中-俄概率论及其应用研讨会

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中-俄概率论及其应用研讨会

简历

Makhamat Gafurov1967年在塔什干国立大学获得数学学士学位，随后在俄罗斯科学院斯捷克洛夫数学所和鸟兹别克斯坦科学院数学所攻读硕士并于1970年取得硕士学位，1982年在乌兹别克斯坦科学院数学所取得物理和数学博士学位。1989-2024年任塔什干汽车与公路学院数学系教授，2024年至今任塔什干劳动与社会关系学院数学系教授。出版了一部专著和三部教材，发表了70余篇学术论文。目前研究方向为随机游动和离散域的边界问题、随机变量和的极限定理、统计方法与风险分析等。获得了15项优秀科研和教学奖励，为乌兹别克斯坦数学会理事，是包括Bernoulli概率论及数理统计世界大会等多个组织委员会的委员。与波兰、德国、蒙古、中国等国家知名高校进行了卓有成效的学术交流与合作。

报告题目

Asymptotic estimates for a small parameter in Hartmann-Wintner law of the iterated logarithm

报告

主要观点

The work is devoted to a further refinement of the classical Hartman–Wintner theorem on the law of the iterated logarithm for a sequence of independent and identically distributed random variables. We establish estimates for the rate of convergence in the form of convergent series of weighted probabilities of large deviations by deriving exact asymptotes, with respect to a small parameter, of the series . These results refine the Theorem 2 in (Gafurov M.U, Lecture notes in Math, vol.1021, 1983). Regarding the assertion of the Theorem 2, academician Y.V. Prokhorov stated the problem of finding further terms of asymptotic representation . In this work, we provide the solution of the stated problem. Analogs of the obtained results were proved for a family of independent and identically distributed random variables indexed on sectors (O. Klesov, Springer, 2014) of the d-dimensional integer lattice of the Euclidean space.

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简历

向开南，1993年6月本科毕业于湘潭大学数学系；1993.9-1996.6在北京师范大学数学系读硕士；1996.9-1999.6在中国科学院应用数学研究所读博士；1999.7-2001.6在北京大学数学科学学院做博士后；2001年6月博士后出站后进入湖南师范大学工作；2007年3月调往南开大学；2019年3月回湘潭大学工作；当前研究兴趣是群和图上的概率与几何(渗流、Ising模型、随机图、概率组合、随机游走、几何群论、无穷图论)；在Comm. Pure Appl. Math.、Ann. Probab.、Tran. AMS.、J. Comb. Th. Ser. B.、J. Stat. Phys.、Ann. Inst. H. Poincare Probab. Stat.、Bernoulli等上发表论文。

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Local weak convergence of uniform random fullerenes

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主要观点

We prove that the uniform random fullerenes converge locally weakly towards the planar hexagonal lattice, which confirms Conjecture 1 in A. Bille, V. Buchstaber, S. Coste, S. Kuriki and E. Spodarev [(2025). Random eigenvalues of graphenes and the triangulation of plane. J. Phys. A: Math. Theor. 58(2), paper no. 025212.]. This talk is based on the following paper:

Liu Jing, Xiang Kainan and Zou Lang. (2026). Local weak convergence of uniform random fullerenes.

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简历

Alexey Khartov 2008年在莫斯科国立通讯与信息技术大学获得学士学位，2012年在圣彼得堡国立大学获得硕士学位，2014年在圣彼得堡斯捷克洛夫数学研究所(俄罗斯科学院)获得博士学位，现为俄罗斯科学院哈尔科维奇信息传输问题研究所高级研究员。研究方向为无穷可分分布、极限定理、多变量随机过程逼近等。在Electron.J.Probab., Bernoulli, Theory Probab. Appl., Pacific Journal of Mathematics等国际知名期刊发表论文30多篇。

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The rational-infinitely divisible distributions: limit theorems and denseness

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主要观点

We consider a new class  of distribution functions  that have the property of rational-infinite divisibility: there exist some infinitely divisible distribution functions  and  such that =F*. Characteristic functions of elements of  admit  the Lévy-Khinchine type representations with ``signed spectral measures''. This class   is a wide natural extension of the fundamental class of  infinitely divisible distribution functions and it is actively studied now.  In this talk, we will review recent important results on limit and compactness theorems for discrete distributions of the class  with respect to convergence in total variation. We also discuss general results   concerning the weak convergence  for arbitrary representatives of the class . In addition, we will consider the denseness properties of the class   in the family of all distribution functions with respect to convergence in the weak sense and in total variation.

