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Nhan-Phu Chung

Publications and source records attributed to Nhan-Phu Chung.

18 recordsLinked to original sources
arXiv2025

Dynamical Properties of Tokens in Self-Attention and Effects of Positional Encoding

This paper investigates the dynamical properties of tokens in pre-trained Transformer models and explores their application to improving Transformers. To this end, we analyze the dynamical system governing the continuous-time limit of the pre-trained model and characterize the asymptotic behavior of its solutions. Specifically, we characterize when tokens move closer to or farther from one another over time, depending on the model parameters. We provide sufficient conditions, based on these parameters, to identify scenarios where tokens either converge to zero or diverge to infinity. Unlike prior works, our conditions are broader in scope and more applicable to real-world models. Furthermore, we investigate how different forms of positional encoding -- specifically absolute and rotary -- affect these dynamical regimes. Empirical evidence reveals that the convergence scenario adversely impacts model performance. Motivated by these insights, we propose simple refinements to Transformer architectures that mitigate convergence behavior in models with absolute or rotary positional encoding. These findings support theoretical foundations and design principles for improving Transformer models.

cs.LG↗
arXiv2025

Weak Optimal Entropy Transport Problems

In this paper, we introduce weak optimal entropy transport problems that cover both optimal entropy transport problems and weak optimal transport problems introduced by Liero, Mielke, and Savaré [27]; and Gozlan, Roberto, Samson and Tetali [20], respectively. Under some mild assumptions of entropy functionals, we establish a Kantorovich type duality for our weak optimal entropy transport problem. We also introduce martingale optimal entropy transport problems, and express them in terms of duality, homogeneous marginal perspective functionals and homogeneous constraints.

math.FA↗
arXiv2024

Algebraic and topological properties of Riordan groups over finite fields

In this paper, we investigate algebraic and topological properties of the Riordan groups over finite fields. These groups provide a new class of topologically finitely generated profinite groups with finite width. We also introduce, characterize index-subgroups of our Riordan groups, and finally we show exactly the range of Hausdorff dimensions of these groups. The latter results are analogous to the work of Barnea and Klopsch for the Nottingham groups.

math.GR↗
arXiv2023

cc-differential uniformity, (almost) perfect cc-nonlinearity, and equivalences

In this article, we introduce new notions $cc$-differential uniformity, $cc$-differential spectrum, PccN functions and APccN functions, and investigate their properties. We also introduce $c$-CCZ equivalence, $c$-EA equivalence, and $c1$-equivalence. We show that $c$-differential uniformity is invariant under $c1$-equivalence, and $cc$-differential uniformity and $cc$-differential spectrum are preserved under $c$-CCZ equivalence. We characterize $cc$-differential uniformity of vectorial Boolean functions in terms of the Walsh transformation. We investigate $cc$-differential uniformity of power functions $F(x)=x^d$. We also illustrate examples to prove that $c$-CCZ equivalence is strictly more general than $c$-EA equivalence.

cs.IT↗
arXiv2021

Event-triggered bipartite consensus for multiagent system with general linear dynamics: an integral-type event triggered control

In this paper, we propose an integral based event-triggering controller for bipartite consensus of the multi-agent systems whose dynamics are described by general linear system. We prove that the system achieves the bipartite consensus in asymptotic regime and there is a positive minimum inter-event time (MIET) between two consecutive triggering times of each agent. The proof of the asymptotive stability involves a novel argument used to bound the norm of the difference between the true state and its estimated state (for each time) by an integration of its square. Numerical results are provided supporting the effectiveness of the proposed controller.

math.OC↗
arXiv2021

Weak optimal total variation transport problems and generalized Wasserstein barycenters

In this paper, we establish a Kantorovich duality for weak optimal total variation transport problems. As consequences, we recover a version of duality formula for partial optimal transports established by Caffarelli and McCann; and we also get another proof of Kantorovich-Rubinstein Theorem for generalized Wasserstein distance $\widetilde{W}_1^{a,b}$ proved before by Piccoli and Rossi. Then we apply our duality formula to study generalized Wasserstein barycenters. We show the existence of these barycenters for measures with compact supports. Finally, we prove the consistency of our barycenters.

math.OC↗
arXiv2020

Barycenters in the Hellinger-Kantorovich space

Recently, Liero, Mielke and Savaré introduced Hellinger-Kantorovich distance on the space of nonnegative Radon measures of a metric space $X$ [19,20]. We prove that Hellinger-Kantorovich barycenters always exist for a class of metric spaces containing of compact spaces, and Polish $CAT(1)$ spaces; and if we assume further some conditions on starting measures, such barycenters are unique. We also introduce homogeneous multimarginal problems and illustrate some relations between their solutions with Hellinger-Kantorovich barycenters. Our results are analogous to the work of Agueh and Carlier [1] for Wassertein barycenters.

math.OC↗
arXiv2020

Gromov-Hausdorff distances for dynamical systems

We study equivariant Gromov-Hausdorff distances for general continuous actions which are not necessarily isometric as Fukaya introduced. We prove that if an action is expansive and has pseudo-orbit tracing property then it is stable under our adapted equivariant Gromov-Hausdorff topology. Finally, using Lott and Villani's ideas of optimal transport in studying curvature-dimension conditions, we investigate equivariant Gromov-Hausdorff convergence for actions of locally compact amenable groups on Wasserstein spaces.

