arXiv research

Sharp lower bound for first Neumann eigenvalue found in terms of diameter and width.

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

Study bounds Urysohn width of manifolds under surgeries.

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

Study shows bound on Uryson width for specific 3D manifolds.

Sharp bounds on neural network approximation rates and widths.

The paper proves finite Urysohn 1-width for 4 and 5 manifolds with positive biRicci curvature.

Minimum width for ReLU networks to approximate L^p functions is max(d_x+1, d_y).

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

Develops a new theory of width for embedded circles in Riemannian manifolds.

Smooth manifolds can be triangulated with graphs of bounded twin-width.

Improved bounds for neural network approximations of functions.

Complex-valued neural networks can approximate any continuous function with bounded widths and depths.

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

A Riemannian n-manifold M has k-dimensional Uryson width bounded by a constant c >0 if there exists a continuous map f from M to an k-dimensional polyhedral space P, such that the pullbacks f^{-1}(p) of all points p in P have diameters bounded by c. We prove that an n-dimensional Riemannian manifold M with at least n-k…

Width trees link link invariants and bridge number.

Chow and Liu (1968) studied the problem of learning a maximumlikelihood Markov tree. We generalize their work to more complexMarkov networks by considering the problem of learning a maximumlikelihood Markov network of bounded complexity. We discuss howtree-width is in many ways the appropriate measure of complexity and…

This research proves that two min-max theories for hypersurfaces are equivalent.

Study Gaussian-process limits of neural networks using tensor programs.

New approach finds minimum width for deep, narrow MLPs.

Study on spherical bodies of constant width on the unit sphere, proving bounds on their relative effective radius.

The expressivity of neural networks as a function of their depth, width and type of activation units has been an important question in deep learning theory. Recently, depth separation results for ReLU networks were obtained via a new connection with dynamical systems, using a generalized notion of fixed points of a con…

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…

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

Study on ball widths and minimal submanifolds in space forms.

The paper proves unique geodesics on hyperbolic surfaces and finds lower bounds.

Study on 1-Uryson width of polyhedra and their covers.

New findings show depth is more important than width in neural networks.

Quantitative CLTs show neural network distributions converge to Gaussian as width increases.

Sine activation functions enable two-layer neural networks to learn modular addition more efficiently.

New insights into $p$-widths of surfaces, proving optimality and calculating constants.

Proves properties of neural network basins of attraction and their expressiveness.

The paper sets limits on neural network sizes based on dataset shapes.

Gradient methods improve deep network training with tighter bounds and faster convergence.

We provide an upper bound for the Gromov width of compact homogeneous Hodge manifolds $(M, ω)$ with $b_2(M)=1$. As an application we obtain an upper bound on the Seshadri constant $ε(L)$ where $L$ is the ample line bundle on $M$ such that $c_1(L)=[\fracωπ]$.

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

New optimizers control network width scaling, improving stability and transfer across different model sizes.

This paper tightens bounds on the smallest eigenvalue of NTK for deep ReLU networks.

Study proves deep narrow RNNs can approximate any function, with minimum width independent of data length.

We study a notion of "width" for Jordan curves in $\mathbb{CP}^1$, paying special attention to the class of quasicircles. The width of a Jordan curve is defined in terms of the geometry of its convex hull in hyperbolic three-space. A similar invariant in the setting of anti de Sitter geometry was used by Bonsante-Schle…

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

Uniform convergence of interpolators proven for Gaussian data.

The study proves a tube theorem for complex hyperbolic manifolds.

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

Deep neural networks' infinite-width behavior approximated by Gaussian models.

Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.

The volume spectrum of fiber bundles is bounded by the product of the base's volume spectrum and the fiber's volume.