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简历

Vladimir Vatutin 1974年毕业于莫斯科罗蒙诺索夫国立大学力学数学系，1977年在斯捷克洛夫数学研究所获得副博士学位（论文：分支过程的极限定理，导师：B.A.谢瓦斯季扬诺夫），1987年在该所获得物理数学科学博士学位（论文：具有正则变化生成函数的临界分支过程）。1977年起任职于斯捷克洛夫数学研究所至今，现为离散数学部主任、教授。研究方向为分支过程、随机环境中的分支过程、随机置换、随机树、局部极限定理。1988年获苏联科学院重要数学成果奖，现任《Theory of Probability and its Applications》《Markov Processes and Related Fields》《Discrete Mathematics and Applications》等期刊编委。

报告题目

Functional limit theorem for critical branching processes in extreme non-favorable random environment

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主要观点

Let

be a critical branching process in random environment and

be its associated random walk. Assuming that the increments distribution of the associated random walk belongs without centering to the domain of attraction of an α-stable law we prove a conditional functional limit theorem describing, as  the distribution of the number of particles in the process

Given  and .

Joint work with Elena DYAKONOVA.

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Pavel Gumenyuk 2005年博士毕业于俄罗斯萨拉托夫国立大学，先后挪威、西班牙、意大利、挪威从事博士后、研究院、访问(副)教授和访问教授，现为意大利米兰理工大学数学系副教授。研究方向为复分析、全纯动力系统、几何函数论、Loewner理论与随机过程等。在国际著名期刊Mathematische Annalen,Trans.Amer. Math.Soc., J. Geom. Anal., J. Lond. Math. Soc, Internat. J. Math.等发表论文36篇，为Transactions of the American Mathematical Society等近30个国际学术期刊的审稿人。在西班牙、瑞典、德国、挪

威、意大利、芬兰、日本等国家长期进行学术访问与交流，获得了俄罗斯、挪威、西班牙、意大利等国家的荣誉和奖项。

报告题目

Inhomogeneous branching processes with constant Denjo-Wolff point

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主要观点

This talk is based on a joint paper [Ann. Appl. Probab. 36 (3), 2163-2198, (June 2026) DOI: 10.1214/25-AAP2276] coauthored with Takahiro Hasebe (Hokkaido University, Sapporo, JAPAN) and José Luis Pérez Garmendia (Centro de Investigación en Matemáticas, Guanajuato, MÉXICO). We consider inhomogeneous branching processes with continuous time and state. Transition probabilities of such processes are described by the so-called Laplace exponents, which can be extended to holomorphic self-mappings of the right half-plane. This allows to associate to each Laplace exponent the so-called Denjoy – Wolff point (or DW-point for short; this notion comes from the iteration theory for holomorphic functions in hyperbolic domains). For homogeneous processes, the position of the DW-point does not depend on time. We study inhomogeneous processes possessing the same property. We characterize such processes in terms of their branching mechanisms and, depending on the position of the DW-point, obtain various results concerning the extinction and explosion probabilities.

In addition, for continuous-time branching processes with the position of the DW-point turns out to determine whether a given processes can be embedded into a continuous state branching process or not.

Our method is heavily based on recent developments in non-autonomous holomorphic dynamics, and in particular, on results we obtained in [Constr. Approx. 61, 379–412 (2025) DOI: https://doi.org/10.1007/s00365-023-09675-9].

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Dmitry Zaporozhets 2001年毕业于圣彼得堡国立大学获得数学专家学位，2005年在斯捷克洛夫数学研究所圣彼得堡分所获得物理数学副博士学位（论文：随机多项式理论中的几何方法），2017年在该所获得物理数学科学博士学位（论文：随机多项式的零点、代数数的分布与随机过程的凸包）。现为俄罗斯科学院斯捷克洛夫数学研究所圣彼得堡分所科研副所长、俄罗斯科学院通讯院士。研究方向为概率论、随机几何、凸几何、数的几何。在《Annals of Probability》《Advances in Mathematics》《Geometric and Functional Analysis》等顶级期刊发表论文40余篇。

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On the Steiner entire function

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主要观点

The classical Steiner polynomial is one of the central objects of convex geometry, encoding fundamental metric characteristics of a convex body. In this talk, we discuss its infinite-dimensional analogue: the Steiner entire function associated with a compact convex set in a Hilbert space. This analytic object serves as a natural generating function for intrinsic volumes and reveals unexpected connections between convex geometry, Gaussian processes, and the theory of entire functions.

