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Tianyi Zheng

Publications and source records attributed to Tianyi Zheng.

At least 37 records · Page 2Linked to original sources
arXiv2024

Prime orbit theorems for expanding Thurston maps: Genericity of strong non-integrability condition

In the second paper [LZ24b] of this series, we obtained an analog of the prime number theorem for a class of branched covering maps on the $2$-sphere $S^2$ called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness or holomorphicity assumption. More precisely, the number of primitive periodic orbits, ordered by a weight on each point induced by a non-constant (eventually) positive real-valued Hölder continuous function on $S^2$ satisfying the $α$-strong non-integrability condition, is asymptotically the same as the well-known logarithmic integral, with an exponential error bound. In this third and last paper of the series, we show that the $α$-strong non-integrability condition is generic in the class of $α$-Hölder continuous functions.

math.DS↗
arXiv2024

Confined subgroups in groups with contracting elements

In this paper, we study the growth of confined subgroups through boundary actions of groups with contracting elements. We establish that the growth rate of a confined subgroup is strictly greater than half of the ambient growth rate in groups with purely exponential growth. Along the way, several results are obtained on the Hopf decomposition for boundary actions of subgroups with respect to conformal measures. In particular, we prove that confined subgroups are conservative, and examples of subgroups with nontrivial Hopf decomposition are constructed. We show a connection between Hopf decomposition and quotient growth and provide a dichotomy on quotient growth of Schreier graphs for subgroups in hyperbolic groups.

math.GR↗
arXiv2024

Prime orbit theorems for expanding Thurston maps: Dirichlet series and orbifolds

We obtain an analog of the prime number theorem for a class of branched covering maps on the $2$-sphere $S^2$ called expanding Thurston maps, which are topological models of some non-uniformly expanding rational maps without any smoothness or holomorphicity assumptions. More precisely, we show that the number of primitive periodic orbits, ordered by a weight on each point induced by an (eventually) positive real-valued Hölder continuous function on $S^2$ that is not cohomologous to a constant, is asymptotically the same as the well-known logarithmic integral. In particular, our results apply to postcritically-finite rational maps for which the Julia set is the whole Riemann sphere.

math.DS↗
arXiv2023

On the spectrum of asymptotic entropies of random walks

Given a random walk on a free group, we study the random walks it induces on the group's quotients. We show that the spectrum of asymptotic entropies of the induced random walks has no isolated points, except perhaps its maximum.

math.GR↗
arXiv2023

Isoperimetric inequalities, shapes of Følner sets and groups with Shalom's property ${H_{\mathrm{FD}}}$

We prove an isoperimetric inequality for groups. As an application, we obtain lower bound on Følner functions in various nilpotent-by-cyclic groups. Under a regularity assumption, we obtain a characterization of Følner functions of these groups. As another application, we evaluate the asymptotics of the Følner function of $Sym(\mathbb{Z})\rtimes {\mathbb{Z}}$. We construct new examples of groups with Shalom's property $H_{\mathrm{FD}}$, in particular among nilpotent-by-cyclic and lacunary hyperbolic groups. Among these examples we find groups with property $H_{\mathrm{FD}}$, which are direct products of lacunary hyperbolic groups and have arbitrarily large Følner functions.

math.GR↗
arXiv2023

Furstenberg entropy spectra of stationary actions of semisimple Lie groups

We determine Furstenberg entropy spectra of ergodic stationary actions of $SL(d,\mathbb{R})$ and its lattices. The constraints on entropy spectra are derived from a refinement of the Nevo-Zimmer projective factor theorem. The realisation part is achieved by means of building Poisson bundles over stationary random subgroups.

math.DS↗
arXiv2023

EnfoMax: Domain Entropy and Mutual Information Maximization for Domain Generalized Face Anti-spoofing

The face anti-spoofing (FAS) method performs well under intra-domain setups. However, its cross-domain performance is unsatisfactory. As a result, the domain generalization (DG) method has gained more attention in FAS. Existing methods treat FAS as a simple binary classification task and propose a heuristic training objective to learn domain-invariant features. However, there is no theoretical explanation of what a domain-invariant feature is. Additionally, the lack of theoretical support makes domain generalization techniques such as adversarial training lack training stability. To address these issues, this paper proposes the EnfoMax framework, which uses information theory to analyze cross-domain FAS tasks. This framework provides theoretical guarantees and optimization objectives for domain-generalized FAS tasks. EnfoMax maximizes the domain entropy and mutual information of live samples in source domains without using adversarial learning. Experimental results demonstrate that our approach performs well on extensive public datasets and outperforms state-of-the-art methods.

cs.CV↗
arXiv2023

Towards Generalizable Data Protection With Transferable Unlearnable Examples

Artificial Intelligence (AI) is making a profound impact in almost every domain. One of the crucial factors contributing to this success has been the access to an abundance of high-quality data for constructing machine learning models. Lately, as the role of data in artificial intelligence has been significantly magnified, concerns have arisen regarding the secure utilization of data, particularly in the context of unauthorized data usage. To mitigate data exploitation, data unlearning have been introduced to render data unexploitable. However, current unlearnable examples lack the generalization required for wide applicability. In this paper, we present a novel, generalizable data protection method by generating transferable unlearnable examples. To the best of our knowledge, this is the first solution that examines data privacy from the perspective of data distribution. Through extensive experimentation, we substantiate the enhanced generalizable protection capabilities of our proposed method.

