arXiv research

Characterizes when differential forms have weak exterior derivatives based on limiting behavior of integration over simplices.

Duality result connects bounded cohomology to relative Gromov seminorm.

Unified framework for solving fixed-point equations in deterministic and stochastic settings.

We show that the Thurston seminorms of all finite covers of an aspherical 3-manifold determine whether it is a graph manifold, a mixed 3-manifold or hyperbolic.

The analysis of classical consensus algorithms relies on contraction properties of adjoints of Markov operators, with respect to Hilbert's projective metric or to a related family of seminorms (Hopf's oscillation or Hilbert's seminorm). We generalize these properties to abstract consensus operators over normal cones, w…

Bayesian nonparametric LABS model adapts to function smoothness in Besov spaces.

Kolmogorov-Arnold Networks improve deep learning adaptivity and can approximate Besov functions optimally.

This paper studies continuum-armed bandits under Besov smoothness conditions and derives minimax rates.

Deep learning has shown high performances in various types of tasks from visual recognition to natural language processing, which indicates superior flexibility and adaptivity of deep learning. To understand this phenomenon theoretically, we develop a new approximation and estimation error analysis of deep learning wit…

Signed seminorms linked to real tropical spaces and matroids.

Bayesian neural networks achieve optimal posterior contraction rates in Besov spaces with intrinsic dimensionality.

Wavelet-based online learning adapts to noisy Besov spaces with high probability.

New cyclicity measures defined in weighted Besov spaces, with stability and geometric analysis.

Study analyzes deep learning's performance on variable exponent Besov space, highlighting adaptivity benefits.

We consider pairs of a non-empty compact connected and locally connected Hausdorff space and a real-valued continuous function. Our aim is to measure the difference between this kind of the pairs. In this notes we introduce new pseudodistances between pairs associated with reparametrization invariant seminorms. We fini…

Study minimax rates for density estimation under Huber contamination and Besov IPM losses.

Geometric analysis on real analytic manifolds using seminorms.

Bayesian neural network achieves nearly optimal performance in Besov space.

New spatiotemporal Besov process improves CT image reconstruction and other inverse problems.

Deep ReLU networks approximate functions in Sobolev and Besov spaces efficiently.

Bayesian KANs achieve near-minimax posterior contraction rates in anisotropic Besov spaces.

We describe a method to compute the Culler-Shalen seminorms of a knot, using the (-3,3,4) pretzel knot as an illustrative example. We deduce that the SL2(C)-character variety of this knot consists of three algebraic curves and that it admits no non-trivial cyclic or finite surgeries. We also summarize similar results f…

We show that the SL(2,C)-character variety of the (-2,3,n) pretzel knot consists of two (respectively three) algebraic curves when 3 does not divide n (respectively 3 divides n) and give an explicit calculation of the Culler-Shalen seminorms of these curves. Using this calculation, we describe the fundamental polygon a…

Deep ReLU networks can efficiently approximate Sobolev and Besov functions.

ConvResNets approximate Besov functions and classify on low-dimensional manifolds.

Researchers found abnormal extremals on specific Lie groups.

We determine the total Culler-Shalen seminorms for the 3-manifolds W_{p/q}:=W(p/q,-) obtained by Dehn filling with slope p/q on one boundary component of the Whitehead link exterior W when p is odd. As part of the proof, we use an explicit parametrization of the eigenvalue variety of W to find a one-variable polynomial…

Study finds abnormal paths on specific Lie groups using algebraic structures.

Adaptive algorithm minimizes online prediction errors for irregular data.

In this paper we propose a function space approach to Representation Learning and the analysis of the representation layers in deep learning architectures. We show how to compute a weak-type Besov smoothness index that quantifies the geometry of the clustering in the feature space. This approach was already applied suc…

Study of $p$-adic simplicial volumes and their properties.

An explicit seminorm $||f||_{#}$ on the vector space of Chow vectors of projective varieties is introduced, and shown to be a generalized Mabuchi energy functional for Chow varieties. The singularities of the Chow varieties give rise to currents supported on their singular loci, while the regular parts are shown to rep…

New Q-learning method achieves optimal sample complexity for average-reward problems.

Study shows global invertibility in nonlinear elasticity with vanishing self-repulsion term.

Study on distributed nonparametric function estimation with optimal rate and cost of adaptation.

Consider the exterior M of a hyperbolic knot lying in a closed, connected, orientable 3-manifold. Culler and Shalen defined norm on H_1(dM;R) using the SL(2,C) character variety of pi_1(M). The Culler-Shalen norm encodes many topological properties of M; in particular it provides information about Dehn fillings of M. T…

Measure homology is a variation of singular homology designed by Thurston in his discussion of simplicial volume. Zastrow and Hansen showed independently that singular homology (with real coefficients) and measure homology coincide algebraically on the category of CW-complexes. It is the aim of this paper to prove that…

Extends Onsager's conjecture to Besov spaces on manifolds with boundary.

Deep learning has exhibited superior performance for various tasks, especially for high-dimensional datasets, such as images. To understand this property, we investigate the approximation and estimation ability of deep learning on anisotropic Besov spaces. The anisotropic Besov space is characterized by direction-depen…

Deep ReLU networks can approximate and learn smooth functions efficiently.

Paper analyzes sample complexity for offline RL with deep ReLU networks.

Deep learning models can adaptively estimate functions with varying smoothness using regularization.

Introduces HTV to measure function complexity in learning schemes.

We present a new proof of Thurston's theorem that the unit ball of a seminorm on $\mathbf{R}^d$ taking integer values on $\mathbf{Z}^d$ is a polyhedra defined by finitely many inequalities with integer coefficients.

The stable 4-genus of a knot K in 3-space is the limiting value of g_4(nK)/n, where g_4 denotes the 4-genus and n goes to infinity. This induces a seminorm on CQ, the concordance group tensored with the rational numbers. Basic properties of the stable genus are developed, as are examples focused on understanding the un…

Study on stochastic covariant derivatives in curved space-time.

In this paper, we consider various tensorial estimates in geometric Besov-type norms on a one-parameter foliation of surfaces with evolving geometries. Moreover, we wish to do this with only very weak control on these geometries. Several of these estimates were established in previous works by S. Klainerman and I. Rodn…

We classify Dehn surgeries on (p,q,r) pretzel knots resulting in a manifold M(s) having cyclic fundamental group and analyze those leading to a finite fundamental group. The proof uses the theory of cyclic and finite surgeries developed by Culler, Shalen, Boyer, and Zhang. In particular, Culler-Shalen seminorms play a …