arXiv research

Wave trace singularity formula for fibre bundles generalizes Poisson summation.

New techniques prove quantum modularity for various functions.

A new discrete privacy mechanism for federated learning.

Paper extends trigonometric summation formula with weights.

The study shows how certain ODEs and integrals are regular under Borel summation.

Combining differential privacy and federated learning improves data security.

In recent work, Cheu et al. (Eurocrypt 2019) proposed a protocol for $n$-party real summation in the shuffle model of differential privacy with $O_{ε, δ}(1)$ error and $Θ(ε\sqrt{n})$ one-bit messages per party. In contrast, every local model protocol for real summation must incur error $Ω(1/\sqrt{n})$, and there exist …

This paper presents a deep learning method for faster magnetic resonance imaging (MRI) by reducing k-space data with sub-Nyquist sampling strategies and provides a rationale for why the proposed approach works well. Uniform subsampling is used in the time-consuming phase-encoding direction to capture high-resolution im…

New theorem shows every integer can be represented by knot summation.

Study geodesics on flat tori, focusing on convex bodies.

Mathematical structures link Gromov-Witten to Donaldson-Thomas invariants.

We study the length, weak length and complex length spectrum of closed geodesics of a compact flat Riemannian manifold, comparing length-isospectrality with isospectrality of the Laplacian acting on p-forms. Using integral roots of the Krawtchouk polynomials, we give many pairs of p-isospectral flat manifolds having di…

Improved greedy 2-coordinate updates for optimization problems with constraints.

Stochastic gradient descent updates parameters with summation gradient computed from a random data batch. This summation will lead to unbalanced training process if the data we obtained is unbalanced. To address this issue, this paper takes the error variance and error mean both into consideration. The adaptively adjus…

The paper analyzes how the one-dimensional Wasserstein distance captures pointwise density differences in finite samples.

Kernel density estimation (KDE) is a popular statistical technique for estimating the underlying density distribution with minimal assumptions. Although they can be shown to achieve asymptotic estimation optimality for any input distribution, cross-validating for an optimal parameter requires significant computation do…

A new method slices and sums radial kernels faster.

We provide faster algorithms for the problem of Gaussian summation, which occurs in many machine learning methods. We develop two new extensions - an O(Dp) Taylor expansion for the Gaussian kernel with rigorous error bounds and a new error control scheme integrating any arbitrary approximation method - within the best …

We consider fast kernel summations in high dimensions: given a large set of points in $d$ dimensions (with $d \gg 3$) and a pair-potential function (the {\em kernel} function), we compute a weighted sum of all pairwise kernel interactions for each point in the set. Direct summation is equivalent to a (dense) matrix-vec…

Quantum dilogarithm function proven from a linear difference equation.

Dirichlet-Neumann duality for Riemannian submersions

For an eigenfunction of the Laplacian on a hyperbolic Riemann surface, the coefficients of the Fourier expansion are described as intertwining functionals. All intertwiners are classified. A refined growth estimate for the coefficients is given and a summation formula is proved.

We prove an explicit formula of the Berezin star product on Kaehler manifolds. The formula is expressed as a summation over certain strongly connected digraphs. The proof relies on a combinatorial interpretation of Englis' work on the asymptotic expansion of the Laplace integral.

Finet uses FBN for efficient, lightweight neural networks.

For the Möbius spheres $S^{q,p}$, we give alternative elementary proofs of the recursive formulas for GJMS-operators and $Q$-curvatures due to the first author [Geom. Funct. Anal. 23, (2013), 1278-1370; arXiv:1108.0273]. These proofs make essential use of the theory of hypergeometric series.

Geometrically proves twisted Poincaré duality for orientable Poisson manifolds.

In this paper, we study some vanishing identities for Gromov-Witten invariants conjectured by K. Liu and H. Xu. We will prove these conjectures in the case that the summation range is large compare to genus. In fact, in such cases, we can obtain a vanishing identity which is stronger than their conjectures. Moreover we…

This paper is an introduction to Khovanov homology, starting with the Kauffman bracket state summation, emphasizing the Bar-Natan Canopoloy and tangle cobordism approach. The paper discusses a simplicial approach to Khovanov homology and a quantum model for it so that the graded Euler characteristic that produces the J…

We propose a Poisson-Lie analog of the symplectic induction procedure, using an appropriate Poisson generalization of the reduction of symplectic manifolds with symmetry. Having as basic tools the equivariant momentum maps of Poisson actions, the double group of a Poisson-Lie group and the reduction of Poisson manifold…

New Poisson structures on algebras linked to derivatives.

VRSMD improves SMD convergence and has implicit regularization.

New Lie groups found for Poisson diffeomorphisms.

The paper connects 3D manifold invariants to hyperbolic cone metrics and discrete Fourier transforms.

Given a measured lamination on a finite area hyperbolic surface we consider a natural measure Mon the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure M gives summation id…

We semiclassicalise the theory of quantum group principal bundles to the level of Poisson geometry. The total space $X$ is a Poisson manifold with Poisson-compatible contravariant connection, the fibre is a Poisson-Lie group in the sense of Drinfeld with bicovariant Poisson-compatible contravariant connection, and the …

Popular deep neural networks (DNNs) spend the majority of their execution time computing convolutions. The Winograd family of algorithms can greatly reduce the number of arithmetic operations required and is present in many DNN software frameworks. However, the performance gain is at the expense of a reduction in float…

We construct a Poisson isomorphism between the formal Poisson manifolds g^* and G^*, where g is a finite dimensional quasitriangular Lie bialgebra. Here g^* is equipped with its Lie-Poisson (or Kostant-Kirillov-Souriau) structure, and G^* with its Poisson-Lie structure. We also quantize Poisson-Lie dynamical r-matrices…

The paper establishes a Poisson Poincaré-Dulac theorem for Poisson-flat connections.

The first cohomology of Poisson algebras is described and conditions for its vanishing are established.

The paper normalizes Poisson saturation of coregular submanifolds.

Develops reduction method for strong Dirac maps.

We reformulate the Poisson structure discovered by Fock and Rosly on moduli spaces of flat connections over marked surfaces in the framework of Poisson structures defined by Lie algebra actions and quasitriangular $r$-matrices, and we show that it is an example of a mixed product Poisson structure associated to pairs o…

A new Poisson bracket defined on Poisson structures with applications to fixed points and cohomology.

Abstract: Homotopy Poisson algebra models for reduced spaces derived from Poisson structures.

Study Poisson algebras for Hamiltonian systems linearization.

We show that there exists an integrable function on the $n$-sphere $(n\ge 2)$, whose Cesàro (C,$\frac{n-1}{2}$) means with respect to the spherical harmonic expansion diverge unboundedly almost everywhere. By studying equivalence theorems, we also obtain the corresponding results for Riesz and Bochner-Riesz means. This…

This chapter is an attempt to present a mathematical theory of compound fractional Poisson processes. The chapter begins with the characterization of a well-known Lévy process: The compound Poisson process. The semi-Markov extension of the compound Poisson process naturally leads to the compound fractional Poisson proc…

Gaussian surrogates improve Poisson imaging performance at low doses.