The talk is based on joint work with D. Dospolova and M. Germanskov.

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简历

王龙敏，南开大学统计与数据科学学院教授，分别于2002年与2007年在南开大学数学学院获得本科与博士学位。主要研究群上随机游走、统计物理、非局部算子及位势理论。在《Communications on Pure and Applied Mathematics》、《Bernoulli》、《Annals of the Institute of Henri Poincaré》等国际知名期刊发表论文近20篇。

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Branching random walks and percolation on hyperbolic graphs

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主要观点

In this talk, we will consider branching random walks and Bernoulli bond percolation on a nonamenable, transitive hyperbolic graph, and show that the Hausdorff dimension of the limit set (the accumulation points on the boundary) is determined by the growth rate of the two-point functions.

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简历

Evgeny Burnaev ，2006年毕业于莫斯科物理技术学院，2008年在俄罗斯科学院信息传输问题研究所获得信息学理论基础副博士学位，2021年在莫斯科物理技术学院获得数学建模方向科学博士学位（俄罗斯最高学术学位）。曾任俄罗斯科学院信息传输问题研究所数据分析与预测建模实验室主任，现为斯科尔科沃科技研究院教授、人工智能中心主任，兼任人工智能发展副总裁。研究方向为机器学习、生成式建模、流形学习、工业预测分析、异常检测及3D计算机视觉。在ICML、ICLR、NeurIPS、CVPR等顶级会议发表论文多篇，指导10名博士生完成论文答辩，曾获莫斯科政府青年科学家奖、俄罗斯联邦政府科学技术奖等多项荣誉。

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Mutual information estimation via bridge matching based on diffusion processes

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主要观点

Diffusion bridge models have recently become a powerful tool in the field of generative modeling. In this work, we leverage them to address another important problem in machine learning and information theory, the estimation of the mutual information (MI) between two random variables. Neatly framing MI estimation as a domain transfer problem, we construct an unbiased estimator for data posing difficulties for conventional MI estimators. We showcase the performance of our estimator on three standard MI estimation benchmarks, i.e., low-dimensional, image-based and high MI, and on real-world data, i.e., protein language model embeddings.

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简历

俞锦炯，华东师范大学统计学院助理教授。2011年本科毕业于北京大学数学科学学院，2017年于新加坡国立大学数学系获得博士学位。2017年至2020年期间，在上海纽约大学从事博士后研究。其主要研究兴趣为概率论、交互粒子系统及统计物理模型。

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Universality of the Brownian net

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主要观点

The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from every point in space and time, while the Brownian net is an extension that also allows branching. We show here that the Brownian net is the universal scaling limit of one-dimensional branching-coalescing random walks with weak binary branching and arbitrary increment distributions that have finite (3+ε)-th moment. This gives the first example in the domain of attraction of the Brownian net where paths can cross without coalescing. Joint work with Rongfeng Sun and Jan Swart.

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周晓文，现任加拿大康考迪亚大学数学与统计系终身教授。1988年及1991年在中山大学获得本科及硕士学位，1999年于美国加州大学伯克利分校获统计学博士学位。长期从事概率论与随机过程理论的研究，主要研究兴趣包括测度值随机过程（超过程）、Levy过程及其在种群遗传学和风险理论中的应用。在《Annals of Probability》《Probability Theory and Related Fields》《Annals of Applied Probability》《Bernoulli》等国际概率刊物发表论文80余篇。

报告题目

Yaglom limits of continuous-state branching processes in Brownian environment

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主要观点

We investigate the asymptotic behavior of continuous-state branching processes in Brownian random environment. In weakly, intermediately and strongly subcritical regimes, we prove, respectively, the existence of the Yaglom limit and derive an explicit representation of its Laplace transform using Kummer confluent hypergeometric functions. We show that the Yaglom limit does not depend on the initial state of the process across all the subcritical regimes.

