Paper tackles spurious vanishing problem in approximate vanishing ideals.
problem Capturing nonlinear structure of perturbed data points leads to spurious vanishing problem.
method Proposes a general method integrating coefficient normalization and iterative basis construction.
result Proposed method overcomes spurious vanishing problem, resulting in shorter feature vectors.
Triality connects three polynomial bases in Lie algebra studies.
problem Understanding relationships between invariant polynomials.
method Triality of mSO(8,C) and Hitchin fibrations. result Explicit automorphisms linking polynomial bases identified.
Proposes a novel approach for deep neural network initialization using polynomial approximations.
problem Improving deep neural network training from initialization.
method Uses polynomial-based approximations to initialize deep neural networks.
result Networks initialized with polynomial approximations are more likely to achieve a desirable local minimum during training.
The extragradient method accelerates convergence in complex game dynamics.
problem Complex interactions in game dynamics cause simple methods to diverge, necessitating more sophisticated approaches.
method A polynomial-based analysis to identify three scenarios for accelerated convergence of the momentum extragradient method.
result The momentum extragradient method achieves faster convergence under specific eigenvalue conditions.
Improved pseudo-label accuracy in semi-supervised learning with Hermite polynomials.
problem Improving pseudo-label accuracy in semi-supervised learning.
method Substituting ReLU activations with Hermite polynomial activations in deep networks.
result Hermite polynomial activations yield significant improvements in pseudo-label accuracy and financial savings.
EPGP surrogate outperforms finite elements in solving wave equations.
problem Benchmarking Gaussian Process surrogates vs. finite elements for wave equation solutions.
method EPGP uses penalized least squares and exponential-polynomial bases; CN-FEM employs Crank--Nicolson time stepping.
result EPGP achieves lower error than CN-FEM under matched degrees-of-freedom.
New formula simplifies interior polynomial calculation.
problem Calculating interior polynomial efficiently.
method New recursion formula based on non-expanding sets.
result Clearer combinatorial interpretation of interior polynomial.
Consider a network of agents connected by communication links, where each agent holds a real value. The gossip problem consists in estimating the average of the values diffused in the network in a distributed manner. We develop a method solving the gossip problem that depends only on the spectral dimension of the netwo…
Novel approach to financial derivatives pricing using rough path theory.
problem No-arbitrage conditions in financial markets necessitating precise integration methods.
method Developed a polynomial-based approximation class for rough path functionals, extending to non-geometric rough paths.
result Motivated a hypothesis for payoff functionals in financial markets, facilitating analysis.
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 (n/2,n/2) 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 K3 surface, in which the intersection form takes the canonical form 2E8(−1)⊕3H. 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 Z2 systolic freedom. We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to [n,n,n]: 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.
Recently there has been renewed interest in the mapping-class group of a compact surface of genus g≥2 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 g. Here we give the details of the proof that there is…
Paper proposes a method to speed up DNNs by quantizing Winograd/Toom-Cook convolutions.
problem Speeding up convolution computations in DNNs with reduced time consumption and improved accuracy.
method Application of base change technique for quantized Winograd-aware training model.
result 8-bit quantized network achieves nearly the same accuracy as direct quantized convolution with minimal additional operations.
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…
I-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…
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…
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 G-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.
Near-convex archetypal analysis improves interpretability and fitting error in NMF.
problem High data fitting error in traditional archetypal analysis.
method Introduces near-convex archetypal analysis (NCAA) that combines AA and NMF.
result NCAA achieves lower data fitting error than state-of-the-art methods.
Hedonic models predict 84-92% of U.S. real estate prices, highlighting environmental factors' impact.
problem Predicting real estate prices using hedonic models with environmental factors.
method P-spline generalized additive models for real estate prices, contrasting with linear and polynomial models.
result GAM models explain 84-92% of U.S. real estate price variance, with environmental factors contributing minimally.
HaKAN uses Hahn-KAN blocks to forecast multivariate time series.
problem Long-term time series forecasting challenges with high complexity and spectral bias.
method HaKAN integrates channel independence, patching, and a stack of Hahn-KAN blocks with residual connections. It uses Hahn polynomial-based learnable activation functions.
result HaKAN consistently outperforms state-of-the-art methods on various forecasting benchmarks.
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 Σ, we consider the abelian group K(Σ) generated by bordered 3-manifolds with boundary Σ, modulo the relation that the three manifolds involved in any surgery triangle sum to zero. We show that K(Σ) is a f…
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.
Reformulates RBF networks for graph-based data.
problem Applying RBF networks to graph data.
method Reformulate RBF networks for adjacency matrices, derive gradient updates.
result Guaranteed same responses as vector-based RBF networks.
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.
Data-driven method solves multiscale elliptic PDEs with random coefficients.
problem Solving multiscale elliptic PDEs with random coefficients.
method Data-driven approach based on intrinsic dimension reduction.
result Efficient solution of multiscale elliptic PDEs with random coefficients.
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.
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 (β,2)-fibered torus. result Found a new basis for KBSM of (β,2)-fibered torus. In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed PGL2-local systems on a punctured surface S. The moduli space is birational to a cluster X-variety, whose positive real points recover the enhanced Teichmüller space of S. 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.
Upper bounds on neural network complexity for PDE solutions.
problem Approximating solutions of parametric PDEs without knowing their exact form.
method Using low-dimensionality of solution manifolds and a small reduced basis.
result Neural networks can approximate PDE solutions with sizes dependent only on the reduced basis.
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.