math.DS↗
arXiv2019

Duality and quotient spaces of generalized Wasserstein spaces

In this article, using ideas of Liero, Mielke and Savaré in [21], we establish a Kantorovich duality for generalized Wasserstein distances $W_1^{a,b}$ on a generalized Polish metric space, introduced by Picolli and Rossi. As a consequence, we give another proof that $W_1^{a,b}$ coincide with flat metrics which is a main result of [25], and therefore we get a result of independent interest that $\left(\mathcal{M}(X), W^{a,b}_1\right)$ is a geodesic space for every Polish metric space $X$. We also prove that $(\mathcal{M}^G(X),W_p^{a,b})$ is isometric isomorphism to $(\mathcal{M}(X/G),W_p^{a,b})$ for isometric actions of a compact group $G$ on a Polish metric space $X$; and several results of Gromov-Hausdorrf convergence and equivariant Gromov-Hausdorff convergence of generalized Wasserstein spaces. The latter results were proved for standard Wasserstein spaces in [22],[14] and [8] respectively.

math.MG↗
arXiv2017

Divergence, Undistortion and Hölder Continuous Cocycle Superrigidity for Full Shifts

In this article, we will prove a full topological version of Popa's measurable cocycle superrigidity theorem for full shifts. More precisely, we prove that every Hölder continuous cocycle for the full shifts of every finitely generated group $G$ that has one end, undistorted elements and sub-exponential divergence function is cohomologous to a group homomorphism via a Hölder continuous transfer map if the target group is complete and admits a compatible bi-invariant metric. Using the ideas of Behrstock, Dru\c {t}u, Mosher, Mozes and Sapir, we show that the class of our acting groups is large including wide groups having undistorted elements and one-ended groups with strong thick of finite orders. As a consequence, irreducible uniform lattices of most of higher rank connected semisimple Lie groups, mapping class groups of $g$-genus surfaces with $p$-punches, $g\geq 2, p\geq 0$, Thompson group, $Aut(F_n)$, $Out(F_n)$, $n\geq3$, certain (2 dimensional)-Coxeter groups, and one-ended right-angled Artin groups are in our class. This partially extends the main result in our previous paper.

math.DS↗
arXiv2016

Topological stability and pseudo-orbit tracing property of group actions

In this note we extend the concept of topological stability from homeomorphisms to group actions on compact metric spaces, and prove that if an action of a finitely generated group is expansive and has the pseudo-orbit tracing property then it is topologically stable. This represents a group action version of the Walter's stability theorem. Moreover we give a class of group actions with topological stability or pseudo-orbit tracing property. On the other hand, we also provide a characterization of subshifts of finite type over finitely generated groups in term of pseudo-orbit tracing property.

math.DS↗
arXiv2016

Continuous Cocycle Superrigidity for Shifts and Groups with One End

In this article, we prove that if a finitely generated group $G$ is not torsion then a necessary and sufficient condition for every full shift over $G$ has (continuous) cocycle superrigidity is that $G$ has one end. It is a topological version of the well known Popa's measurable cocycle superrigidity theorem. For the proof of sufficient condition, we introduce a new specification property for shifts over general groups which play a similar role as malleable property in the measurable setting. This new specification property is good enough for us to extend the method using homoclinic equivalence relation that was introduced by Klaus Schmidt to study cocycle rigidity for $\mathbb{Z}^d$-shifts. Indeed, in this direction we prove this superrigidity result for more certain general systems. And for the converse, we apply Specker's characterization for ends of groups via the associated first cohomology groups to get the result. Finally, combining our results with a result of Xin Li, we have an application in continuous orbit equivalence rigidity.

math.DS↗
arXiv2014

Weak expansiveness for actions of sofic groups

In this paper, we shall introduce $h$-expansiveness and asymptotical $h$-expansiveness for actions of sofic groups. By the definitions, each $h$-expansive action of sofic groups is asymptotically $h$-expansive. We show that each expansive action of sofic groups is $h$-expansive, and, for any given asymptotically $h$-expansive action of sofic groups, the entropy function (with respect to measures) is upper semi-continuous and hence the system admits a measure with maximal entropy. Observe that asymptotically $h$-expansive property was firstly introduced and studied by Misiurewicz for $\mathbb{Z}$-actions using the language of topological conditional entropy. And thus in the remaining part of the paper, we shall compare our definitions of weak expansiveness for actions of sofic groups with the definitions given in the same spirit of Misiurewicz's ideas when the group is amenable. It turns out that these two definitions are equivalent in this setting.

math.DS↗
arXiv2014

Homoclinic groups, IE groups, and expansive algebraic actions

We give algebraic characterizations for expansiveness of algebraic actions of countable groups. The notion of p-expansiveness is introduced for algebraic actions, and we show that for countable amenable groups, a finitely presented algebraic action is 1-expansive exactly when it has finite entropy. We also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms, and show that the IE group determines the Pinsker factor for such actions. For an expansive algebraic action of a polycyclic-by-finite group on X, it is shown that the entropy of the action is equal to the entropy of the induced action on the Pontryagin dual of the homoclinic group, the homoclinic group is a dense subgroup of the IE group, the homoclinic group is nontrivial exactly when the action has positive entropy, and the homoclinic group is dense in X exactly when the action has completely positive entropy.

math.DS↗
arXiv2013

Some remarks on the entropy for algebraic actions of amenable groups

In this short note we study the entropy for algebraic actions of certain amenable groups. The possible values for this entropy are studied. Various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant. We provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to infinity. These results extend previous work by Lind and Ward on p-adic entropy.

math.DS↗
arXiv2012

Topological pressure and the variational principle for actions of sofic groups

In this paper, we introduce topological pressure for continuous actions of countable sofic groups on compact metrizable spaces. This generalizes the classical topological pressure for continuous actions of countable amenable groups on such spaces. We also establish the variational principle for topological pressure in this sofic context.

math.DS↗