arXiv research

Stochastic gradient descent's long-term fluctuations are described by a diffusion limit.

Uniqueness of asymptotic limits for Ricci-flat manifolds with linear volume growth is proven.

Proves limit curve theorem for incomplete metric spaces, applies to null distance in Lorentzian manifolds.

Solves asset allocation for investors with utility functions and limits.

Study reveals three limiting regimes for neural network functionals.

The study shows that limited liability can make banks more stable by choosing less risky assets.

We study the volume distribution of nodal domains of random band-limited functions on generic manifolds, and find that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and…

The study proves a quantitative functional CLT for neural networks with smooth activation functions.

Many graph clustering quality functions suffer from a resolution limit, the inability to find small clusters in large graphs. So called resolution-limit-free quality functions do not have this limit. This property was previously introduced for hard clustering, that is, graph partitioning. We investigate the resolution-…

The objective of this paper is to introduce the notion of generalized almost statistical (briefly, GAS) convergence of bounded real sequences, which generalizes the notion of almost convergence as well as statistical convergence of bounded real sequences. As a special kind of Banach limit functional, we also introduce …

Study of linear classifiers in infinite imbalance scenarios.

This paper constructs a function on Gromov-Hausdorff limits of 2-surfaces with curvature constraints.

The abstract proves every knot type can be parametrized by smooth functions and studies limit knot types.

The large-N limit of Segal-Bargmann transform on spheres is studied.

Proves continuum limits of Lipschitz learning using Γ-convergence.

The paper improves confidence regions for band-limited functions using tighter norm bounds and majority voting.

The paper creates nonparametric confidence bands for band-limited functions.

The study examines Hawkes processes and their long-term behavior.

The study extends classical results on harmonic functions to Riemannian manifolds with non-tangential boundary limits.

The paper explores properties of functions on Teichmüller space, proving theorems about limits and non-ergodicity.

The paper connects neural networks to physics using probability theory.

Functional central limit theorem for kernel gradient flow and infinitesimal gradient boosting

Accurate approximations to density functionals have recently been obtained via machine learning (ML). By applying ML to a simple function of one variable without any random sampling, we extract the qualitative dependence of errors on hyperparameters. We find universal features of the behavior in extreme limits, includi…

Study of deep Stable neural networks with various activation functions.

New framework resolves central limit behavior in differential privacy.

We consider the Willmore functional on graphs, with an additional penalization of the area where the curvature is non-zero. Interpreting the penalization parameter as a Lagrange multiplier, this corresponds to the Willmore functional with a constraint on the area where the graph is flat. Sending the penalization parame…

A $W^{1,p}$-metric on an $n$-dimensional closed Riemannian manifold naturally induces a distance function, provided $p$ is sufficiently close to $n$. If a sequence of metrics $g_k$ converges in $W^{1,p}$ to a limit metric $g$, then the corresponding distance functions $d_{g_k}$ subconverge to a limit distance function …

We extend the calculus of adiabatic pseudo-differential operators to study the adiabatic limit behavior of the eta and zeta functions of a differential operator $δ$, constructed from an elliptic family of operators indexed by $S^1$. We show that the regularized values $η(δ_t,0)$ and $tζ(δ_t,0)$ are smooth functions of …

Study on LOB dynamics using mean-field game theory.

The paper explores the correspondence between gradient flow lines of a function and its Lagrange multiplier functional.

Link signature limit depends on linking matrix under specific polynomial condition.

The aim of the paper is to investigate the relation between inverse limit of branched manifolds and codimension zero laminations. We give necessary and sufficient conditions for such an inverse limit to be a lamination. We also show that codimension zero laminations are inverse limits of branched manifolds. The inverse…

We analyze multi-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations. We rigorously establish the limiting behavior of the multi-layer neural network output. The limit procedure is valid for any number of hidde…

Researchers approximate partition functions on Riemannian spaces in the large N limit.

Using the reformulation in divergence form of the Euler-Lagrange equation for the Willmore functional as it was developed in "Analysis of the Willmore Functional" by T. Riviere (Invent. Math. 174), we study the limit of a local Palais-Smale sequence of weak Willmore immersions with locally square-integrable second fund…

Let $(M,g)$ be a smooth compact Riemannian surface with no boundary. Given a smooth vector field $V$ with finitely many zeroes on $M$, we study the distribution of the number of tangencies to $V$ of the nodal components of random band-limited functions. It is determined that in the high-energy limit, these obey a unive…

In order to choose a neural network architecture that will be effective for a particular modeling problem, one must understand the limitations imposed by each of the potential options. These limitations are typically described in terms of information theoretic bounds, or by comparing the relative complexity needed to a…

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network functio…

Examines challenges and proposes new approaches in machine learning theory.

Study on functions computed by deep-layered machines finds same distribution in neural networks and Boolean circuits.

Study rigidity of Ricci flow limits on nilpotent bundles with zero curvature.

In this paper we derive a scaling limit for an infinite dimensional limit order book model driven by Hawkes random measures. The dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator. With our choice of scaling the dynamics converges to a coupled SDE-ODE s…

The study examines exceptional sets for radial limits of superharmonic functions on curved manifolds.

Corrected Monti's blow-up analysis for H-minimizing sets in Heisenberg group.

We establish decoupled functional CLTs for two-time-scale stochastic approximation.

Finite-precision learning of $anh$ networks is limited by the Monte Carlo rate.

We show how the smooth geometry of Calabi-Yau manifolds emerges from the thermodynamic limit of the statistical mechanical model of crystal melting defined in our previous paper arXiv:0811.2801. In particular, the thermodynamic partition function of molten crystals is shown to be equal to the classical limit of the par…

Optimal first-order methods are shown to be fundamental limits in functional estimation.