The paper establishes a continuous embedding between two types of Barron spaces in neural networks.
problem Understanding the relationship between two types of Barron spaces in neural networks.
method Introduced a continuous embedding inequality between Barron and spectral Barron spaces.
result The embedding inequality holds for any function in the spaces, with constants independent of the input dimension.
The paper analyzes neural networks for solving high-dimensional Schrödinger eigenvalue problems.
problem Analyzing generalization error of neural networks for high-dimensional Schrödinger eigenvalue problems.
method Proves convergence rate of generalization error independent of dimension d under spectral Barron space assumption. Verifies assumption by proving regularity estimate. result Generalization error rate is independent of dimension d under spectral Barron space assumption. Paper analyzes DRM for solving high-dimensional elliptic PDEs with generalization bounds.
problem Analyzing generalization error of neural network methods for high-dimensional PDEs.
method Developed a new solution theory for spectral Barron space and derived generalization error bounds.
result Generalization error bounds are independent of dimension and solutions lie in spectral Barron space.
Study extends neural network approximation to time-varying PDEs using Fourier-Lebesgue spaces.
problem Limitation to static PDEs and different time-domain regularity.
method Extend spectral Barron spaces to anisotropic weighted Fourier-Lebesgue spaces, measure approximation error in Bochner-Sobolev norm.
result Established bound on approximation rate for functions in anisotropic weighted Fourier-Lebesgue spaces.
Study variation spaces for neural networks, linking them to approximation theory.
problem Understanding the variation spaces of shallow neural networks.
method Examined variation spaces defined by convex hulls and integral representations for a dictionary of functions.
result Found that Barron space, spectral Barron space, and Radon BV space are variation spaces for certain neural networks.
This paper introduces a new Barron space for graph signals and proves its properties for GCNNs.
problem Understanding and optimizing the performance of GCNNs on graph signals.
method Introducing a Barron space on graph signals, proving its properties, and showing the approximation and learning capabilities of GCNNs within this space.
result GCNN outputs are contained in the Barron space and can be well approximated by functions in this space.
New neural network rates for unbounded domains with weighted Sobolev spaces.
problem Improving neural network approximation rates for unbounded domains.
method Embedding results for weighted Fourier-Lebesgue spaces in weighted Sobolev spaces, followed by asymptotic approximation rates.
result Asymptotic approximation rates for shallow neural networks without curse of dimensionality for unbounded domains and Muckenhoupt weights.
Study embeddings between Barron spaces with various activation functions, focusing on RePU.
problem Understanding the influence of activation functions on infinitely wide neural networks.
method Prove embeddings by constructing push-forward maps on measures representing functions.
result Barron spaces with RePU activation have a hierarchical structure similar to Sobolev spaces.
New findings suggest Barron space doesn't defy curse of dimensionality for certain types of smoothness.
problem Understanding the curse of dimensionality in neural networks with different smoothness notions.
method Defined ADZ spaces via Mellin transform to encapsulate nonclassical smoothness, compared to classical smoothness.
result Evidence provided that Barron space doesn't defy curse of dimensionality for certain smoothness types.
Two-layer neural networks can approximate functions with fractal singularities.
problem Characterizing functions that can be represented by infinitely wide two-layer neural networks.
method Representation formulas and pointwise properties analysis.
result Functions with fractal or curved singularities cannot be represented by two-layer networks with finite path-norm.
Paper finds sparse representation of functions using inverse scale space flow.
problem Finding sparse representation of L2 functions. method Inverse scale space flow to minimize L2 loss. result Convergence to optimal solution in ideal and noisy cases.
Study approximates operator learning for PDEs using Fourier multipliers.
problem Approximating operator behavior for PDE simulations.
method Approximation of operator symbols in Fourier domain using semi-norms.
result Identifies conditions for achieving predefined approximation error.
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.
One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation errors by a particular machine learning model. In this paper, we address this iss…
The paper studies how to recover smooth Barron functions from Lp samples efficiently.
problem Recovering smooth Barron functions from limited Lp samples. method Analyzes the optimal reconstruction error of Barron functions using Lp samples. result Establishes bounds on the optimal reconstruction error for Barron functions.
This note explains when neural networks can be seen as Gaussian processes.
problem Understanding the relationship between neural networks and Gaussian processes.
method Formulating a Gaussian process regression based on neural network outputs and analyzing the resulting posterior mean functions.
result The posterior mean functions of neural networks follow a Gaussian process in certain cases, providing an interpretation of reproducing kernel Hilbert spaces.
