arXiv subjects

Rong Tang

Publications and source records attributed to Rong Tang.

At least 37 records · Page 2Linked to original sources

arXiv2024

Formal integration of complete Rota-Baxter Lie algebras

In this paper, first we revisit the formal integration of Lie algebras, which give rise to braces in some special cases. Then we establish the formal integration theory for complete Rota-Baxter Lie algebras, that is, we show that there is a Rota-Baxter group with the underlying group structure given by the Baker-Campbell-Hausdorff formula, associated to any complete Rota-Baxter Lie algebra. In particular, we use the post-Lie Magnus expansion to give the explicit formula of the Rota-Baxter operator. Finally we show that one can obtain a graded Rota-Baxter Lie ring from a filtered Rota-Baxter group.

math-ph↗

arXiv2024

Post-groupoids and quiver-theoretical solutions of the Yang-Baxter equation

The notion of post-groups was introduced by Bai, Guo and the first two authors recently, which are the global objects corresponding to post-Lie algebras, equivalent to skew-left braces, and can be used to construct set-theoretical solutions of the Yang-Baxter equation. In this paper, first we introduce the notion of post-groupoids, which consists of a group bundle and some other structures satisfying some compatibility conditions. Post-groupoids reduce to post-groups if the underlying base is one point. An action of a group on a set gives rise to the natural example of post-groupoids. We show that a post-groupoid gives rise to a groupoid (called the Grossman-Larson groupoid), and an action on the original group bundle. Then we introduce the notion of relative Rota-Baxter operators on a groupoid with respect to an action on a group bundle. A relative Rota-Baxter operator naturally gives rise to a post-groupoid and a matched pair of groupoids. One important application of post-groupoids is that they give rise to quiver-theoretical solutions of the Yang-Baxter equation on the quiver underlying the Grossman-Larson groupoid. We also introduce the notion of a skew-left bracoid, which consists of a group bundle and a groupoid satisfying some compatibility conditions. A skew-left bracoid reduces to a skew-left brace if the underlying base is one point. We give the one-to-one correspondence between post-groupoids and skew-left bracoids. Finally, we show that post-Lie groupoids give rise to post-Lie algebroids via differentiation.

math-ph↗

arXiv2024

Quasi-triangular, factorizable Leibniz bialgebras and relative Rota-Baxter operators

We introduce the notion of quasi-triangular Leibniz bialgebras, which can be constructed from solutions of the classical Leibniz Yang-Baxter equation (CLYBE) whose skew-symmetric parts are invariant. In addition to triangular Leibniz bialgebras, quasi-triangular Leibniz bialgebras contain factorizable Leibniz bialgebras as another subclass, which lead to a factorization of the underlying Leibniz algebras. Relative Rota-Baxter operators with weights on Leibniz algebras are used to characterize solutions of the CLYBE whose skew-symmetric parts are invariant. On skew-symmetric quadratic Leibniz algebras, such operators correspond to Rota-Baxter type operators. Consequently, we introduce the notion of skew-symmetric quadratic Rota-Baxter Leibniz algebras, such that they give rise to triangular Leibniz bialgebras in the case of weight $0$, while they are in one-to-one correspondence with factorizable Leibniz bialgebras in the case of nonzero weights.

math.QA↗

arXiv2024

Conditional Diffusion Models are Minimax-Optimal and Manifold-Adaptive for Conditional Distribution Estimation

We consider a class of conditional forward-backward diffusion models for conditional generative modeling, that is, generating new data given a covariate (or control variable). To formally study the theoretical properties of these conditional generative models, we adopt a statistical framework of distribution regression to characterize the large sample properties of the conditional distribution estimators induced by these conditional forward-backward diffusion models. Here, the conditional distribution of data is assumed to smoothly change over the covariate. In particular, our derived convergence rate is minimax-optimal under the total variation metric within the regimes covered by the existing literature. Additionally, we extend our theory by allowing both the data and the covariate variable to potentially admit a low-dimensional manifold structure. In this scenario, we demonstrate that the conditional forward-backward diffusion model can adapt to both manifold structures, meaning that the derived estimation error bound (under the Wasserstein metric) depends only on the intrinsic dimensionalities of the data and the covariate.

math.ST↗

arXiv2024

On the Computational Complexity of Metropolis-Adjusted Langevin Algorithms for Bayesian Posterior Sampling

