GD monotonically decreases GFS sharpness in neural networks and scalar models.
problem Oscillatory behavior of loss in GD training.
method Analysis of GFS sharpness and empirical validation.
result GFS sharpness decreases monotonically during GD training.
Neural network predicts functional responses from scalar inputs.
problem Regression of functional responses with large scalar predictors and nonlinear relationships.
method Transform functional response to finite dimensions, design feed-forward neural network, modify output via objective functions, apply roughness penalty.
result Proposed neural network outperforms conventional methods in multiple scenarios.
New method uses scalars to approximate physics functions.
problem Designing neural networks that respect physical symmetries.
method Parameterizing polynomial functions equivariant to various symmetries using scalars.
result Universal approximation of polynomial functions under various symmetries using scalars.
Equivariant neural networks improve performance and generalization in complex scalar field theory tasks.
problem Improving performance and generalization in neural networks for complex scalar field theory tasks.
method Incorporating translational equivariance into neural network architectures.
result Equivariant neural networks significantly outperform non-equivariant networks in various tasks, including those beyond the training set and across different lattice sizes.
Evaluation of systemic risk in networks of financial institutions in general requires information of inter-institution financial exposures. In the framework of Debt Rank algorithm, we introduce an approximate method of systemic risk evaluation which requires only node properties, such as total assets and liabilities, a…
Model projection transfers convolutional network properties to feedforward networks.
problem Transferring properties between feedforward and convolutional networks.
method Unified node-level framework with tensor-valued activations, model projection.
result Projected CNN nodes inherit GFFN-style trainable structure.
A new framework for hyperbolic neural networks using the Klein model is introduced.
problem Previous works focused on Poincaré and hyperboloid models, neglecting the Klein model.
method Formulation of operations using the Klein model, study of the Klein linear layer, and comparison with Poincaré ball model.
result The Klein HNN performs similarly to the Poincaré ball model, offering a third option.
Equivariant networks improve geometric prediction without scalar approximations.
problem Efficiently predicting geometric tensors in real-world scenarios.
method Equivariant networks for geometric prediction.
result Equivariant networks can generalize to unseen systems for geometric prediction.
The paper tackles the trade-off between fairness and accuracy in machine learning models.
problem Ensuring fairness in machine learning often reduces model accuracy.
method The paper introduces formal tools for reconciling the fairness-accuracy tension using Pareto optimality from multi-objective optimization.
result The Chebyshev scalarization scheme is superior for finding Pareto optimal solutions compared to the linear scalarization scheme.
Method integrates functional data into neural networks for better interpretability.
problem Integrating functional data into neural networks for improved interpretability.
method Integrates functional data into deep neural networks, visualizes dynamic weights.
result Model performs well in prediction and recovery of functional weights.
Improves neural network regression performance using encoder-decoder approach.
problem Suboptimal regression results from classical methods in neural networks.
method Proposes a new method using an encoder-decoder approach for regression.
result Shows improved performance on various real-world datasets.
Based on the notion of information bottleneck (IB), we formulate a quantization problem called "IB quantization". We show that IB quantization is equivalent to learning based on the IB principle. Under this equivalence, the standard neural network models can be viewed as scalar (single sample) IB quantizers. It is know…
Vector-valued neural learning has emerged as a promising direction in deep learning recently. Traditionally, training data for neural networks (NNs) are formulated as a vector of scalars; however, its performance may not be optimal since associations among adjacent scalars are not modeled. In this paper, we propose a n…
Transforms between neural networks using manifold-learning techniques.
problem Establish equivalence between different neural networks.
method Diffusion maps with a Mahalanobis-like metric to construct transformations between network outputs and internal neuron activations.
result Established equivalence classes between neural networks trained on various data types.
Develops vector-valued RKBS for neural networks and operators.
problem Understanding function spaces of Rd-valued neural networks and neural operators. method Defines and constructs vector-valued RKBS (vv-RKBS) without restrictive assumptions.
result Establishes Representer Theorem for neural architectures.
A universal collection of 4 invariants improves neural network accuracy for molecular dynamics.
problem Improving accuracy of neural networks in molecular dynamics.
method Developed a universal collection of 4 smooth scalar invariants on M(3) x M(3) and evaluated their effectiveness in a PONITA neural network architecture.
result Using a universal collection of invariants significantly improves neural network accuracy.
Algorithm optimizes cascaded functions with known structure.
problem Optimizing a function network with known structure.
method GPN-UCB algorithm with upper confidence bounds and theoretical regret bounds.
result Near-optimal cumulative and simple regret bounds.