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Andrei Zorine，1995年至2001年就读于罗巴切夫斯基下诺夫哥罗德国立大学计算数学与控制论系。自2002年9月起，留校在计算数学与控制论系工作至今。主要研究方向为排队论、随机过程、运筹学与数学规划，特别关注具有周期转移强度的连续时间马尔可夫链和马尔可夫调制输入的轮询系统。在《Queueing Systems》等期刊发表论文多篇，并于2014年发表关于轮询系统遍历性条件的经典论文。曾担任多个国际学术会议的程序委员会委员。

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Jump chain approach to continuous-time Markov chains

with periodic transition intensities

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主要观点

In reliability and queuing theory, continuous-time Markov chains (CTMCs) play important role. If the CTMC is multivariate, or its transition intensities are time-dependent, analysis of the limiting behavior can be fairly involved. In this case, a jump chain of (an augmented) process can be studied. In the talk a CTMC {X(t); t ≥ 0} is considered which is a number in the queue process of a M(t)/M(t)/1/∞ queue with arrival intensity λ(t) and service intensities μ1(t) for queue holding 1 job, μ2(t) for any larger queue. We assume that all of the functions λ(t), μ1(t), and μ2(t) are periodic with period 1. Let J(t) = {t} where {·} stands for the fractional part function, then {(X(t), J(t)); t ≥ 0} be a time-homogeneous Markov process whose invariant measure’s densityare in a certain relations to the Kolmogorov differential equations for the original CTMC. Set τ0 = 0, let τn+1 = inf{t: t > τn , X(t) ≠ X(τn)}, n =0, 1, …, be the jump instants of the Markov process {X(t); t ≥ 0}. In the talk we analyze a general-state time-homogeneous Markov chain {(Xn, Jn); n = 0, 1, …} where Xn = X(τn), Jn = J(τn). This General Markov chain is proved to be ψ-irreducible, some of its small sets are found, and conditions for the invariant measure existence are established by means of a new iterative dominating technique [1, 2]. The stationary solution for the Kolmogorov partial differential equations for the process {(X(t), J(t)); t ≥ 0} is given in terms of the invariant measure of the general Markov chain {(Xn, Jn); n =0, 1, …}.

References.

1. Zorine A. V. On ergodicity conditions in a polling model with Markov modulated input and state-dependent routing. Queueing Systems. 2014. V. 76, No 2. P. 223–241.

2. Zorine A. V. Controlled M/M/1−RQ system with randomized acceptance from the orbit. Queueing Theory and Network Applications. QTNA 2019. Lecture Notes in Computer Science. V. 11688. Cham: Springer, 2019. P. 64–76.

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简历

石权，中科院数学与系统科学研究院副教授，本科毕业于清华大学，2013-2016年在瑞士苏黎世大学攻读数学博士学位，法国巴黎十三大博士后，2017-2019年英国牛津大学访问研究员，2019-2021年德国曼海姆大学研究助理，2021年至今，中国科学院数学与系统科学研究院副研究员。研究方向为增长分裂过程、随机树、列维过程和分枝粒子系统，在概率著名期刊The Annals of Applied Probability, Bernoulli, Electronic Journal of Probability, Annales de I' Institut Henri Poincare (Probabilites et Statistiques), Stochastic Processes and Their Applications等发表研究论文10余篇。

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From multitype branching Brownian motions to branching Markov additive processes

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主要观点

We study a class of multitype branching Lévy processes, where particles move according to type-dependent Lévy processes, switch types via an irreducible Markov chain, and branch according to type-dependent laws. This framework generalizes multitype branching Brownian motions. Using techniques of Markov additive processes, we develop a spine decomposition. This approach further enables us to prove convergence results for the additive martingales and derivative martingales, and establish the existence and uniqueness of travelling wave solutions to the corresponding multitype FKPP equations.

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Evgeny Prokopenko，2012年在新西伯利亚国立大学获得学士学位，2014年获得硕士学位，2018年在索伯列夫数学所(新西伯利亚)获得数学(概率)博士学位，2019-2021在法国埃塞克高等商学院进行博士后研究工作，现为索伯列夫数学所高级研究员。研究方向为马氏过程、更新过程、极限定理以及随机过程的应用建模和算法等，2022年被华为公司授予技术合作优秀伙伴奖。在Stochastic Processes and their Appications, Statistics and Probability Letters, Markov Processes Relat. Fields等国际刊发表论20余篇。

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Trajectory analysis of stochastic gradient descent near local minima

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主要观点

The talk addresses limit theorems for additive stochastic gradient descent (SGD) employing a constant step size that eventually decays to zero. The local asymptotic behavior of the SGD process is examined, and sufficient conditions are derived under which the iterates remain within a neighborhood of the local minimum.