Shallow diffusion models learn hidden low-dimensional structures effectively.
problem Learning from high-dimensional signals like images and video.
method Analysis of shallow diffusion models over the Barron space of single layer neural networks.
result Shallow diffusion models can adapt to simple low-dimensional structures, overcoming the curse of dimensionality.
Deep neural networks achieve optimal learning rates for high-dimensional classification.
problem Learning classification functions from noisy data with smooth boundaries.
method Empirical risk minimization over deep neural networks for locally Barron-regular decision boundaries.
result Optimal estimation rates are independent of dimension and can be achieved by deep neural networks.
This paper studies neural networks with bounded norms to avoid the curse of dimensionality.
problem The curse of dimensionality in approximating functions by neural networks.
method Investigates over-parameterized two-layer neural networks with norm constraints in RKHS.
result Improved sample complexity and generalization bounds for neural networks with bounded norms.
New analysis shows a gap between Gaussian RKHS and neural networks on unbounded domains.
problem Understanding the function space bias of neural networks compared to Gaussian RKHS.
method Infinite-center asymptotic analysis of neural network Banach space and Gaussian RKHS on unbounded domains.
result Certain functions in Gaussian RKHS have infinite norm in neural network Banach space on unbounded domains.
Neural networks can approximate high-dimensional classifiers with ReLU networks under margin conditions.
problem Approximating high-dimensional discontinuous classifiers with neural networks.
method Using ReLU neural networks with three hidden layers, approximating a classifier with a Barron-regular decision boundary.
result High-dimensional discontinuous classifiers can be approximated with a rate of n−1 under strong margin conditions. Group-invariant neural networks improve approximation accuracy for symmetric functions.
problem Improving approximation accuracy for symmetric functions using neural networks.
method Investigates the generalization error of group-invariant neural networks within the Barron framework.
result Group invariance introduces a factor δ that can significantly improve approximation accuracy when it is small.
This paper analyzes shallow ReLU networks in L^p and Sobolev spaces, focusing on approximation and generalization.
problem Approximation and generalization of shallow ReLU networks in L^p and Sobolev spaces.
method Spherical harmonic analysis and embeddings into spectral Barron spaces for L^p spaces, path-norm control for Sobolev spaces.
result Minimax-optimal rates for nonparametric regression with shallow ReLU networks under path-norm control.
Study wormholes in surface moduli space, proving conjecture.
problem Wormholes in moduli space of surfaces of general type.
method KSBA compactification, extremal P-resolutions, general wormhole conjecture.
result Proved wormhole conjecture for many cases.
The paper constructs exotic complex projective surfaces using rational blowdowns.
problem Finding exotic complex projective surfaces with specific invariants.
method Rational blowdowns and construction of Kollár--Shepherd-Barron--Alexeev surfaces.
result First exotic surfaces constructed, including minimal and symplectic ones.
We algebraically prove K-stability of polarized Calabi-Yau varieties and canonically polarized varieties with mild singularities. In particular, the} "stable varieties" introduced by Kollar-Shepherd-Barron and Alexeev, which form compact moduli space, are proven to be K-stable although it is well known that they are \t…
We establish L∞ and L2 error bounds for functions of many variables that are approximated by linear combinations of ReLU (rectified linear unit) and squared ReLU ridge functions with ℓ1 and ℓ0 controls on their inner and outer parameters. With the squared ReLU ridge function, we show th…
Sharp lower bounds on shallow neural networks' approximation rates are derived.
problem The efficiency of shallow neural networks in approximating functions.
method Lower bounding the L2-metric entropy and Kolmogorov n-widths of the convex hull of neural network basis functions. result Sharp lower bounds on the approximation rates for shallow neural networks are provided.
Study shows Bergman kernels match averages on quotient spaces, proving non-vanishing of Poincaré series.
problem Proving non-vanishing of Poincaré series on finite-volume quotients of Hermitian symmetric spaces.
method Using Bergman kernels and averaging over discrete groups, proving non-vanishing of Poincaré series.
result Large class of relative Poincaré series does not vanish on general locally symmetric spaces of finite volume.
Improved bounds for neural network approximations of functions.
problem Bounding the width of neural networks for function approximation.
method Extending Radon-based norms to bounded open sets and deriving new approximation bounds.
result Improved sparse approximation bounds for neural networks.