In this paper, we examine the computational complexity of sampling from a Bayesian posterior (or pseudo-posterior) using the Metropolis-adjusted Langevin algorithm (MALA). MALA first employs a discrete-time Langevin SDE to propose a new state, and then adjusts the proposed state using Metropolis-Hastings rejection. Most existing theoretical analyses of MALA rely on the smoothness and strong log-concavity properties of the target distribution, which are often lacking in practical Bayesian problems. Our analysis hinges on statistical large sample theory, which constrains the deviation of the Bayesian posterior from being smooth and log-concave in a very specific way. In particular, we introduce a new technique for bounding the mixing time of a Markov chain with a continuous state space via the $s$-conductance profile, offering improvements over existing techniques in several aspects. By employing this new technique, we establish the optimal parameter dimension dependence of $d^{1/3}$ and condition number dependence of $κ$ in the non-asymptotic mixing time upper bound for MALA after the burn-in period, under a standard Bayesian setting where the target posterior distribution is close to a $d$-dimensional Gaussian distribution with a covariance matrix having a condition number $κ$. We also prove a matching mixing time lower bound for sampling from a multivariate Gaussian via MALA to complement the upper bound.

math.ST↗

arXiv2024

Deformation maps of quasi-twilled Lie algebras

In this paper, we provide a unified approach to study the cohomology theories and deformation theories of various types of operators in the category of Lie algebras, including modified $r$-matrices, crossed homomorphisms, derivations, homomorphisms, relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators and deformation maps of matched pairs of Lie algebras. The main ingredients are quasi-twilled Lie algebras. We introduce two types of deformation maps of a quasi-twilled Lie algebra. Deformation maps of type I unify modified $r$-matrices, crossed homomorphisms, derivations and homomorphisms between Lie algebras, while deformation maps of type II unify relative Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators and deformation maps of matched pairs of Lie algebras. We further give the controlling algebras and cohomologies of these two types of deformation maps, which not only recover the existing results for crossed homomorphisms, derivations, homomorphisms, relative Rota-Baxter operators, twisted Rota-Baxter operators and Reynolds operators, but also leads to some new results which are unable to solve before, e.g. the controlling algebras and cohomologies of modified $r$-matrices and deformation maps of matched pairs of Lie algebras.

math-ph↗

arXiv2024

VSHPIC: A Particle-In-Cell Algorithm Based On Vector Spherical Harmonics Expansion

The Particle-in-Cell (PIC) simulation has been a widely used method for studying plasma physics. However, fully three-dimensional PIC simulations always require huge computational resources. For problems with near azimuthal symmetry, recent work has shown that expanding all the quantities defined on the grid in azimuthal harmonics and truncating the expansion can improve the code efficiency. In this paper, we describe a novel parallel algorithm for efficiently simulating three-dimensional near-spherical symmetry problems. Our approach expands all physical quantities in the $θ$ and $ϕ$ directions in spherical coordinates using vector spherical harmonics. The code is capable of simulating three-dimensional asymmetric scenarios by accurately tracking the evolution of distinct individual modes while preserving the charge conservation law. The fundamental dispersion relation of EM waves in the plasma has been obtained using VSHPIC simulation results. The code also shows a well strong scalability up to more than 1000 cores.

physics.plasm-ph↗

arXiv2023

Cohomology and Homotopy of Lie triple systems

In this paper, first we give the controlling algebra of Lie triple systems. In particular, the cohomology of Lie triple systems can be characterized by the controlling algebra. Then using controlling algebras, we introduce the notions of homotopy Nambu algebras and homotopy Lie triple systems. We show that $2$-term homotopy Lie triple systems is equivalent to Lie triple $2$-systems, and the latter is the categorification of a Lie triple system. Finally we study skeletal and strict Lie triple $2$-systems. We show that skeletal Lie triple $2$-systems can be classified the third cohomology group, and strict Lie triple $2$-systems are equivalent to crossed modules of Lie triple systems.

math.RA↗

arXiv2023

Relative Rota-Baxter operators of weight 0 on groups, pre-groups, braces, the Yang-Baxter equation and $T$-structures

In this paper, we study relative Rota-Baxter operators of weight $0$ on groups and give various examples. In particular, we propose different approaches to study Rota-Baxter operators of weight $0$ on groups and Lie groups. We establish various explicit relations among relative Rota-Baxter operators of weight $0$ on groups, pre-groups, braces, set-theoretic solutions of the Yang-Baxter equation and $T$-structures.

