arXiv research

Extends width estimates to family case using index theory.

Proves principles and estimates for initial data sets in Einstein equations.

Paper proves a noncompact version of Gromov's band-width estimate.

Study bounds Urysohn width of manifolds under surgeries.

Study shows upper limit for torical band width with spectral curvature bounds.

Estimates bandwidth for CMC initial data sets.

Estimates neural network error approximating compact sets.

The macroscopic version of Urysohn width for scalar curvature is disproven in high dimensions.

Establish optimal Lipschitz lower bounds for functions on manifolds with negative curvature, revealing interplay between width, boundary area, and topology.

We focus on estimating \emph{a priori} generalization error of two-layer ReLU neural networks (NNs) trained by mean squared error, which only depends on initial parameters and the target function, through the following research line. We first estimate \emph{a priori} generalization error of finite-width two-layer ReLU …

Fisher width is a geometric measure of complexity on statistical manifolds.

This article concerns the expressive power of depth in neural nets with ReLU activations and bounded width. We are particularly interested in the following questions: what is the minimal width $w_{\text{min}}(d)$ so that ReLU nets of width $w_{\text{min}}(d)$ (and arbitrary depth) can approximate any continuous functio…

Study shows neural networks trained with GD converge to Gaussian processes with polynomial decay.

The paper bounds the min-max width of embedded circles on spheres and manifolds.

The study explores positive scalar curvature metrics on non-orientable manifolds and their covers.

This work optimizes neural network bit-width and layer-width for efficiency.

This article concerns the expressive power of depth in deep feed-forward neural nets with ReLU activations. Specifically, we answer the following question: for a fixed $d_{in}\geq 1,$ what is the minimal width $w$ so that neural nets with ReLU activations, input dimension $d_{in}$, hidden layer widths at most $w,$ and …

Sharp bounds on neural network approximation rates and widths.

We prove that a bounded open set U in Euclidean n-space has k-width less than C(n) Volume(U)^{k/n}. Using this estimate, we give lower bounds for the k-dilation of degree 1 maps between certain domains in Euclidean space. In particular, we estimate the smallest (n-1)-dilation of any degree 1 map between two n-dimension…

This paper studies the expressive power of graph neural networks falling within the message-passing framework (GNNmp). Two results are presented. First, GNNmp are shown to be Turing universal under sufficient conditions on their depth, width, node attributes, and layer expressiveness. Second, it is discovered that GNNm…

Study shows polynomial-width neural networks can closely approximate infinite-width networks in polynomial time.

This paper establishes the (nearly) optimal approximation error characterization of deep rectified linear unit (ReLU) networks for smooth functions in terms of both width and depth simultaneously. To that end, we first prove that multivariate polynomials can be approximated by deep ReLU networks of width $\mathcal{O}(N…

Better uncertainty estimates for neural networks using Gaussian process priors.

Given a Riemannian metric on a homotopy $n$-sphere, sweep it out by a continuous one-parameter family of closed curves starting and ending at point curves. Pull the sweepout tight by, in a continuous way, pulling each curve as tight as possible yet preserving the sweepout. We show: Each curve in the tightened sweepout …

Plots show miscalibration directly as slopes of secant lines.

New framework analyzes deep learning optimization with finite width networks, revealing generalization gaps and excess risks.

We prove a sharp area estimate for catenoids that allows us to rule out the phenomenon of multiplicity in min-max theory in several settings. We apply it to prove that i) the width of a three-manifold with positive Ricci curvature is realized by an orientable minimal surface ii) minimal genus Heegaard surfaces in such …

Let $M$ be a closed connected spin manifold such that its spinor Dirac operator has non-vanishing (Rosenberg) index. We prove that for any Riemannian metric on $V = M \times [-1,1]$ with scalar curvature bounded below by $σ> 0$, the distance between the boundary components of $V$ is at most $C_n/\sqrtσ$, where $C_n = \…

We define the Wirtinger width of a knot. Then we prove the Wirtinger width of a knot equals its Gabai width. The algorithmic nature of the Wirtinger width leads to an efficient technique for establishing upper bounds on Gabai width. As an application, we use this technique to calculate the Gabai width of approximately …

The paper explores geometry and positive scalar curvature on non-compact manifolds.

Study of manifolds with specific curvature properties using capillary surfaces.

Empirical study compares finite- and infinite-width BNNs, revealing performance differences under model mismatch.

We propose a Generalized Dantzig Selector (GDS) for linear models, in which any norm encoding the parameter structure can be leveraged for estimation. We investigate both computational and statistical aspects of the GDS. Based on conjugate proximal operator, a flexible inexact ADMM framework is designed for solving GDS…

Paper characterizes gradient descent dynamics for neural networks with finite width.

In the framework of Multifractal Diffusion Entropy Analysis we propose a method for choosing an optimal bin-width in histograms generated from underlying probability distributions of interest. The method presented uses techniques of Rényi's entropy and the mean squared error analysis to discuss the conditions under whi…

The isospectral problem for p-widths is solved using Zoll metrics on S^2.

The paper explores rigidity theorems for spectral curvature bounds in 3-manifolds.

Width trees link link invariants and bridge number.

Lectures on deep learning properties in infinite and large-width networks.

Computed p-widths for hemisphere, first for manifolds with boundary.

Polygon $p$-widths are found via billiard trajectories.

A number of results for C$^2$-smooth surfaces of constant width in Euclidean 3-space ${\mathbb{E}}^3$ are obtained. In particular, an integral inequality for constant width surfaces is established. This is used to prove that the ratio of volume to cubed width of a constant width surface is reduced by shrinking it along…

The study proves a tube theorem for complex hyperbolic manifolds.

Computed p-widths for real projective plane.

Residual networks with block width max(d_x, d_y) approximate all functions.

Analysis of non-asymptotic estimation error and structured statistical recovery based on norm regularized regression, such as Lasso, needs to consider four aspects: the norm, the loss function, the design matrix, and the noise model. This paper presents generalizations of such estimation error analysis on all four aspe…

While studying the existence of closed geodesics and minimal hypersurfaces in compact manifolds, the concept of width was introduced in different contexts. Generally, the width is realized by the energy of the closed geodesics or the volume of minimal hypersurfaces, which are found by the Minimax argument. Recently, Ma…

Proves conjecture about sphere widths under rotational symmetry.