arXiv subjects

Jianchao Wu

Publications and source records attributed to Jianchao Wu.

28 records · Page 2Linked to original sources

arXiv2019

Learning Actor Relation Graphs for Group Activity Recognition

Modeling relation between actors is important for recognizing group activity in a multi-person scene. This paper aims at learning discriminative relation between actors efficiently using deep models. To this end, we propose to build a flexible and efficient Actor Relation Graph (ARG) to simultaneously capture the appearance and position relation between actors. Thanks to the Graph Convolutional Network, the connections in ARG could be automatically learned from group activity videos in an end-to-end manner, and the inference on ARG could be efficiently performed with standard matrix operations. Furthermore, in practice, we come up with two variants to sparsify ARG for more effective modeling in videos: spatially localized ARG and temporal randomized ARG. We perform extensive experiments on two standard group activity recognition datasets: the Volleyball dataset and the Collective Activity dataset, where state-of-the-art performance is achieved on both datasets. We also visualize the learned actor graphs and relation features, which demonstrate that the proposed ARG is able to capture the discriminative relation information for group activity recognition.

cs.CV↗

arXiv2019

Isomorphism of the cubical and categorical cohomology groups of a higher-rank graph

We use category-theoretic techniques to provide two proofs showing that for a higher-rank graph $Λ$, its cubical (co-)homology and categorical (co-)homology groups are isomorphic in all degrees, thus answering a question of Kumjian, Pask and Sims in the positive. Our first proof uses the topological realization of a higher-rank graph, which was introduced by Kaliszewski, Kumjian, Quigg, and Sims. In our more combinatorial second proof, we construct, explicitly and in both directions, maps on the level of (co-)chain complexes that implement said isomorphism. Along the way, we extend the definition of cubical (co-)homology to allow arbitrary coefficient modules.

math.OA↗

arXiv2018

Amenability of coarse spaces and K-algebras

In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over commutative fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra.

math.RA↗

arXiv2018

Amenability and uniform Roe algebras

Amenability for groups can be extended to metric spaces, algebras over commutative fields and $C^*$-algebras by adapting the notion of Følner nets. In the present article we investigate the close ties among these extensions and show that these three pictures unify in the context of the uniform Roe algebra $C_u^*(X)$ over a metric space $(X,d)$ with bounded geometry. In particular, we show that the following conditions are equivalent: (1) $(X,d)$ is amenable; (2) the translation algebra generating $C_u^*(X)$ is algebraically amenable (3) $C_u^*(X)$ has a tracial state; (4) $C_u^*(X)$ is not properly infinite; (5) $[1]_0\neq [0]_0$ in the $K_0$-group $K_0(C_u^*(X))$; (6) $C_u^*(X)$ does not contain the Leavitt algebra as a unital $*$-subalgebra; (7) $C_u^*(X)$ is a Følner $C^*$-algebra in the sense that it admits a net of unital completely positive maps into matrices which is asymptotically multiplicative in the normalized trace norm. We also show that every possible tracial state of the uniform Roe algebra $C_u^*(X)$ is amenable.

math.OA↗

arXiv2018

Localization C*-algebras and K-theoretic duality

Based on the localization algebras of Yu, and their subsequent analysis by Qiao and Roe, we give a new picture of KK-theory in terms of time-parametrized families of (locally) compact operators that asymptotically commute with appropriate representations.

math.KT↗

arXiv2017

Rokhlin dimension for actions of residually finite groups

We introduce the concept of Rokhlin dimension for actions of residually finite groups on C*-algebras, extending previous notions of Rokhlin dimension for actions of finite groups and the integers, as introduced by Hirshberg, Winter and the third author. If the group has a box space of finite asymptotic dimension, then actions with finite Rokhlin dimension preserve the property of having finite nuclear dimension, when passing to the crossed product C*-algebra. A detailed study of the asymptotic dimension of box spaces shows that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a reasonably large class of examples. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group C*-algebras have finite nuclear dimension. This extends an analogous result about $\mathbb{Z}^m$-actions by the first author. We also provide some results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing C*-algebra.

math.OA↗

arXiv2016

Rokhlin dimension for flows

We introduce a notion of Rokhlin dimension for one parameter automorphism groups of C*-algebras. This generalizes Kishimoto's Rokhlin property for flows, and is analogous to the notion of Rokhlin dimension for actions of the integers and other discrete groups introduced by the authors and Zacharias in previous papers. We show that finite nuclear dimension and absorption of a strongly self-absorbing C*-algebra are preserved under forming crossed products by flows with finite Rokhlin dimension, and that these crossed products are stable. Furthermore, we show that a flow on a commutative C*-algebra arising from a free topological flow has finite Rokhlin dimension, whenever the spectrum is a locally compact metrizable space with finite covering dimension. For flows that are both free and minimal, this has strong consequences for the associated crossed product C*-algebras: Those containing a non-zero projection are classified by the Elliott invariant (for compact manifolds this consists of topological K-theory together with the space of invariant probability measures and a natural pairing given by the Ruelle-Sullivan map).

math.OA↗

arXiv2016

The nuclear dimension of C*-algebras associated to homeomorphisms

We show that if X is a finite dimensional locally compact Hausdorff space, then the crossed product of C_0(X) by any automorphism has finite nuclear dimension. This generalizes previous results, in which the automorphism was required to be free. As an application, we show that group C*-algebras of certain non-nilpotent groups have finite nuclear dimension.

math.OA↗

arXiv2016

Ergodic invariant states and irreducible representations of crossed product $C^*$-algebras

Motivated by reformulating Furstenberg's $\times p,\times q$ conjecture via representations of a crossed product $C^*$-algebra, we show that in a discrete $C^*$-dynamical system $(A,Γ)$, the space of (ergodic) $Γ$-invariant states on $A$ is homeomorphic to a subspace of (pure) state space of $A\rtimesΓ$. Various applications of this in topological dynamical systems and representation theory are obtained. In particular, we prove that the classification of ergodic $Γ$-invariant regular Borel probability measures on a compact Hausdorff space $X$ is equivalent to the classification a special type of irreducible representations of $C(X)\rtimes Γ$.

math.OA↗

arXiv2015

The Asymptotic Dimension of Box Spaces for Elementary Amenable Groups

We show that the asymptotic dimension of box spaces behaves (sub)additively with respect to extensions of groups. As a result, we obtain that for an elementary amenable group, the asymptotic dimension of any of its box spaces is bounded above by its Hirsch length. This bound is shown to be an equality for a large subclass of groups including all virtually polycyclic groups.

math.MG↗