The main object of study is the Markov chain defined by the recurrence

,

where f : R →R is a suitable function; {} is a sequence of independent and identically distributed random variables representing noise;  is the step size, which tends to zero;  is the initial point. The central problem addressed in this talk is to determine the conditions on f , { },  , , and  under which

as

in probability or almost surely, where m is a local minimum of f(x).

For nontrivial dynamics, we require  , ensuring the process evolves

rather than remaining near the initial point with high probability. The condition E[] = 0 is essential; otherwise, the process effectively minimizes f(x) E[]·x rather than f (x). We consider the case when the noise distribution is has regularly varying tail with index α > 2.

This is joint work with Dmitry Dudukalov.

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简历

黎怀谦，2011年获得北京师范大学与法国勃艮第大学（Université de Bourgogne）双博士学位，师从著名概率论学者。2011年至2013年期间，他先后师从吴黎明教授从事博士后研究，并在澳洲国立大学和麦考瑞大学进行学术访问。他曾就职于四川大学数学学院，2018年起任职于天津大学应用数学中心至今。主要研究领域为随机分析，特别是热核估计、特征值估计以及Fokker-Planck方程等。在《Journal de Mathématiques Pures et Appliquées》、《Journal d'Analyse Mathématique》、《Science China Mathematics》等国际知名期刊发表多篇论文。

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Limiting Formulas for Negative Sobolev Norms

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主要观点

This talk presents an asymptotic formula for negative Sobolev norms as the fractional order tends to zero. Using heat-kernel convolutions to realize these spaces on , we show that, under a mild boundedness assumption, the order times the -th power of the norm converges to a dimension-free constant times the  norm. The proof combines heat-kernel regularization, monotonicity, weak compactness, and an Abelian–Tauberian argument. We further extend this result to general measure spaces equipped with a family of sub-additive, bounded, and continuous operators. This gives a negative-order analogue of the classical Bourgain–Brezis–Mironescu and Maz'ya–Shaposhnikova formulas.

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KHOMIDOV MUKHRIDDIN，乌兹别克斯坦罗曼诺夫斯基数学研究所随机分析实验室高级研究员，2023年获乌兹别克斯坦国立大学物理与数学博士学位。主要研究概率论与随机过程、动力系统、混沌动力学及统计物理，重点关注动力系统返回时间与击中时间的极限定理等问题，已在国内外学术期刊和国际会议发表论文10余篇。

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On the limit distribution of return times for circle maps

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主要观点

In this talk, we study limit distributions of return times for circle homeomorphisms with a single critical point and an irrational rotation number. We investigate return-time distributions associated with intervals generated by dynamical partitions and show that the corresponding limit distribution is singular with respect to the Lebesgue measure. Furthermore, we consider a family of parameter-dependent limit distributions and compare the corresponding probability measures. The obtained results reveal quasi-random statistical phenomena arising in deterministic dynamical systems.

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严晓东，西安交通大学数学与统计学院教授，博士生导师，入选国家级青年人才项目和校内青拔A类支持计划，荣获“华为火花奖"，"滴滴盖亚学者"，研究方为统计决策、统计推断和统计计算等。学术成果发表在著名期刊JRSSB,AOS,JASA,JOE以及人工智能顶级会议NeurlPS,ICMI,AAAI等50余篇。在“高等教育出版社出版”以独立主编出版了《机器学习》、《数据科学实践基础-基于R》两部教材。

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Asymptotic theory and sequential test for adaptive multi-armed bandit process

报告

主要观点

Multi-armed bandit (MAB) processes constitute a foundational subclass of. reinforcement learning problems and represent a central topic in statistical decision theory, but are limited to simultaneous adaptive allocation and sequential test, because of the absence of asymptotic theory under non-i.i.d sequence and sublinear information. To address this open challenge, we propose Urn Bandit (UNB) process to integrate the reinforcement mechanism of urn probabilistic models with MAB principles, ensuring almost sure convergence of resource allocation to optimal arms. We establish the joint functional central limit theorem (FCLT) for consistent estimators of expected rewards under non-i.i.d., non-sub-Gaussian and sublinear reward samples with pairwise correlations across arms. To overcome the limitations of existing methods that focus mainly on cumulative regret, we establish the asymptotic theory along with adaptive allocation that serves powerful sequential test, such as arms comparison, A/B testing, and policy valuation. Simulation studies and real data analysis demonstrate that UNB maintains statistical test performance of equal randomization (ER) design but obtain more average rewards like classical MAB processes.

