arXiv research

We study the bilipschitz equivalence type of tree-graded spaces, showing that asymptotic cones of relatively hyperbolic groups (resp. asymptotic cones of groups containing a cut-point) only depend on the bilipschitz equivalence types of the pieces in the standard (resp. minimal) tree-graded structure. In particular, th…

The paper studies properties of group relations induced by compatible coarse structures.

New equivalences found between subsampling and ridge regularization methods.

Dropout and RaM become equivalent in large ResNets as depth and width increase.

Proposes new rule for ranking investment prospects over long horizons.

The study connects norms and filtrations on section rings of projective manifolds.

The paper introduces a new equivalence relation for finitely presented groups based on their asymptotic topology.

We give a parametrization to the asymptotic Teichmuller space of the open unit disk through equivalent classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of the unit disk. Then using the parametrization, we define a new metric on the asymptotic Teichmuller space. Two other rela…

Random Transformers behave like polynomial models in ICL with asymptotic growth.

High-dimensional data analysis has motivated a spectrum of regularization methods for variable selection and sparse modeling, with two popular classes of convex ones and concave ones. A long debate has been on whether one class dominates the other, an important question both in theory and to practitioners. In this pape…

Authors prove an asymptotic expansion for spectral zeta functions on discrete tori.

The basic model for high-frequency data in finance is considered, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam's sense asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function $σ$. A…

We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with boundary, in dimension $n\geq3$. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric $g$, there is a conformally equivalent asymptotically flat scal…

In [16], a new family of vector-valued risk measures called multivariate expectiles is introduced. In this paper, we focus on the asymptotic behavior of these measures in a multivariate regular variations context. For models with equivalent tails, we propose an estimator of these multivariate asymptotic expectiles, in …

We prove in a simple and coordinate-free way the equivalence bteween the classical definitions of the mass or the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asympto…

Algorithm decides if pseudo-Anosov flows have perfect fits.

The paper proves a non-asymptotic test error approximation for KRR.

Paper proves injectivity of non-abelian X-ray transform on certain spaces.

This work analyzes self-attention matrices using random matrix theory.

The paper analyzes the statistical properties of GANs using $f$-divergence.

New proof shows D-SGD and SAM are equivalent, revealing advantages of decentralization.

In the context of the Heston model, we establish a precise link between the set of equivalent martingale measures, the ergodicity of the underlying variance process and the concept of asymptotic arbitrage proposed in Kabanov-Kramkov and in Follmer-Schachermayer.

This study presents new analytic approximations of the stochastic-alpha-beta-rho (SABR) model. Unlike existing studies that focus on the equivalent Black-Scholes (BS) volatility, we instead derive the equivalent constant-elasticity-of-variance (CEV) volatility. Our approach effectively reduces the approximation error i…

The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with nonzero mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch-Hawking mass. Th…

We study the asymptotic behaviour of doubly periodic instantons with square-integrable curvature. Then, we establish the equivalence given by the Nahm transform between the doubly periodic instantons with square integrable curvature and the wild harmonic bundles on the dual torus.

This article investigates the eigenspectrum of the inner product-type kernel matrix $\sqrt{p} \mathbf{K}=\{f( \mathbf{x}_i^{\sf T} \mathbf{x}_j/\sqrt{p})\}_{i,j=1}^n$ under a binary mixture model in the high dimensional regime where the number of data $n$ and their dimension $p$ are both large and comparable. Based on…

Study on Teichmüller rays' asymptotic behavior and distances.

We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in particular, we determine the volume entropy, the spectrum and the relative densities of …

Study of two-layer NNs under Gaussian mixtures data, proving polynomial models equivalent to neural networks.

Defines CAMC discrete nets and their properties.

Two operators are equivalent in geometric scattering theory under certain conditions.

Random branched covers of groups are homotopy equivalent to geometrically small cancellation complexes.

Two curves in hyperbolic space share a bounded distance, leading to a minimizing surface.

Deep random feature models are analyzed for their performance with exact asymptotic expressions.

Minimal surfaces' area bounds proven equivalent, extending known results.

We consider the Bochner Laplacian on high tensor powers of a positive line bundle on a closed symplectic manifold (or, equivalently, the semiclassical magnetic Schrödinger operator with the non-degenerate magnetic field). We assume that the operator has discrete wells. The main result of the paper states asymptotic exp…

In this article we consider asymptotically harmonic manifolds which are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature $h$. We prove the following equivalences for asymptotically harmonic manifolds $X$ under the additional assumpti…

We investigate the asymptotics of the total number of simple $4a+1$-knots with Alexander polynomial of the form $mt^2 +(1-2m) t + m$ for some $m \in [-X, X]$. Using Kearton and Levine's classification of simple knots, we give equivalent algebraic and arithmetic formulations of this counting question. In particular, thi…

We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic g…

Directed acyclic graph (DAG) models are popular for capturing causal relationships. From observational and interventional data, a DAG model can only be determined up to its \emph{interventional Markov equivalence class} (I-MEC). We investigate the size of MECs for random DAG models generated by uniformly sampling and o…

The study shows how geometric Weyl bulk-density exponent rigidifies spectral encodings in O-regularly varying classes.

In this note, we use Chern's magic form $Φ_k$ in his famous proof of the Gauss-Bonnet theorem to define a mass for asymptotically flat manifolds. It turns out that the new defined mass is equivalent to the one that we introduced recently by using the Gauss-Bonnet-Chern curvature $L_k$. Moreover, this equivalence implie…

The main result of the paper is a version of the fundamental theorem of asset pricing (FTAP) for large financial markets based on an asymptotic concept of no market free lunch for monotone concave preferences. The proof uses methods from the theory of Orlicz spaces. Moreover, various notions of no asymptotic arbitrage …

Study shows double descent curve in high-dimensional linear regression with random projections.

Paper introduces a medoid-based approach for efficient Fréchet regression.

Uniqueness proven for specific types of geometric structures.

We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil-Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equ…

Geometric models help classify infinite-type surface mapping class groups.