arXiv research

Local X-ray transform works well near boundaries in hyperbolic spaces.

New compact examples of pseudo-Kähler manifolds with essential conformal transformations found.

Local invertibility of higher order tensor transforms on compact manifolds.

Study of Lorentzian manifolds with specific transformations.

Local invertibility of ray transforms on convex manifolds.

In this paper we consider the local X-ray transform for general flows. We extend the results on the local and global invertibility of the geodesic ray transform proved by Uhlmann and Vasy \cite{UV} to the X-ray transform for a general flow. The key improvement is that our argument for the ellipticity of the conjugated …

We define Radon transform and its inverse on the two-dimensional anti-de Sitter space over local fields using a novel construction through a quadratic equation over the local field. We show that the holographic bulk reconstruction of quantum fields in this space can be formulated as the inverse Radon transform, general…

This thesis explores GNNs, categorizing them into local and global approaches.

ABHT boosts regression by filtering regions with different smoothness.

Under a convexity assumption on the boundary we solve a local inverse problem, namely we show that the geodesic X-ray transform can be inverted locally in a stable manner; one even has a reconstruction formula. We also show that under an assumption on the existence of a global foliation by strictly convex hypersurfaces…

CutMix training technique improves spatial locality in Vision Transformers.

We study strict local martingales via h-transforms, a method which first appeared in Delbaen-Schachermayer. We show that strict local martingales arise whenever there is a consistent family of change of measures where the two measures are not equivalent to one another. Several old and new strict local martingales are i…

MLDL preserves manifold geometry in vector transformations.

Researchers classify and characterize Bäcklund transformations for hyperbolic Monge-Ampère systems.

Local Hardy spaces defined for Riemannian manifolds with bounded geometry.

We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and th…

Transformers with multiple layers learn to estimate bigram distributions, while single-layer models often get stuck in unigram local minima.

In many Lagrangian field theories one has a Poisson bracket defined on the space of local functionals. We find necessary and sufficient conditions for a transformation on the space of local functionals to be canonical in three different cases. These three cases depend on the specific dimensions of the vector bundle of …

Paper studies Transformer learning theory for Euclidean and Riemannian domains.

A new model detects and localizes anomalies in multivariate time series data.

Study S-duality and supersymmetry on curved manifolds using localization.

For many tasks and data types, there are natural transformations to which the data should be invariant or insensitive. For instance, in visual recognition, natural images should be insensitive to rotation and translation. This requirement and its implications have been important in many machine learning applications, a…

We construct a local action of the group of rational maps from $S^2$ to $GL(n,C)$ on local solutions of flows of the ZS-AKNS $sl(n,C)$-hierarchy. We show that the actions of simple elements (linear fractional transformations) give local Bäcklund transformations, and we derive a permutability formula from different fact…

ETC improves Transformer models for long and structured inputs.

Local supertwistors help study 6D conformal supergravity.

Probabilistic STNs improve image classification and robustness.

In this paper we study the local magnetic ray transform of symmetric tensor fields up to rank two on a Riemannian manifold of dimension $\geq 3$ with boundary. In particular, we consider the magnetic ray transform of the combinations of tensors of different orders due to the nature of magnetic flows. We show that such …

Time series forecasting is an important problem across many domains, including predictions of solar plant energy output, electricity consumption, and traffic jam situation. In this paper, we propose to tackle such forecasting problem with Transformer [1]. Although impressed by its performance in our preliminary study, …

We study the integral transform over a general family of broken rays in $\mathbb{R}^2$. It is natural for broken rays to have conjugate points, for example, when they are reflected from a curved boundary. If there are conjugate points, we show that the singularities cannot be recovered from local data and therefore art…

The expression (-1/u) times the Hessian of u transforms as a symmetric (0,2) tensor under projective coordinate transformations, so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold M, the section u can be regarded as a metric potential analogous to the local potential …

DNFS trains efficient samplers for discrete distributions using locally equivariant Transformers.

We study the Penrose transform for the `quaternionic objects' whose twistor spaces are complex manifolds endowed with locally complete families of embedded Riemann spheres with positive normal bundles.

Parametric spatial transformation models have been successfully applied to image registration tasks. In such models, the transformation of interest is parameterized by a fixed set of basis functions as for example B-splines. Each basis function is located on a fixed regular grid position among the image domain, because…

We illustrate how to compute local risk minimization (LRM) of call options for exponential Lévy models. We have previously obtained a representation of LRM for call options; here we transform it into a form that allows use of the fast Fourier transform method suggested by Carr & Madan. In particular, we consider Merton…

Study of affine transformations on topological manifolds, focusing on local freeness and solvability.

In this paper we develop the vectorial Ribaucour transformation for Euclidean submanifolds. We prove a general decomposition theorem showing that under {appropriate} conditions the composition of two or more vectorial Ribaucour transformations is again a vectorial Ribaucour transformation. An immediate consequence of t…

The paper models musical motif transformations in Beethoven's works.

We study two kinds of transformation groups of a compact locally conformally Kahler (l.c.K.) manifold. First we study compact l.c.K. manifolds with parallel Lee form by means of the existence of a holomorphic l.c.K. flow. Next, we introduce the Lee-Cauchy-Riemann (LCR) transformations as a class of diffeomorphisms pres…

Improves local learning models for complex feature extraction.

We consider the problem of developing a method to reconstruct a potential $q$ from the partial data Dirichlet-to-Neumann map for the Schrödinger equation $(-Δ_g+q)u=0$ on a fixed admissible manifold $(M,g)$. If the part of the boundary that is inaccessible for measurements satisfies a flatness condition in one directio…

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric…

We give a description of local and global moves on a class of locally planar trivalent graphs and we show that it contains $λ$-Scale calculus, therefore in particular untyped lambda calculus. Surprisingly, the beta reduction rule comes from a local "sewing" transformation of trivalent locally planar graphs.

Consider a Riemannian manifold in dimension $n\geq 3$ with strictly convex boundary. We prove the local invertibility, up to potential fields, of the geodesic ray transform on tensor fields of rank four near a boundary point. This problem is closely related with elastic \textit{qP}-wave tomography. Under the condition …

New method makes CP intervals locally adaptive using trainable transformations.

GridPyM handles grid diagrams for knot theory.

Let y''' = f(x, y, y', y'') be a 3rd order ODE. By Cartan equivalence method, we will study the local equivalence problem under the transformations group of time-fixed coordinates.

This paper studies transformer learning dynamics and initialization.

Study shows neural network parameters converge to ridgelet spectrum.