Study minimax rates for binary classifier estimation with margin conditions.
problem Estimating binary classifiers with geometric margin conditions.
method Derive lower bounds for worst-case learning rates over various function classes.
result Identify optimal rates close to O(n−1) for different function classes. We analyze random feature and two-layer neural networks using duality framework.
problem Understanding and analyzing functions within Fp,π and Barron spaces. method Duality framework based on information-based complexity (IBC).
result Sharp bounds for learning Fp,π using RFMs without curse of dimensionality for p>1. New algorithm reduces regret in online portfolio and quantum state learning.
problem Efficiently learning portfolios and quantum states online with minimal regret.
method BISONS algorithm for online portfolio selection, SCHRODINGER'S BISONS for quantum states, with polylogarithmic regret.
result First efficient algorithm with polylogarithmic regret for online portfolio selection and quantum states.
Study spectral distances on compact RCD spaces.
problem Understanding spectral convergence in RCD spaces.
method Established relationships between different spectral convergences and constructed a spectral approximation map.
result Found canonical spectral approximation map for RCD spaces.
We construct a simply connected minimal complex surface of general type with pg=0 and K2=2 which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with pg=0 and K2=1. In order to construct the example, we combin…
Classifies surfaces with T-singularities and ample canonical class.
problem Classifying surfaces with T-singularities and specific properties.
method Using KSBA moduli space and techniques for surfaces with T-singularities.
result Identifies surfaces with only T-singularities in the KSBA space and proves non-smoothability conditions.
The minimum description length (MDL) principle in supervised learning is studied. One of the most important theories for the MDL principle is Barron and Cover's theory (BC theory), which gives a mathematical justification of the MDL principle. The original BC theory, however, can be applied to supervised learning only …
This work defines a new function space for multi-layer neural networks.
problem Characterizing the function space of multi-layer neural networks.
method Defining a neural Hilbert ladder (NHL) as an infinite union of reproducing kernel Hilbert spaces (RKHSs).
result Established theoretical properties of the new function space, including generalization guarantees and dynamics of random fields.
Spheres' spectral structure converges to Gaussian space's as dimensions grow.
problem Understanding spectral convergence between high-dimensional spheres and Gaussian spaces.
method Proving spectral convergence using projections and eigenvalues.
result Spectral structure on high-dimensional spheres converges to Gaussian space's as dimensions increase.
Defines spectral selectors on lens spaces for contactomorphisms.
problem Understanding the geometry of contactomorphism groups on lens spaces.
method Using Givental's non-linear Maslov index, defines spectral selectors.
result Standard Reeb flow is a geodesic for specific lens spaces.
Study spectral settings of generalized Laplacians on homogeneous spaces.
problem Understanding the spectral properties of generalized Laplacians on compact homogeneous spaces.
method Investigates the generic spectral configuration of operators on G-invariant metrics on M=G/K. result The spectral setting depends on G-isometries and hidden symmetries. The paper generalizes spectral section concepts to non-compact spaces.
problem Generalizing spectral sections to non-compact base spaces.
method Generalization to arbitrary base spaces, applications to cobordism theorems, investigation of Riesz continuity.
result If a family of operators has a spectral section, it is Riesz continuous.
In this paper we give an explicit parametrisation of the moduli space of equivariant harmonic maps from a 2-torus to the 3-sphere. As Hitchin proved, a harmonic map of a 2-torus is described by its spectral data, which consists of a hyperelliptic curve together with a pair of differentials and a line bundle. The space …
This paper proposes a spectral clustering algorithm for hyperbolic spaces, improving efficiency over Euclidean methods.
problem Inefficient clustering in Euclidean spaces for complex data structures.
method Developed a spectral clustering algorithm using hyperbolic similarity matrices.
result The algorithm converges at least as fast as Euclidean spectral clustering and performs better on complex datasets.
Researchers describe a spectral sequence for knots in 3D space.
problem Understanding the Sinha spectral sequence for knots in R^3.
method Explicit description using Fox Neuwirth chain complexes and multicomplex structure.
result A non-trivial third page differential found, contradicting the rational case.
New methods avoid spectral pollution in transfer operators for accurate analysis.
problem Spectral pollution in finite-dimensional approximations of transfer operators.
method Algorithms for computing spectral properties of transfer operators without spectral pollution.
result Accurate spectral estimation across various applications, including protein folding models.
New reservoir computing approach handles infinite-dimensional systems.
problem Approximating and generalizing complex input/output systems.
method Randomly generated echo state networks with neural networks.
result Proves universal approximation properties for new class of systems.
New estimates show spectral gap stability in RCD spaces, close to Beta distribution.
problem Stability of spectral gap bounds in metric-measure spaces.
method Combines L1-functional inequality and Stein's method. result Sharp quantitative estimate for spectral gap stability.