arXiv research

In this paper, we propose an implicit gradient descent algorithm for the classic $k$-means problem. The implicit gradient step or backward Euler is solved via stochastic fixed-point iteration, in which we randomly sample a mini-batch gradient in every iteration. It is the average of the fixed-point trajectory that is c…

If the cyclic sequence of faces for all the vertices in a map are of same type, then the map is said to be a semi-equivelar map. In this article, we classify all the types of semi-equivelar maps on the surface of Euler genus 3, $i.e.$, on the surface of Euler characteristic $-1$. That is, we present {a complete map typ…

Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A cr…

A classical result says that a free action of the circle $\Bbb{S}^1$ on a topological space $X$ is geometrically classified by the orbit space $B$ and by a cohomological class ${H}^{^{2}}{(B,\Bbb{Z})}$, the Euler class. When the action is not free we have a difficult open question: $Π$ : "Is the space $X$ determined by…

Given any diagram of a link, we define on the cube of Kauffman's states a "2-complex" whose homology is an invariant of the associated framed links, and such that the graded Euler characteristic reproduces the unnormalized Kauffman bracket. This includes a categorification of brackets skein relation. Then we incorporat…

New method improves Euler approximation for local stochastic volatility models.

Over the last decade, both the neural network and kernel adaptive filter have successfully been used for nonlinear signal processing. However, they suffer from high computational cost caused by their complex/growing network structures. In this paper, we propose two random Euler filters for complex-valued nonlinear filt…

EuSN uses Euler discretization for stable, non-dissipative reservoir computing.

ThiopheneIV is a new solver for implied volatility with proven monotonicity.

Paper studies particle method for LSV model calibration, proving convergence and error bounds.

We prove that a representation from the fundamental group of a closed surface of negative Euler characteristic with values in the isometry group of a Riemannian manifold of sectional curvature bounded by -1 can be dominated by a Fuchsian representation. Moreover, we prove that the domination can be made strict, unless …

New time series generation models improve accuracy and correlation identification.

Extends transversality to supergeometry, proving stability and genericity.

Diffusion models simulate molecular dynamics with adjustable accuracy.

Study proves convergence of interest rate model approximations.

In this paper we investigate the following question: under what conditions can a second-order homogeneous ordinary differential equation (spray) be the geodesic equation of a Finsler space. We show that the Euler-Lagrange partial differential system on the energy function can be reduced to a first order system on this …

We take a first step towards understanding the relationship between foliations and universally tight contact structures on hyperbolic 3-manifolds. If a surface bundle over a circle has pseudo-Anosov holonomy, we obtain a classification of "extremal" tight contact structures. Specifically, there is exactly one contact s…

The correspondence between residual networks and dynamical systems motivates researchers to unravel the physics of ResNets with well-developed tools in numeral methods of ODE systems. The Runge-Kutta-Fehlberg method is an adaptive time stepping that renders a good trade-off between the stability and efficiency. Can we …

Rectified flows achieve optimal sample complexity for generating data.

This paper provides a description of an algebraic setting for the Lagrangian formalism over graded algebras and is intended as the necessary first step towards the noncommutative C-spectral sequence (variational bicomplex). A noncommutative version of integration procedure, the notion of adjoint operator, Green's formu…

This research smooths out fluid equations to avoid sudden shocks.

Developed unbiased estimators for Heston model with stochastic interest rates.

Enhances learning of structured distributions using nonlinear denoising score matching.

We consider the stochastic volatility model $dS_t = σ_t S_t dW_t,dσ_t = ωσ_t dZ_t$, with $(W_t,Z_t)$ uncorrelated standard Brownian motions. This is a special case of the Hull-White and the $β=1$ (log-normal) SABR model, which are widely used in financial practice. We study the properties of this model, discretized in …

Recent advances in Bayesian learning with large-scale data have witnessed emergence of stochastic gradient MCMC algorithms (SG-MCMC), such as stochastic gradient Langevin dynamics (SGLD), stochastic gradient Hamiltonian MCMC (SGHMC), and the stochastic gradient thermostat. While finite-time convergence properties of th…

Corrected samplers reduce discretization error in discrete flow models without additional computational cost.

Langevin MCMC samples efficiently from Riemannian manifolds with geometric Euler-Murayama analysis.

Defines discrete differential geometry concepts in homotopy type theory.

New Euler characteristics for groupoids generalize orbifold Euler characteristics.

Applying standard Markov chain Monte Carlo (MCMC) algorithms to large data sets is computationally infeasible. The recently proposed stochastic gradient Langevin dynamics (SGLD) method circumvents this problem in three ways: it generates proposed moves using only a subset of the data, it skips the Metropolis-Hastings a…

The paper defines and calculates Euler characteristics for quandles.

ISALT uses inference to simulate SDEs with large time-steps, improving efficiency.

Poisson Midpoint Method improves Langevin Dynamics for diffusion models.

Two types of nonvanishing results are presented for compact Kähler varieties.

Efficiently simulates the Heston model with large time steps using a novel method.

Proves Euler characteristic of collapsing Alexandrov spaces.

Odd-dimensional orbifolds' Euler characteristic equals half of their boundary's.

New evidence supports the Euler class one conjecture for tight contact structures.

RFM uses tangent vector fields to match data on manifolds, analyzing TV convergence for Euler discretization.

Summarizes connections between Euler characteristic theorems and conjectures.

We present a dynamical system framework for understanding Nesterov's accelerated gradient method. In contrast to earlier work, our derivation does not rely on a vanishing step size argument. We show that Nesterov acceleration arises from discretizing an ordinary differential equation with a semi-implicit Euler integrat…

Expands Euler-Poincare characteristic to supergeometry.

It is well known that the Euler characteristic of an odd dimensional compact manifold is zero. An Euler complex is a combinatorial analogue of a compact manifold. We present here an elementary proof of the corresponding result for Euler complexes.

In this paper, we study Euler classes in groups of homeomorphisms of Seifert fibered 3-manifolds. We show that, in contrast to the familiar Euler class for $\mathrm{Homeo}_0(S^1)^δ$, these Euler classes for $\mathrm{Homeo}_0(M^3)^δ$ are unbounded classes. In fact, we give examples of flat topological M bundles over a g…

Extends Zeitlin's model to 3-D axisymmetric Euler equations.

Schwarzian derivative connects to Euler-Lagrange equations in variational calculus.

Co-Euler structures were studied by Burghelea and Haller on closed manifolds as dual objects to Euler structures. We extend the notion of co-Euler structures to the situation of compact manifolds with boundary. As an application, by studying their variation with respect to smooth changes of the Riemannian metric, co-Eu…

Shows Euler-like vector fields come from specific embeddings.