Quaternion-Kähler manifolds' stability and rigidity of scalar curvature studied.
problem Stability and rigidity of scalar curvature in quaternion-Kähler manifolds.
method Analysis of stability and rigidity conditions using Einstein manifold properties.
result Quaternion-Kähler manifolds of negative scalar curvature are stable and scalar curvature rigid.
Proves curvature comparison for Riemannian bands in low dimensions.
problem Curvature comparison in Riemannian bands with lower bounds.
method Uses warped products over scalar-flat manifolds with log-concave warping.
result Scalar and mean curvature comparison results proven.
Study on scalar curvature deformations in pseudohermitian manifolds.
problem Deformation of scalar curvature in pseudohermitian manifolds.
method Analogy with Riemannian manifolds, introduction of R-singular spaces, stability conditions, partial infinitesimal rigidity. result Partial infinitesimal rigidity result for scalar curvature of compact pseudohermitian manifolds.
Small Weyl infimum on 4-manifolds with positive scalar curvature.
problem Understanding scalar curvature on 4-manifolds.
method Analyzing the Weyl functional and comparing scalar and self-dual Weyl curvatures.
result The infimum of the Weyl functional is small on many 4-manifolds with positive scalar curvature.
The paper examines Randers metrics with isotropic scalar curvature properties.
problem Characterizing Randers metrics with specific scalar curvature properties.
method Analyzes properties of Randers metrics with isotropic scalar curvature.
result Proves that Randers metrics with weakly isotropic scalar curvature have isotropic S-curvature and are either Minkowskian or Riemannian. In this paper we present results on dynamic multivariate scalar risk measures, which arise in markets with transaction costs and systemic risk. Dual representations of such risk measures are presented. These are then used to obtain the main results of this paper on time consistency; namely, an equivalent recursive form…
The paper establishes bounds on scalar curvature on asymptotically flat manifolds.
problem Establishing scalar curvature bounds on asymptotically flat manifolds.
method Using Ricci-DeTurck flow and distributional scalar curvature, the paper derives bounds on scalar curvature.
result The scalar curvature lower bound under Ricci-DeTurck flow depends on the scalar curvature lower bound in the β-weak sense and time.
Minimal splitting factors help study scalar curvature constraints.
problem Scalar curvature constraints in geometry.
method Introducing minimal splitting factors with positive scalar curvature.
result Minimal splitting factors have properties similar to area minimizing hypersurfaces.
Sharp bounds on scalar curvature spectrum and rigidity theorems.
problem Understanding scalar curvature bounds and rigidity on manifolds.
method Sharp upper bounds for the bottom spectrum of the Beltrami Laplacian, scalar curvature rigidity theorem.
result Sharp upper bound for the bottom spectrum of the Beltrami Laplacian and scalar curvature rigidity theorem.
Study finds open manifolds without complete metrics with positive scalar curvature.
problem Topological obstruction to positive scalar curvature on open manifolds.
method Defined Schoen-Yau-Schick and weak Schoen-Yau-Schick manifolds to prove the absence of complete metrics with positive scalar curvature.
result Proved no complete metric with positive scalar curvature on open Schoen-Yau-Schick manifolds.
In this paper we investigate complete critical metrics of the L2-norm of the scalar curvature. We prove that any complete critical metric with positive scalar curvature has constant scalar curvature and we characterize critical metrics with nonnegative scalar curvature in dimension three and four.
The aim of the present paper is to provide an \emph{intrinsic} investigation of special Finsler spaces of Hp-scalar curvature and of Hp-constant curvature. Characterizations of such spaces are shown. Sufficient condition for Finsler space of Hp-scalar curvature to be of perpendicular scalar curvature i…
The study restricts scalar curvature on certain noncompact manifolds.
problem Ruling out complete metrics with nonnegative scalar curvature.
method Conditions and examples of positive scalar curvature metrics.
result Examples of manifolds with positive scalar curvature.
Sharp volume growth ratio for 3D manifolds with positive scalar curvature.
problem Volume growth and scalar curvature in non-compact Riemannian manifolds.
method Analyzing 3D complete, non-compact manifolds with non-negative Ricci and positive scalar curvature.
result Obtained sharp linear volume growth ratio and rigidity.
