arXiv research

TOLD++ improves convergence of diffusion models by critically damping the forward transition matrix.

New Langevin method achieves third order convergence for strongly log-concave distributions.

We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find all fundamental invariants of a system of third order ODEs and, in particular, de…

Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points su…

We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution $ν= e^{-f}$ on $\mathbb{R}^n$. We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming $ν$ satisfies a log-Sobolev inequality and the Hessian of $f$ is bounded. Notably, we do not assume convexity or boun…

This paper is devoted to a third order study of the end-point map in sub-Riemannian geometry. We first prove third order open mapping results for maps from a Banach space into a finite dimensional manifold. In a second step, we compute the third order term in the Taylor expansion of the end-point map and we specialize …

Third-order symmetric Lorentzian manifolds, i.e. Lorentzian manifold with zero third derivative of the curvature tensor, are classified. These manifolds are exhausted by a special type of pp-waves, they generalize Cahen-Wallach spaces and second-order symmetric Lorentzian spaces.

A third-order approximation for close-to-the-money European option prices under an infinite-variation CGMY Lévy model is derived, and is then extended to a model with an additional independent Brownian component. The asymptotic regime considered, in which the strike is made to converge to the spot stock price as the ma…

Minimal surfaces in third-order ODEs identified for linear second-order ODEs.

Recently, we proposed to transform the outputs of each hidden neuron in a multi-layer perceptron network to have zero output and zero slope on average, and use separate shortcut connections to model the linear dependencies instead. We continue the work by firstly introducing a third transformation to normalize the scal…

NR retraction approximates geodesics on submanifolds efficiently.

Study natural invariants for third order nonlinear operators on 2D manifolds.

New variational principles found for conformal geodesics.

FedAc accelerates Federated Averaging for distributed optimization.

We characterize Lorentzian three-dimensional hyper-CR Einstein-Weyl structures in terms of invariants of the associated third order ordinary differential equations.

We study the Proximal Langevin Algorithm (PLA) for sampling from a probability distribution $ν= e^{-f}$ on $\mathbb{R}^n$ under isoperimetry. We prove a convergence guarantee for PLA in Kullback-Leibler (KL) divergence when $ν$ satisfies log-Sobolev inequality (LSI) and $f$ has bounded second and third derivatives. Thi…

The thesis clarifies when local updates outperform centralized methods in heterogeneous data environments.

We construct a family of split signature Einstein metrics in four dimensions, corresponding to particular classes of third order ODEs considered modulo fiber preserving transformations of variables.

A second order variational description of the autoparallel curves of some differential-geometric connection for the third order Mathisson's 'new mechanics' of a relativistic free spinning particle is suggested starting from general requirements of invariance and 'variationality'.

This paper is devoted to apply the equivariant moving frame method to study the local equivalence problem of third order ordinarily differential equation under the pseudo-group of fiber preserving transformations.

In this paper, we provide local and global convergence guarantees for recovering CP (Candecomp/Parafac) tensor decomposition. The main step of the proposed algorithm is a simple alternating rank-$1$ update which is the alternating version of the tensor power iteration adapted for asymmetric tensors. Local convergence g…

Paper proves Simon's third gap conjecture for minimal surfaces in spheres.

We study differential invariants of the third order linear differential operators and use them to find conditions for equivalence of differential operators acting in line bundles on two dimensional manifolds with respect to groups of authomorphisms.

Study third order Einstein deformations for Kähler-Einstein metrics on compact manifolds.

In this paper we define $n$th order Hessian structures on manifolds and study them. In particular, when $n = 3$, we make a detailed study and establish a one-to-one correspondence between {\it third-order Hessian structures} and a {\it certain class of connections} on the second-order tangent bundle of a manifold. Furt…

For CR structures in dimension three, the CR pluriharmonic functions are characterized by the vanishing of a third order operator. This third order operator, after composition with the divergence operator, gives the fourth order analogue of the Paneitz operator. In this short note, we give criteria under which the kern…

Improved fourth-order compact scheme for option valuation with Robin boundary condition.

We introduce an alternative approach to the third order helicity of a volume preserving vector field $B$, which leads us to a lower bound for the $L^2$-energy of $B$. The proposed approach exploits correspondence between the Milnor $\barμ_{123}$-invariant for 3-component links and the homotopy invariants of maps to con…

We discuss contact invariant structures on the space of solutions of a third-order ordinary differential equation. Associated to any third-order differential equation modulo contact transformations, Chern introduced a degenerate conformal Lorentzian metric on the space of 2-jets of functions of one variable. When the W…

Study shows the second fundamental form of pseudospherical surfaces is universal and not dependent on specific solutions.

Mathisson's 'new mechanics' of a relativistic spinning particle is shown to follow, in the case of planar motion, from only general requirements of relativistic invariance and of the dependence on third order derivatives along with the 'variationality' feature. The Hamiltonian counterpart ultimately recovers the Dixon …

The linearization problem by use of the Cartan equivalence method for scalar third-order ODEs via point transformations was solved partially in [1,2]. In order to solve this problem completely, the Cartan equivalence method is applied to provide an invariant characterization of the linearizable third-order ordinary dif…

New tensor completion method converges linearly and is highly practical.

Based on the theory of Poisson vertex algebras we calculate skew-symmetry conditions and Jacobi identities for a class of third-order nonlocal operators of differential-geometric type. Hamiltonian operators within this class are defined by a Monge metric and a skew-symmetric two-form satisfying a number of differential…

The theory of the vortex filament in three-dimensional fluid dynamics, consisting mainly of the models up to the third-order approximation, is an attractive subject in both physics and mathematics. Many efforts have been devoted to the extension of the theory to higher-dimensional symmetric Lie algebras. However, such …

A variational equation of the third order in three-dimensional space is proposed which describes autoparallel curves of some connection.

This article is dedicated to solve the equivalence problem for two third order differential operators on the line under general fiber--preserving transformation using the Cartan method of equivalence. We will do three versions of the equivalence problems: first via the direct equivalence problem, second equivalence pro…

A dynamical system on the total space of the fibre bundle of second order accelerations, $T^2M$, is defined as a third order vector field $S$ on $T^2M$, called semispray, which is mapped by the second order tangent structure into one of the Liouville vector field. For a regular Lagrangian of second order we prove that …

Enhances gradient estimates for Hermitian Monge-Ampère equations.

Develops high-order approximations for financial models, proving convergence and regularity.

The Cartan equivalence method is utilized to deduce an invariant characterization of the scalar third-order ordinary differential equation $u'"=f(x,u,u',u")$ which admits the maximal seven-dimensional point symmetry Lie algebra. The method provides auxiliary functions which can be used to efficiently obtain the point t…

We address the problem of local geometry of third order ODEs modulo contact, point and fibre-preserving transformations of variables. Several new and already known geometries are described in a uniform manner by the Cartan method of equivalence. This includes conformal, Weyl and metric geometries in three and six dimen…

This dissertation advances the theoretical foundation of local optimization methods in Federated Learning.

Study non-standard bi-orders on punctured torus bundles, matching standard ones in key subgroups.

Let $V$ be a vector space of dimension $n+1$. We demonstrate that $n$-component third-order Hamiltonian operators of differential-geometric type are parametrised by the algebraic variety of elements of rank $n$ in $S^2(Λ^2V)$ that lie in the kernel of the natural map $S^2(Λ^2V)\to Λ^4V$. Non-equivalent operators corres…

The paper bounds eigenvalues of specific operators on certain manifolds.

Sharp inequality found on three-balls for fourth order Sobolev traces.

Proposes efficient sampling methods for solving linear inverse problems.