主讲人照片

简历

Alexander Shklyaev自2002年本科进入世界知名的莫斯科大学后于2011年在该校获得博士学位并留校工作至今，现为莫斯科大学力学与数学学院高级研究员。研究方向为随机环境中的分枝过程、大偏差理论、卡方检验和统计应用等，在知名国际期刊Theory of Probability and Its Applications, Discrete Mathematics and Applications, Journal of Mathematical Sciences等发表20余篇sci论文.

报告题目

How to grow like a product of means in a varying environment

报告

主要观点

Let  be independent random variables with the given distributions . The process {}defined by

is called a branching process in a varying environment (BPVE).

In [1] G. Kersting gives sufficient conditions for BPVE to survive and to converge in  and a.s. to a non-degenerate limit. However, an important condition for this work was uniformly boundness of

(1)

This condition is nice. However, it is very restrictive in the case of a random environment. We want to obtain new conditions for BPVE, that can't be applied to different models of branching process in a random environment from a quenched point of view.

The key problem is to obtain the following important result:

where W is the limit of a natural martingale {This fact is a key step to study the limit behavior of {} conditioned to survive in terms of the "associated random walk" (in the case of BPVE this sequence is non-random).

In the recent preprint [2] we obtain this result under two conditions:

1. standard condition

2. condition

(2)

Condition (2) is easily verified in the case of regenerative random environment (for example, i.i.d. random environment, random environment). In this case this condition is much weaker than uniform condition (1).

Also, our results do not use the p.g.f. technique, thus, we can relax  condition to  or even . In that case we need a stronger version of (2). In the report we discuss the conditions, apply it to particular cases of BPVE and present some generalizations.

主讲人照片

简历

Alisher Jalilov，研究方向为一维动力系统与分析。他于韩国亚洲大学（Ajou University）获得数学博士学位，并曾在该校数学系担任助教。他有超过12年的大学教学经验，来美国科技大学前曾在塔什干的Kimyo国际大学和阿米提大学任教。

报告题目

On the support of stationary distributions for AR(1)-type iterated function systems

报告

主要观点

We consider a class of AR(1)-type random processes on the unit interval of the form

where  is a contraction and () is an i.i.d. sequence taking finitely many values with positive probabilities. Under natural invariance assumptions this process is generated by the finite iterated function system consisting of the maps , where  are the possible values of the noise.

The aim of the talk is to study the support of the stationary distribution of this process. We show that it coincides with the Hutchinson attractor of the associated iterated function system. We then discuss sufficient conditions under which this support is a Cantor-type set. The main emphasis is on injectivity, separation of images, and contraction coefficients. Examples related to Bernoulli perturbations and nonlinear AR(1) models are also discussed.

主讲人照片

简历

Nikita Puchkin 分别于2016年和2018年以荣誉学位获得莫斯科物理技术学院应用数学与物理学学士和硕士学位，并于2018年同时获得斯科尔科沃科技学院数学与计算机科学荣誉硕士学位。2023年在俄罗斯高等经济大学获得数学博士学位。自2018年起任职于高等经济大学，现任计算机科学学院副教授，并自2024年9月起担任人工智能模型理论基础实验室主任。研究方向为高维统计推断、机器学习理论、生成式扩散模型，专注于score函数估计、流形学习、深度神经网络逼近能力等。曾荣获2020年度俄罗斯青年数学家奖和2024年度青年领军学者基金。在《Journal of Machine Learning Research》《IEEE Transactions on Information Theory》等期刊及COLT、AISTATS等顶级会议发表论文多篇。

报告题目

On estimation of a score function and its derivatives without the curse of dimensionality

报告

主要观点

We study the problem of estimating the score function using both implicit score matching and denoising score matching. Assuming that the data distribution exhibits a low-dimensional structure, we prove that implicit score matching is able not only to adapt to the intrinsic dimension, but also to achieve the same rates of convergence as denoising score matching in terms of the sample size. Furthermore, we demonstrate that both methods allow us to estimate log-density Hessians without the curse of dimensionality by simple differentiation. This justifies convergence of ODE-based samplers for generative diffusion models. Our approach is based on Gagliardo-Nirenberg-type inequalities relating weighted -norms of smooth functions and their derivatives.

主讲人照片