Riemannian manifolds with bounded Ricci curvature have finite Uryson width.
problem Bounding Uryson width for manifolds with Ricci curvature constraints.
method Continuous map to a polyhedral space with controlled diameter.
result Riemannian manifolds with specific Ricci curvature bounds have finite Uryson width.
Proves properties of neural network basins of attraction and their expressiveness.
problem Characterize the properties of basins of attraction in neural networks.
method Analyzes width-bounded neural networks, proving properties of basins of attraction.
result Boundedness and path-connectedness of basins of attraction under certain conditions.
Minimum width for ReLU networks to approximate L^p functions is max(d_x+1, d_y).
problem Characterizing the minimum width for ReLU networks to approximate L^p functions.
method Analyzing networks with ReLU activation functions and proving the minimum width required.
result The minimum width required for the universal approximation of L^p functions is exactly max(d_x+1, d_y).
This paper explores a novel sparse shortcut topology in neural networks.
problem Understanding the effectiveness and characteristics of shortcut connections in neural networks.
method Investigates a novel sparse shortcut topology, demonstrating its expressivity and generalizability.
result The proposed topology enables a one-neuron-wide deep network to approximate any univariate continuous function and shows excellent generalizability.
Develops a new theory of width for embedded circles in Riemannian manifolds.
problem Defining and understanding the width of embedded circles in Riemannian manifolds.
method Morse-Lusternik-Schnirelmann theory applied to geodesics and minimising configurations.
result Classifies configurations of minimising geodesics intersecting embedded circles.
Infinite width ReLU networks can approximate functions with bounded Euclidean norm.
problem Functions that can be approximated by ReLU networks with bounded Euclidean norm.
method Analyzing the minimal network norm required to approximate a given function.
result The minimal network norm for representing a function \( f \) is \( \max(\int |f''(x)| dx, |f'(-\infty) + f'(+\infty)|) \).
The paper proves rigidity for certain product spaces and bounds for band widths.
problem Proving rigidity for product spaces and bounds for band widths.
method Combining stable weighted slicing with a spectral Dirac operator argument.
result Closed spin (Mn,g) is isometrically covered by Sn−mimesRm under certain conditions. Paper analyzes NAC with neural networks for efficient policy optimization.
problem Improving sample and iteration complexity in policy optimization.
method Entropy regularization, averaging, neural network approximation, and optimization techniques.
result Entropy regularization and averaging ensure stability and sharp sample complexity bounds.
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.
Gradient descent optimally trains RNNs without overparameterization.
problem Training recurrent neural networks (RNNs) with gradient descent.
method Nonasymptotic analysis of gradient descent for RNNs with diagonal weight matrices.
result Gradient descent can achieve optimality in RNNs with a network size scaling logarithmically with the number of samples.
Extends spectral torus band inequalities for compact manifolds with scalar curvature bounds.
problem Proving upper bounds for the width of compact manifolds with boundary.
method Utilizes spacetime harmonic functions, μ-bubbles, and spinorial Callias operators.
result Generalizes Schoen-Yau black hole existence theorem to higher dimensions.
Characterizes functions representable by infinite-width ReLU networks with bounded weights.
problem Understanding function representation in overparameterized neural networks.
method Analyzes functions in Ws,1(R) spaces and their Radon transform. result All functions in Ws,1(R) can be represented with bounded norm. The paper defines conic reach and shows polynomial parallel volume in the plane.
problem Understanding geometric properties of sets in the plane.
method Introducing conic reach and using local Steiner formula to show polynomial volume.
result Polynomial parallel volume of sets in the plane with conic reach.
This work establishes the equivalence between neural networks and support vector machines.
problem Establishing the equivalence between neural networks and support vector machines.
method Proposed a method to establish the equivalence between infinitely wide neural networks trained by soft margin loss and standard soft margin SVMs with NTK trained by subgradient descent.
result The equivalence between NN and SVM is established, enabling practical applications such as non-vacuous generalization bounds and robustness certificates.
Decision trees and shallow neural networks have different geometric complexities, impacting their interpretability and accuracy.
problem The geometric simplicity of decision boundaries in decision trees conflicts with the approximation capabilities of shallow neural networks.
method Analysis of the Radon total variation (RTV) seminorm to compare geometric complexity of decision regions and neural network approximations.
result Smooth barrier scores can approximate decision regions with finite RTV, but their performance depends on the tube-mass condition near the decision boundary.
Unified approach learns Ising models from various dynamics and data types.
problem Efficiently learning Ising model parameters from data under diverse conditions.
method Simple logistic regression approach, generalizing existing algorithms.
result Logistic regression succeeds in multiple new settings where assumptions are violated.