatic selection of the best solver and basis-adaptive scheme improves surrogate model accuracy.
The paper defines constraints for commuting endomorphisms in generalized tangent bundles.
problem Identifying constraints for commuting endomorphisms in generalized tangent bundles.
method Using Gröbner basis techniques to construct and study tensors forming ideals.
result Explicit construction and study of tensors forming ideals of commuting endomorphisms.
Graphs derived from cohomology help reconstruct defining graphs of Artin groups.
problem Reconstructing the defining graph from the cohomology of Artin groups.
method Constructing cohomology basis graphs from cup products of cohomology groups.
result The cohomology basis graph always contains the defining graph as a subgraph.
Study of semi-principal bundles using group actions and wreath products.
problem Understanding bundles with fibers as free -spaces.
method Defining semi-principal bundles, bases, and frame bundles; using wreath products and functors.
result Semi-principal bundles can be retracted to principal bundles, preserving parallel transport.
A number of fundamental quantities in statistical signal processing and information theory can be expressed as integral functions of two probability density functions. Such quantities are called density functionals as they map density functions onto the real line. For example, information divergence functions measure t…
Surgery triangles are an important computational tool in Floer homology. Given a connected oriented surface , we consider the abelian group generated by bordered 3-manifolds with boundary , modulo the relation that the three manifolds involved in any surgery triangle sum to zero. We show that is a f…
Geometrically interprets symplectic structure in 3-manifold triangulations.
problem Understanding symplectic structures in 3-manifold triangulations.
method Geometric interpretation and algorithm construction for symplectic basis.
result Algorithm constructs curves forming a symplectic basis.
Paper develops a two-population model to assess longevity basis risk.
problem Mismatch between hedger's liability and hedging instrument causes longevity basis risk.
method Develops a two-population mortality model using Lee-Carter model and renewal process.
result Proposed model provides significant risk reduction when mortality jumps and sampling risk are considered.
New bases found for Kauffman bracket skein module of fibered torus.
problem Computing Kauffman bracket skein module of fibered 3-manifolds.
method Constructed bases for Kauffman bracket skein module of product annulus and circle, then applied to -fibered torus.
result Found a new basis for KBSM of -fibered torus.
Neural Chaos uses neural networks instead of polynomials for stochastic modeling.
problem Challenges in constructing surrogate models with uncertainty quantification for complex or high-dimensional stochastic processes.
method Adopting spectral expansion formalism with neural network basis functions, identifying them data-drivenly without prior assumptions.
result Demonstrates effectiveness of the proposed scheme through numerical examples of varying complexity.
In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed -local systems on a punctured surface . The moduli space is birational to a cluster -variety, whose positive real points recover the enhanced Teichmüller space of . Their b…
We consider the problem of designing a sparse Gaussian process classifier (SGPC) that generalizes well. Viewing SGPC design as constructing an additive model like in boosting, we present an efficient and effective SGPC design method to perform a stage-wise optimization of a predictive loss function. We introduce new me…
New basis for quantum gl_N invariants derived from Macdonald polynomials.
problem Constructing new bases for quantum gl_N invariants.
method Using interpolation Macdonald polynomials and Okounkov's results.
result Cyclotomic expansions for gl_N invariants and knot invariants.
DBKs enable scalable GPs with tractable inference for large datasets.
problem Scaling Gaussian processes to large and complex datasets while maintaining tractable inference.
method DBKs constructed from neural-network-parameterized basis functions with explicit low-rank structure, enabling linear-complexity inference.
result DBKs provide a unified perspective and improve predictive accuracy, uncertainty quantification, and computational efficiency.
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…
On Convergence Rate of Adaptive Multiscale Value Function Approximation For Reinforcement Learningcs.LG
In this paper, we propose a generic framework for devising an adaptive approximation scheme for value function approximation in reinforcement learning, which introduces multiscale approximation. The two basic ingredients are multiresolution analysis as well as tree approximation. Starting from simple refinable function…
In this paper we define an explicit basis for the -web algebra (the generalization of Khovanov's arc algebra) using categorified -skew Howe duality. Our construction is a -web version of Hu--Mathas' graded cellular basis and has two major application…
We use the Jones-Wenzl idempotents to construct a basis of Temperley-Lieb algebra TL_n. This allows a short calculation for a Gram determinant of Lickorish's bilinear form on the Temperley-Lieb algebra.
This paper develops efficient surrogate models for optimization of complex dynamical systems.
problem Computational expense in solving complex dynamical systems through numerical simulation.
method Combination of proper orthogonal decomposition and radial basis functions for constructing low-dimensional surrogate models.
result Surrogate models reduce computational time for optimization problems while maintaining accuracy.
Dual Bayesian Affine Estimators for Wiener-type state-space models
problem Estimating parameters in Wiener-type state-space models
method Fixed-point architecture combining two affine estimators
result Dual basis-parameter estimator achieves comparable parameter MSE to purely affine estimator
Gradient-enhanced GSA uses Poincaré chaos expansions for accurate sensitivity analysis.
problem Accurately estimating Sobol' indices with limited data.
method Integrates sparse, gradient-enhanced regression with Poincaré chaos expansions for derivative-based sensitivity analysis.
result Accurately estimated Sobol' indices using limited data.
BEKAN uses RBFs and evolutionary methods to solve PDEs with boundary conditions.
problem Enforcing boundary conditions in neural networks for PDE solutions.
method Boundary condition-guaranteed evolutionary Kolmogorov-Arnold Network (BEKAN) with radial basis functions (RBFs). Incorporates Dirichlet, periodic, and Neumann conditions.
result BEKAN outperforms MLP and B-splines KAN in solving PDEs with boundary conditions.
New method improves speed of estimating bivariate functional data.
problem Estimating bivariate functional data at faster rates.
method Adapting to directional regularity of bivariate processes.
result Faster rates of convergence achieved through change-of-basis.
New coordinates for SL3-web graphs on surfaces defined by Fock-Goncharov.
problem Characterizing functions on SL3-character varieties of surfaces.
method Introduced tropical Fock-Goncharov coordinates based on Knutson-Tao rhombus inequalities and congruence conditions.
result Tropical coordinates naturally index the commutative algebra of functions on SL3-character varieties.
Characterizes a specific type of neural network for alternating group equivariance.
problem Understanding and characterizing neural networks with alternating group equivariance.
method Characterization of all possible -equivariant neural networks using tensor powers of .
result Found a basis of matrices for learnable, linear -equivariant layer functions.
This paper proposes a new method to adapt ROMs for new parameter settings.
problem ROMs lack robustness when applied to new parameter settings.
method Regression trees on Grassmann Manifold to learn the mapping between parameters and POD bases.
result The proposed method is capable of establishing the mapping between parameters and POD bases, thus adapting ROMs for new parameters.
We prove Transformers can learn diverse Gröbner bases.
problem Training Transformers for Gröbner basis computation.
method Prove generality of dataset generation algorithm; propose extended algorithm.
result Datasets are sufficiently general for diverse Gröbner bases learning.