Quantum K-theory of quintic 3-fold conjectured with non-polynomial coefficients.
problem Reconstructing quantum K-theory for quintic 3-fold.
method Formulated explicit conjecture for small J-function and its q-difference equation.
result Coefficients of q-difference equations are non-polynomial functions of Gopakumar-Vafa invariants.
The paper shows neural networks can approximate functions over non-compact domains with non-polynomial activation.
problem Approximating functions over non-compact domains using neural networks.
method Using single-hidden-layer feedforward neural networks with non-polynomial activation functions over non-compact subsets of Euclidean spaces.
result Neural networks can approximate functions in weighted Ck-spaces and weighted Sobolev spaces over unbounded domains. This work presents a non-intrusive model reduction method to learn low-dimensional models of dynamical systems with non-polynomial nonlinear terms that are spatially local and that are given in analytic form. In contrast to state-of-the-art model reduction methods that are intrusive and thus require full knowledge of t…
Non-polynomial growth harmonic maps from the complex plane to the hyperbolic space are studied. Some non-surjectivity results are obtained. Moreover, images of such harmonic maps are investigated with reference to their Hopf differentials.
New bounds show complex neural networks need many queries to learn.
problem Learning non-polynomial activation functions with Gaussian marginals.
method Gradient boosting procedure to amplify lower bounds on SQ dimension of neural networks.
result Statistical-query lower bounds for ReLU regression with 2ncε queries. The action of the mapping class group of the thrice-punctured projective plane on its GL(2,C) character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corres…
We study the uniqueness of a vortex equation involving an entire function on the complex plane. As geometric applications, we show that there is a unique harmonic map u:C→H2 satisfying ∂u=0 with prescribed polynomial Hopf differential; there is a unique affine spherical imm…
Study examines null vector fields on Lorentzian manifolds.
problem Understanding the structure of null vector fields on Lorentzian manifolds.
method Investigates the bundle structure and ternary product of nowhere vanishing null vector fields.
result Null tangent bundle is a non-polynomial graded bundle with a para-associative ternary product.
Neural networks can approximate functions uniformly across various measures.
problem Universal approximation of functions across different probability measures.
method Proving neural networks are dense in Orlicz spaces, extending classical theorems.
result Neural networks uniformly approximate functions for weakly compact families of measures.
Data symmetries in neural networks can generate conserved quantities.
problem Conservation laws in neural networks
method Using tensorizable networks
result Data augmentation can induce conserved quantities
Wide neural networks can be closely approximated by Gaussian processes, with rates depending on the activation function's properties.
problem Approximating the behavior of wide neural networks using Gaussian processes.
method Established convergence rates for the central limit theorem in an infinite-dimensional functional space, using a transportation distance metric.
result Explicit convergence rates for neural networks approximated by Gaussian processes, varying based on the activation function's properties.
Much combinatorial optimisation problems constitute a non-polynomial (NP) hard optimisation problem, i.e., they can not be solved in polynomial time. One such problem is finding the shortest route between two nodes on a graph. Meta-heuristic algorithms such as A∗ along with mixed-integer programming (MIP) methods …
The paper proves deep neural networks with analytic activation can approximate any function.
problem Approximating functions with neural networks using analytic activation functions.
method Elementary proofs for real and complex networks, Stone-Weierstrass theorem, Mergelyan's theorem.
result Closure of neural network classes equals space of polynomials for analytic activation.
Ehlers-Kundt conjecture is a physical assertion about the fundamental role of plane waves for the description of gravitational waves. Mathematically, it becomes equivalent to a problem on the Euclidean plane R2 with a very simple formulation in Classical Mechanics: given a non-necessarily autonomous potent…
New framework establishes positivity of DNTK for PINNs.
problem Establishing positivity of NTK for PINNs with multiple differential operators.
method Proposed Differential Neural Tangent Kernel (DNTK) for PINNs.
result Positivity of infinite width DNTK for various activation functions and differential operators.
UDENet and ResNet can approximate any function, with ODENet showing UAP for continuous functions.
problem Approximating any function using ODENet and ResNet.
method Proved UAP for ODENet and ResNet, derived gradient, and applied to various problems.
result UDENet and ResNet can approximate any function, with ODENet showing UAP for continuous functions.