math-ph↗

arXiv2023

Symplectic structure, product structures and complex structures on Leibniz algebras

In this paper, a symplectic structure on a Leibniz algebra is defined to be a {\em symmetric} nondegenerate bilinear form satisfying certain compatibility condition, and a phase space of a Leibniz algebra is defined to be a symplectic Leibniz algebra satisfying certain conditions. We show that a Leibniz algebra has a phase space if and only if there is a compatible Leibniz-dendriform algebra, and phase spaces of Leibniz algebras one-to-one corresponds to Manin triples of Leibniz-dendriform algebras. Product (paracomplex) structures and complex structures on Leibniz algebras are studied in terms of decompositions of Leibniz algebras. A para-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a paracomplex structure satisfying a compatibility condition. We show that a symplectic Leibniz algebra admits a para-Kähler structure if and only if the Leibniz algebra is the direct sum of two isotropic subalgebras as vector spaces. A complex product structure on a Leibniz algebra consists of a complex structure and a product structure satisfying a compatibility condition. A pseudo-Kähler structure on a Leibniz algebra is defined to be a symplectic structure and a complex structure satisfying a compatibility condition. Various properties and relations of complex product structures and pseudo-Kähler structures are studied. In particular, Leibniz-dendriform algebras give rise to complex product structures and pseudo-Kähler structures naturally.

math.RA↗

arXiv2023

From relative Rota-Baxter operators and relative averaging operators on Lie algebras to relative Rota-Baxter operators on Leibniz algebras: a uniform approach

In this paper, first we construct two subcategories (using symmetric representations and antisymmetric representations) of the category of relative Rota-Baxter operators on Leibniz algebras, and establish the relations with the categories of relative Rota-Baxter operators and relative averaging operators on Lie algebras. Then we show that there is a short exact sequence describing the relation between the controlling algebra of relative Rota-Baxter operators on a Leibniz algebra with respect to a symmetric (resp. antisymmetric) representation and the controlling algebra of the induced relative Rota-Baxter operators (resp. averaging operators) on the canonical Lie algebra associated to the Leibniz algebra. Finally, we show that there is a long exact sequence describing the relation between the cohomology groups of a relative Rota-Baxter operator on a Leibniz algebra with respect to a symmetric (resp. antisymmetric) representation and the cohomology groups of the induced relative Rota-Baxter operator (resp. averaging operator) on the canonical Lie algebra.

math.RA↗

arXiv2023

Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation

The notions of a post-group and a pre-group are introduced as a unification and enrichment of several group structures appearing in diverse areas from numerical integration to the Yang-Baxter equation. First the Butcher group from numerical integration on Euclidean spaces and the $\mathcal{P}$-group of an operad $\mathcal{P}$ naturally admit a pre-group structure. Next a relative Rota-Baxter operator on a group naturally splits the group structure to a post-group structure. Conversely, a post-group gives rise to a relative Rota-Baxter operator on the sub-adjacent group. Further a post-group gives a braided group and a solution of the Yang-Baxter equation. Indeed the category of post-groups is isomorphic to the category of braided groups and the category of skew-left braces. Moreover a post-Lie group differentiates to a post-Lie algebra structure on the vector space of left invariant vector fields, showing that post-Lie groups are the integral objects of post-Lie algebras. Finally, post-Hopf algebras and post-Lie Magnus expansions are utilized to study the formal integration of post-Lie algebras. As a byproduct, a post-group structure is explicitly determined on the Lie-Butcher group from numerical integration on manifolds.

math.QA↗

arXiv2023

Nonlocal Filtration Equations with Rough Kernels in the Heisenberg Group

Motivated by the extensive investigations of integro-differential equations on $\mathbb{R}^n$, we consider nonlocal filtration type equations with rough kernels on the Heisenberg group $\mathbb{H}^n$. We prove the existence and uniqueness of weak solutions corresponding to suitable initial data. Furthermore, we obtain the large time behavior of solutions and the uniform Hölder regularity of sign-changing solutions for the porous medium type equations ($m\geq 1$). Notice that both conformal fractional operators $\mathscr{L}_{α/2}$ and pure power fractional operators $\mathscr{L}^{α/2}$ on the Heisenberg group $\mathbb{H}^n$ have their integral representations with suitable kernels. Therefore, all the results in this paper will hold for these equations with operators $\mathscr{L}_{α/2}$ or $\mathscr{L}^{α/2}$.

