arXiv research

In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere and the isotopies between these positions. She poses several questions about these width complexes; in particular, she asks whether the…

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

New findings on depth vs. width in neural networks, showing depth can improve learnability.

The paper classifies knot Floer complexes of low width, simplifying knot bases.

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

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…

There are many "minimax" complexity functions in mathematics: width of a tree or a link, Heegaard genus of a 3-manifold, the Cheeger constant of a Riemannian manifold. We define such a function w, "width", on countable (or finite) groups and show w(Z^k) = k-1.

We characterize the first min-max width of real projective spaces of any dimension. The width is the minimum area over the Clifford hypersurfaces. We also compute the Morse index of the Clifford hypersurfaces in the complex and quaternionic projective spaces.

This paper investigates the approximation power of three types of random neural networks: (a) infinite width networks, with weights following an arbitrary distribution; (b) finite width networks obtained by subsampling the preceding infinite width networks; (c) finite width networks obtained by starting with standard G…

Deep forests enhance expressiveness exponentially with depth, not width or tree size.

The study proves a tube theorem for complex hyperbolic manifolds.

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…

Develops a new theory for neural systems stability and width effects.

We give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the diagram. In particular, we get a $O(N^{\frac{3}{2} \mathrm{cw}} \mathrm{poly}(n))$ time algorithm to compute any Resh…

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

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

Bayesian neural networks learn efficiently at infinite width, matching polynomial-width performance.

Empirical study shows standard CNNs deviate from NTK predictions.

A remarkable characteristic of overparameterized deep neural networks (DNNs) is that their accuracy does not degrade when the network's width is increased. Recent evidence suggests that developing compressible representations is key for adjusting the complexity of large networks to the learning task at hand. However, t…

To each knot $K\subset S^3$ one can associated its knot Floer homology $\hat{HFK}(K)$, a finitely generated bigraded abelian group. In general, the nonzero ranks of these homology groups lie on a finite number of slope one lines with respect to the bigrading. The width of the homology is, in essence, the largest horizo…

Recent theoretical work has guaranteed that overparameterized networks trained by gradient descent achieve arbitrarily low training error, and sometimes even low test error. The required width, however, is always polynomial in at least one of the sample size $n$, the (inverse) target error $1/ε$, and the (inverse) fail…

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

New method combines HQR and WACI for better time series prediction intervals.

Inspired by the work of G. Lu on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer--Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also c…

Neural Tangents is a library designed to enable research into infinite-width neural networks. It provides a high-level API for specifying complex and hierarchical neural network architectures. These networks can then be trained and evaluated either at finite-width as usual or in their infinite-width limit. Infinite-wid…

Paper calculates topological complexity of robot movement in narrow aisles.

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

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 …

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

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

This work explores feature learning tradeoffs in neural networks.

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

The trunk of a knot in $S^3$, defined by Makoto Ozawa, is a measure of geometric complexity similar to the bridge number or width of a knot. We prove that for any two knots $K_1$ and $K_2$, we have $tr(K_1 \# K_2) = \max\{tr(K_1),tr(K_2)\}$, confirming a conjecture of Ozawa. Another conjecture of Ozawa asserts that any…

Width trees link link invariants and bridge number.

The paper provides examples of keen weakly reducible bridge spheres for links in b-bridge position.

Scharlemann and Thompson define a numerical complexity for a 3-manifold using handle decompositions of the manifold. We show that for compact hyperbolic 3-manifolds this is linearly related to a definition of metric complexity in terms of the areas of level sets of Morse functions.

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.

Transformers capture combinatorial tasks with bounded error and logarithmic sample dependence.

Computed p-widths for real projective plane.

Study bounds Urysohn width of manifolds under surgeries.

Bayesian linear networks reveal optimal depth and width trade-offs.

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

New approach predicts generalization of deep neural networks in proportional-width regime.

Study on Klein bottle's cotangent bundle using contact homology.

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…

New principle reveals how neural networks learn complex interactions.