Detecting correlated trees helps align sparse graphs.
problem Detecting correlation between trees for sparse random graphs.
method MPAlign message-passing algorithm for graph alignment.
result MPAlign succeeds in polynomial time for partial alignment.
Cryptotree enables accurate predictions on encrypted data using Random Forests.
problem Applying machine learning to private data while preserving confidentiality.
method Adapting Neural RF to CKKS scheme for HE operations on encrypted data.
result Cryptotree achieves better prediction results on encrypted data than regular RF.
FNOs learn solution operators of dissipative equations efficiently via spectral methods.
problem Learning and approximation of solution operators for dissipative equations.
method Introducing spectral methods and deriving FNO approximation bounds and sample complexity guarantees.
result Polynomial sample complexity guarantees for FNOs learning solution operators of dissipative equations.
In order to identify important variables that are involved in making optimal treatment decision, Lu et al. (2013) proposed a penalized least squared regression framework for a fixed number of predictors, which is robust against the misspecification of the conditional mean model. Two problems arise: (i) in a world of ex…
New method for MAP inference using Benders' decomposition.
problem Finite-time convergence guarantee for MAP inference.
method Sequentially adding constraints using Benders' decomposition.
result Higher optimal posterior value compared to other methods.
Quantum computing optimizes ESG portfolios efficiently.
problem Optimizing investment portfolios with risk, return, and ESG considerations.
method Formulated discrete Markowitz portfolio theory (DMPT) for quantum annealers, incorporating ESG ratings.
result Discrete portfolios converge to continuous solutions as budgets increase, outperforming traditional methods.
In this short report, we investigate the ability of the DCCA coefficient to measure correlation level between non-stationary series. Based on a wide Monte Carlo simulation study, we show that the DCCA coefficient can estimate the correlation coefficient accurately regardless the strength of non-stationarity (measured b…
Formula connects linking coefficients to Kontsevich integral coefficients.
problem Linking coefficients from Kontsevich integral.
method Purely combinatorial approach.
result Expresses linking coefficients as combinations of Kontsevich integral coefficients.
Computed distortion coefficients for the α-Grushin plane.
problem Analyzing the distortion coefficients of the α-Grushin plane.
method Using generalised trigonometric functions and synthetic curvature conditions.
result Estimates for distortion coefficients and a curvature condition conjecture.
Machine learning predicts Kronecker coefficients with high accuracy.
problem Predicting Kronecker coefficients from tensor products of symmetric group representations.
method Training machine learning models (NN, CNN, GBDT) to classify Kronecker coefficients as zero or non-zero.
result Trained models achieve high accuracy (≈0.98) in classifying Kronecker coefficients. Abstract: Determines thermoelastic coefficients from boundary data.
problem Determining coefficients of thermoelastic system from boundary information.
method Explicit expression for thermoelastic Dirichlet-to-Neumann map with variable coefficients.
result Thermoelastic Dirichlet-to-Neumann map uniquely determines coefficients on the manifold.
New filling functions for groups with coefficients show different asymptotic behavior.
problem Difficulty in filling loops with surfaces in Cayley graphs.
method Defining homological filling functions with coefficients and proving their differences.
result Filling functions for n-cycles with coefficients in different groups have distinct asymptotic behavior. Abstract: Determines Lamé coefficients from boundary measurements.
problem Determining Lamé coefficients from elastic boundary measurements.
method Explicit symbol of elastic Dirichlet-to-Neumann map, partial derivatives determination.
result Elastic Dirichlet-to-Neumann map uniquely determines Lamé coefficients.
Defines and proves properties of weighted renormalized volume coefficients.
problem None explicitly stated; focuses on mathematical definitions and proofs.
method Defines weighted renormalized volume coefficients and proves their variational nature and polynomial representation.
result Weighted renormalized volume coefficients are variational and can be expressed as polynomials of specific tensors.
Improved 3D LiDAR data classification using product coefficients.
problem Enhancing accuracy in 3D LiDAR data classification.
method Introducing product coefficients derived from measure theory as additional features in the classification process, alongside PCA.
result Significant improvement in classification accuracy with product coefficients.