George Cybenko's landmark 1989 paper showed that there exists a feedforward neural network, with exactly one hidden layer (and a finite number of neurons), that can arbitrarily approximate a given continuous function f on the unit hypercube. The paper did not address how to find the weight/parameters of such a networ…
The paper studies Kropina metrics with a specific curvature property.
problem Characterizing Kropina metrics with isotropic scalar curvature.
method Tensor analysis to derive expressions and characterize metrics.
result Characterization of Kropina metrics with isotropic scalar curvature.
Formula for scalar curvature under metric collapse.
problem Finding positive scalar curvature metrics.
method Formula involving scalar curvature and adapted orthonormal frame.
result Effect of metric collapse on scalar curvature.
The paper explores conditions for positive scalar curvature on manifolds with boundaries and their doubles.
problem Conditions for positive scalar curvature on manifolds with boundaries and their doubles.
method Analyzes the relationship between boundary conditions and positive scalar curvature metrics on manifolds and their doubles.
result Provides conditions for positive scalar curvature metrics on manifolds with boundaries and their doubles.
The paper classifies flag manifolds with specific isotropy components and finds conditions for Kähler-like scalar curvature.
problem Classifying flag manifolds with specific isotropy components and finding conditions for Kähler-like scalar curvature.
method Investigating invariant almost Hermitian structures on generalized flag manifolds with two or three irreducible components.
result Classification of flag manifolds admitting Kähler-like scalar curvature and conditions for such structures.
Paper generalizes scalar curvature theorem to weighted manifolds.
problem Generalizing scalar curvature rigidity theorem to weighted manifolds.
method Proves a refinement of Llarull's theorem for P-scalar curvature.
result Establishes a Llarull type theorem for SkimesTn−k. The paper explores geometry and positive scalar curvature on non-compact manifolds.
problem Understanding the relationship between geometry and positive scalar curvature on non-compact manifolds.
method Analysis of volume growth, scalar curvature integral, and width in different dimensions.
result Proves minimal volume growth and integral of scalar curvature in three dimensions, and volume growth with stronger conditions in higher dimensions.
Develops scalar curvature in generalized Kahler geometry and shows constant scalar curvature on compact Lie groups.
problem Defines scalar curvature in generalized Kahler geometry.
method Introduces scalar curvature in terms of pure spinors formalism and develops a moment map framework.
result Scalar curvature is given by the moment map, generalizing results from ordinary Kahler geometry.
In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scala…
Study local properties of Chern-scalar curvature through linearization stability.
problem Local properties of Chern-scalar curvature function.
method Linearization analysis of the Chern-scalar curvature function.
result Stability of linearization and structure of metrics with prescribed curvature.
Survey of nonnegative scalar curvature sequences and their limits.
problem Understanding sequences of manifolds with nonnegative scalar curvature.
method Analyzing sequences of manifolds with nonnegative scalar curvature and proving convergence.
result Proved the GH and SWIF convergence of an extreme example.
FuncNN package enables deep learning with functional covariates.
problem Lack of software for deep learning with functional covariates.
method Developed an R package using keras architecture, introducing functions for model building, predictions, and cross-validation.
result First package for deep learning with functional covariates.
A universal interatomic potential for an arbitrary set of chemical elements is urgently needed in computational materials science. Graph convolution neural network (GCN) has rich expressive power, but previously was mainly employed to transport scalars and vectors, not rank ≥2 tensors. As classic interatomic poten…
Estimates for scalar curvature equations on Kähler manifolds with singularities.
problem Developing estimates for scalar curvature equations with singular metrics.
method Estimates and Laplacian estimates for scalar curvature equations of degenerate Kähler metrics.
result Derivation of estimates for singular constant scalar curvature Kähler metrics and singular Kähler-Einstein metrics.
The paper studies Berwald scalar curvature properties in Finsler geometry.
problem Characterizing Finsler manifolds based on Berwald scalar curvature.
method Analyzes properties of Berwald scalar curvature and its implications for Finsler manifolds.
result Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.
Researchers confirm scalar-flatness for critical metrics in 5-9 dimensions.
problem Verifying scalar-flatness for critical metrics in specific dimensions.
method Analyzing complete Riemannian manifolds with critical metrics of the L2-scalar curvature functional. result The conjecture that all complete noncompact critical metrics with finite energy are scalar-flat is confirmed for dimensions 5 to 9.
Study connects curvature bounds to map existence and flow solutions.
problem Existence of lower scalar curvature bounds and measures.
method Relates curvature bounds to map existence and backward limit of Ricci flow solutions.
result Sufficient condition for existence of limiting scalar curvature measure.