arXiv research

The paper defines strong emergence in field theories and proves it exists between certain theories.

The paper axiomatizes strong emergence in parameterized field theories and proves existence theorems.

In this article we present a natural generalization of Newton's Second Law valid in field theory, i.e., when the parameterized curves are replaced by parameterized submanifolds of higher dimension. For it we introduce what we have called the geodesic $k$-vector field, analogous to the ordinary geodesic field and which …

Study examines infinite limits of transformer dynamics, identifying key parameterizations.

Introduces a neural network-based method for efficient state and parameter estimation in complex systems.

Residual networks with depthwise hyperparameter scaling transfer optimal hyperparameters across width and depth.

A field theory is constructed in the context of parameterized absolute parallelism geometry. The theory is shown to be a pure gravity one. It is capable of describing the gravitational field and a material distribution in terms of the geometric structure of the geometry used (the parallelization vector fields). Three t…

Constructs maps from field theories to complexified K-theory and elliptic cohomology.

Harmonic basis vector fields on surfaces

The importance of Einstein's geometrization philosophy, as an alternative to the least action principle, in constructing general relativity (GR), is illuminated. The role of differential identities in this philosophy is clarified. The use of Bianchi identity to write the field equations of GR is shown. Another similar …

Study bounds graph neural networks' over-parameterized error.

New framework analyzes deep neural networks using feature probabilities.

Chas and Sullivan have defined an intersection-type product on the homology of the free loop space LM of an oriented manifold M. In this paper we show how to extend this construction to a topological conformal field theory of degree d. In particular, we get operations on the homology of LM which are parameterized by th…

Compact parameterization improves Bayesian neural network performance.

We consider four-dimensional gravity coupled to a non-linear sigma model whose scalar manifold is a non-compact geometrically finite surface $Σ$ endowed with a Riemannian metric of constant negative curvature. When the space-time is an FLRW universe, such theories produce a very wide generalization of two-field $α$-att…

In the past decade, deep neural networks (DNNs) came to the fore as the leading machine learning algorithms for a variety of tasks. Their raise was founded on market needs and engineering craftsmanship, the latter based more on trial and error than on theory. While still far behind the application forefront, the theore…

Two novel algorithms for conformal parameterization of multiply-connected surfaces.

This paper organizes recent deep learning theory advances.

Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. On the theoretical side, a long line of works has been focusing on training neural networks with one hidden layer. The theory of multi-layer networks remains largely …

The study analyzes deep linear networks from random initialization, capturing dynamics and hyperparameter effects.

Gradient flows for surface energies with tensor fields are derived and analyzed.

New scaling framework for MoE architectures ensures stability and optimal performance at scale.

New bounds show BBVI's gradient variance matches SGD conditions, improving parameterization efficiency.

Local convergence theory for mildly over-parameterized neural nets.

Enhances quantum sensing by eliminating multiple oscillations in field amplitude estimation.

A new algorithm for learning shallow neural networks with infinite width.

Conformal surface parameterization is useful in graphics, imaging and visualization, with applications to texture mapping, atlas construction, registration, remeshing and so on. With the increasing capability in scanning and storing data, dense 3D surface meshes are common nowadays. While meshes with higher resolution …

New parameterization for $(1,1)$-knots simplifies their study.

A parameterization that is a modified version of a previous work is proposed for the returns and correlation matrix of financial time series and its properties are studied. This parameterization allows easy introduction of non-stationarity and it shows several of the characteristics of the true, observed realizations, …

The class of non-rigid registration methods proposed in the framework of PDE-constrained Large Deformation Diffeomorphic Metric Mapping is a particularly interesting family of physically meaningful diffeomorphic registration methods. PDE-constrained LDDMM methods are formulated as constrained variational problems, wher…

This is a survey of the theory of complex projective (CP^1) structures on compact surfaces. After some preliminary discussion and definitions, we concentrate on three main topics: (1) Using the Schwarzian derivative to parameterize the moduli space (2) Thurston's parameterization of the moduli space using grafting (3) …

New GM layers improve neural network performance.

DeformRS certifies deep networks against various input deformations.

New methods solve tensor-on-tensor regression with unknown rank, revealing benefits of over-parameterization.

Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces o…

This paper introduces a new bound to explain generalization in over-parameterized models.

Bayesian neural networks ignore data in infinite units limit.

The paper explores how over-parameterized linear regression models generalize without violating learning theory principles.

This paper presents a method to compute the {\it quasi-conformal parameterization} (QCMC) for a multiply-connected 2D domain or surface. QCMC computes a quasi-conformal map from a multiply-connected domain $S$ onto a punctured disk $D_S$ associated with a given Beltrami differential. The Beltrami differential, which me…

Surface parameterizations have been widely used in computer graphics and geometry processing. In particular, as simply-connected open surfaces are conformally equivalent to the unit disk, it is desirable to compute the disk conformal parameterizations of the surfaces. In this paper, we propose a novel algorithm for the…

Knoop enhances variable selection with over-parameterization and knockoffs.

Novel PCA method for high-dimensional inverse problems.

New framework for better mapping of surfaces onto ellipsoids.

Develops spherical density-equalizing maps for closed surfaces.

The study examines spectral dynamics in deep neural networks, predicting how outliers evolve during training.

We describe the sample paths of the stochastic field $F = F_t(v,x,q)$ of aggregate utilities parameterized by Pareto weights $v$ and total cash amounts $x$ and stocks' quantities $q$ in an economy. We also describe the sample paths of the stochastic field $G = G_t(u,y,q)$, which is conjugate to $F$ with respect to the …

Numerous methods for computing conformal mesh paramterizations has been developed due to the vast applications in the field of geometry processing. Spectral conformal parameterization (SCP) is one of these methods to computing a quality conformal parameterization based on the spectral technique. SCP focus on a generali…

Determinantal point processes (DPPs), which arise in random matrix theory and quantum physics, are natural models for subset selection problems where diversity is preferred. Among many remarkable properties, DPPs offer tractable algorithms for exact inference, including computing marginal probabilities and sampling; ho…