math.AP↗

arXiv2023

Estimating Distributions with Low-dimensional Structures Using Mixtures of Generative Models

There has been a growing interest in statistical inference from data satisfying the so-called manifold hypothesis, assuming data points in the high-dimensional ambient space to lie in close vicinity of a submanifold of much lower dimension. In machine learning, encoder-decoder pair based generative modelling approaches have been successful in learning complicated high-dimensional distributions such as those over images and texts by explicitly imposing the low-dimensional manifold structure. In this work, we introduce a new approach for estimating distributions on unknown submanifolds via mixtures of generative models. We show that conventional generative modeling approaches using a single encoder-decoder pair are generally unable to capture data distributions under the manifold hypothesis, unless the underlying manifold admits a global parametrization; however, this issue can be solved by using a collection of encoder-decoder pairs for learning different local patches of the data supporting manifold. A rigorous theoretical analysis is developed to demonstrate that the proposed estimator attains the minimax-optimal rate of convergence for the implicit estimation of data distributions with manifold structures. Our experiments show that, by utilizing parameter sharing, the proposed method can significantly improve the performance of conventional auto-encoder based generative modelling approaches with minimal additional computational efforts.

stat.ME↗

arXiv2022

Deformations and homotopy of Rota-Baxter operators and $\mathcal{O}$-operators on Lie algebras

This article gives a brief introduction to some recent work on deformation and homotopy theories of Rota-Baxter operators and more generally $\mathcal{O}$-operators on Lie algebras, by means of the differential graded Lie algebra approach. It is further shown that these theories lift the existing connection between $\mathcal{O}$-operators and pre-Lie algebras to the levels of deformations and homotopy.

math.QA↗

arXiv2022

Nonabelian embedding tensors

In this paper, first we introduce the notion of a nonabelian embedding tensor, which is a nonabelian generalization of an embedding tensor. Then we introduce the notion of a Leibniz-Lie algebra, which is the underlying algebraic structure of a nonabelian embedding tensor, and can also be viewed as a nonabelian generalization of a Leibniz algebra. Next using the derived bracket, we construct a differential graded Lie algebra, whose Maurer-Cartan elements are exactly nonabelian embedding tensors. Consequently, we obtain the differential graded Lie algebra that governs deformations of a nonabelian embedding tensor. Finally, we define the cohomology of a nonabelian embedding tensor and use the second cohomology group to characterize linear deformations.

math.RA↗

arXiv2022

Post-Hopf algebras, relative Rota-Baxter operators and solutions of the Yang-Baxter equation

In this paper, first we introduce the notion of a post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman-Larsson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions of the Yang-Baxter equation. Then we introduce the notion of relative Rota-Baxter operators on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota-Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota-Baxter operator also induces a post-Hopf algebra. Then we show that relative Rota-Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota-Baxter operators give solutions of the Yang-Baxter equation in certain cocommutative Hopf algebras. Finally we characterize relative Rota-Baxter operators on Hopf algebras using relative Rota-Baxter operators on the Lie algebra of primitive elements, graphs and module bialgebra structures.

math-ph↗

arXiv2022

Minimax Rate of Distribution Estimation on Unknown Submanifold under Adversarial Losses

Statistical inference from high-dimensional data with low-dimensional structures has recently attracted lots of attention. In machine learning, deep generative modeling approaches implicitly estimate distributions of complex objects by creating new samples from the underlying distribution, and have achieved great success in generating synthetic realistic-looking images and texts. A key step in these approaches is the extraction of latent features or representations (encoding) that can be used for accurately reconstructing the original data (decoding). In other words, low-dimensional manifold structure is implicitly assumed and utilized in the distribution modeling and estimation. To understand the benefit of low-dimensional manifold structure in generative modeling, we build a general minimax framework for distribution estimation on unknown submanifold under adversarial losses, with suitable smoothness assumptions on the target distribution and the manifold. The established minimax rate elucidates how various problem characteristics, including intrinsic dimensionality of the data and smoothness levels of the target distribution and the manifold, affect the fundamental limit of high-dimensional distribution estimation. To prove the minimax upper bound, we construct an estimator based on a mixture of locally fitted generative models, which is motivated by the partition of unity technique from differential geometry and is necessary to cover cases where the underlying data manifold does not admit a global parametrization. We also propose a data-driven adaptive estimator that is shown to simultaneously attain within a logarithmic factor of the optimal rate over a large collection of distribution classes.

math.ST↗