High-dimensional, large-sample astrophysical databases of galaxy clusters, such as the Chandra Deep Field South COMBO-17 database, provide measurements on many variables for thousands of galaxies and a range of redshifts. Current understanding of galaxy formation and evolution rests sensitively on relationships between…
We study the structure of the stable coefficients of the Jones polynomial of an alternating link. We start by identifying the first four stable coefficients with polynomial invariants of a (reduced) Tait graph of the link projection. This leads us to introduce a free polynomial algebra of invariants of graphs whose ele…
Improved MoE performance through perturbing cosine router.
problem Representation collapse and parameter redundancy in MoE models.
method Least square estimation of cosine router in MoE, followed by noise addition to improve convergence rates.
result Perturbed cosine router leads to polynomial convergence rates for MoE models.
Paper calculates third coefficient in Kaehler-Einstein metric expansion.
problem Understanding Kaehler-Einstein metrics and their epsilon functions.
method Computes the third coefficient in the TYCZ-expansion of the epsilon function.
result Discovers the vanishing of the third coefficient's significance.
Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.
problem Local elliptic regularity for operators with low regularity coefficients in Sobolev-type spaces.
method Rescaling estimates and multiplication results for function spaces.
result Unified set of interior estimates and regularity inference for operators with Sobolev-type coefficients.
CVNNs improve performance in tasks with complex-valued inputs.
problem Improving performance in tasks with complex-valued inputs.
method Analyze the approximation properties of complex-valued neural networks (CVNNs).
result Quantitative approximation bounds for CVNNs, showing error scales as m−k/(2n). The paper uses Floer homology to study twist coefficients and their behavior after capping off.
problem Behavior of twist coefficients after capping off a boundary component.
method Heegaard Floer homology to constrain twist coefficients.
result Results about fractional Dehn twists and Floer homology of cyclic branched covers.
Extends A-type coefficient polynomials to B-type setting, introducing new invariants.
problem Tackles the B-type skein relation and introduces new coefficient polynomials.
method Introduces coefficient polynomials associated with the B-type skein relation and proves their invariance under Reidemeister moves.
result Shows that the generating series of these coefficient polynomials recovers the Kauffman polynomial.
We simplify complex regression coefficients using linearization and feature comparison.
problem Interpreting high-dimensional regression coefficients from nonlinear responses.
method Developed a linearization method to derive feature coefficients and compare them with regression coefficients.
result Shows how regression coefficients relate to linearized feature coefficients and how they change under regularization.
Bounding twist number of surface links using polynomial coefficients.
problem Bounding the twist number of alternating surface links.
method Introducing a generalized homological Kauffman bracket and applying it to surface link diagrams.
result A bound for the twist number of alternating surface links in terms of polynomial coefficients.
Characterizes differential forms and vector fields with constant coefficients on manifolds.
problem Understanding constant coefficient differential forms and vector fields on manifolds.
method Analyzes differential forms and vector fields of specific degrees, proving obstructions and characterizing solutions to partial differential systems.
result Characterizes differential forms and vector fields with constant coefficients of various degrees on smooth manifolds.
Study on gradient descent in Hilbert spaces with Markov chains, focusing on mixing coefficients.
problem Analyzing convergence of gradient descent in Hilbert spaces with stationary Markov chains.
method Examined strictly stationary Markov chains with φ- and β-mixing coefficients, derived probabilistic upper bounds. result Probabilistic upper bounds on convergence behavior of gradient descent algorithm based on mixing coefficients.
In this paper we use wavelet concepts to show that correlation coefficient between two financial data's is not constant but varies with scale from high correlation value to strongly anti-correlation value This studies is important because correlation coefficient is used to quantify degree of independence between two va…
At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network functio…
The conformal powers of the Laplacian of a Riemannian metric which are known as the GJMS-operators admit a combinatorial description in terms of the Taylor coefficients of a natural second-order one-parameter family (˝r;g) of self-adjoint elliptic differential operators. (˝r;g) is a non-Laplace-type perturbation …
Guts determine the leading coefficients of L2-Alexander torsions for 3-manifolds.
problem Determining the leading coefficient of L2-Alexander torsions for 3-manifolds. method Using a new criterion for the convergence of Fuglede-Kadison determinants and the work of Agol and Zhang on guts of 3-manifolds.
result The leading coefficient equals the relative L2-torsion of the guts associated to the cohomology class. Survey of recent measures of association, including a new coefficient.
problem Exploring new measures of association in statistics.
method Survey and introduction of a new correlation coefficient.
result Proposed a new extension of the correlation coefficient to standard Borel spaces.