cs.CR↗
arXiv2022

Long range random walks and associated geometries on groups of polynomial growth

In the context of countable groups of polynomial volume growth, we consider a large class of random walks that are allowed to take long jumps along multiple subgroups according to power law distributions. For such a random walk, we study the large time behavior of its probability of return at time $n$ in terms of the key parameters describing the driving measure and the structure of the underlying group. We obtain assorted estimates including near-diagonal two-sided estimates and the Hölder continuity of the solutions of the associated discrete parabolic difference equation. In each case, these estimates involve the construction of a geometry adapted to the walk.

math.PR↗
arXiv2022

Limit theorems for some long range random walks on torsion free nilpotent groups

We consider a natural class of long range random walks on torsion free nilpotent groups and develop limit theorems for these walks. Given the original discrete group $Γ$ and a random walk $(S_n)_ {n\ge1}$ driven by a certain type of symmetric probability measure $μ$, we construct a homogeneous nilpotent Lie group $G_\bullet(Γ,μ)$ which carries an adapted dilation structure and a stable-like process $(X_t)_{ t\ge0}$ which appears in a Donsker-type functional limit theorem as the limit of a rescaled version of the random walk. Both the limit group and the limit process on that group depend on the measure $μ$. In addition, the functional limit theorem is complemented by a local limit theorem.

math.PR↗
arXiv2022

Growth of groups with linear Schreier graphs

We introduce a new method of proving upper estimates of growth of finitely generated groups and constructing groups of intermediate growth using graphs of their actions. These estimates are of the form $\exp(n^α)$ for some $α<1$, and provide the first examples of such bounds for simple groups of intermediate growth.

math.GR↗
arXiv2020

Law of large numbers for the drift of two-dimensional wreath product

We prove the law of large numbers for the drift of random walks on the two-dimensional lamplighter group, under the assumption that the random walk has finite $(2+ε)$-moment. This result is in contrast with classical examples of abelian groups, where the displacement after $n$ steps, normalised by its mean, does not concentrate, and the limiting distribution of the normalised $n$-step displacement admits a density whose support is $[0,\infty)$. We study further examples of groups, some with random walks satisfying LLN for drift and other examples where such concentration phenomenon does not hold, and study relation of this property with asymptotic geometry of groups.

math.PR↗
arXiv2020

Isoperimetric profiles and random walks on some groups defined by piecewise actions

We study the isoperimetric and spectral profiles of certain families of finitely generated groups defined via actions on labelled Schreier graphs and simple {\em gluing} of such. In one of our simplest constructions---the {\em pocket-extension} of a group $G$---this leads to the study of certain finitely generated subgroups of the full permutation group $\mathbb S(G\cup \{*\})$. Some sharp estimates are obtained while many challenging questions remain.

math.PR↗
arXiv2020

On FC-central extensions of groups of intermediate growth

It is shown that FC-central extensions retain sub-exponential volume growth. A large collection of FC-central extensions of the first Grigorchuk group is provided by the constructions in the works of Erschler and Kassabov-Pak. We show that in these examples subgroup separability is preserved. We introduce two new collections of extensions of the Grigorchuk group. One collection gives first examples of intermediate growth groups with centers isomorphic to $\mathbb{Z}^{\infty}$; and the other provides groups with prescribed oscillating intermediate growth functions.

math.GR↗
arXiv2020

On rigid stabilizers and invariant random subgroups of groups of homeomorphisms

A generalization of the double commutator lemma for normal subgroups is shown for invariant random subgroups of a countable group acting faithfully on a Hausdorff space. As an application, we classify ergodic invariant random subgroups of topological full groups of Cantor minimal $\mathbb{Z}^{d}$-systems. Another corollary is that for an ergodic invariant random subgroup of a branch group, a.e. subgroup $H$ must contain derived subgroups of certain rigid stabilizers. Such results can be applied towards classification of invariant random subgroups of Grigorchuk groups.

math.GR↗
arXiv2019

Growth of periodic Grigorchuk groups

On torsion Grigorchuk groups we construct random walks of finite entropy and power-law tail decay with non-trivial Poisson boundary. Such random walks provide near optimal volume lower estimates for these groups. In particular, for the first Grigorchuk group $G$ we show that its volume growth function $v_{G,S}(n)$ satisfies that $\lim_{n\to\infty}\log\log v_{G,S}(n)/\log n=α_{0}$, where $α_{0}=\frac{\log2}{\logλ_{0}}\approx0.7674$, $λ_{0}$ is the positive root of the polynomial $X^{3}-X^{2}-2X-4$.

math.GR↗
arXiv2018

Random walks among time increasing conductances: heat kernel estimates

For any graph having a suitable uniform Poincare inequality and volume growth regularity, we establish two-sided Gaussian transition density estimates and parabolic Harnack inequality, for constant speed continuous time random walks evolving via time varying, uniformly elliptic conductances, provided the vertex conductances (i.e. reversing measures), increase in time. Such transition density upper bounds apply for discrete time uniformly lazy walks, with the matching lower bounds holding once the parabolic Harnack inequality is proved.

math.PR↗