6,164 research outputs found
Local-To-Global Agreement Expansion via the Variance Method
Agreement expansion is concerned with set systems for which local assignments to the sets with almost perfect pairwise consistency (i.e., most overlapping pairs of sets agree on their intersections) implies the existence of a global assignment to the ground set (from which the sets are defined) that agrees with most of the local assignments.
It is currently known that if a set system forms a two-sided or a partite high dimensional expander then agreement expansion is implied. However, it was not known whether agreement expansion can be implied for one-sided high dimensional expanders.
In this work we show that agreement expansion can be deduced for one-sided high dimensional expanders assuming that all the vertices\u27 links (i.e., the neighborhoods of the vertices) are agreement expanders. Thus, for one-sided high dimensional expander, an agreement expansion of the large complicated complex can be deduced from agreement expansion of its small simple links.
Using our result, we settle the open question whether the well studied Ramanujan complexes are agreement expanders. These complexes are neither partite nor two-sided high dimensional expanders. However, they are one-sided high dimensional expanders for which their links are partite and hence are agreement expanders. Thus, our result implies that Ramanujan complexes are agreement expanders, answering affirmatively the aforementioned open question.
The local-to-global agreement expansion that we prove is based on the variance method that we develop. We show that for a high dimensional expander, if we define a function on its top faces and consider its local averages over the links then the variance of these local averages is much smaller than the global variance of the original function. This decreasing in the variance enables us to construct one global agreement function that ties together all local agreement functions
High Dimensional Random Walks and Colorful Expansion
Random walks on bounded degree expander graphs have numerous applications,
both in theoretical and practical computational problems. A key property of
these walks is that they converge rapidly to their stationary distribution.
In this work we {\em define high order random walks}: These are
generalizations of random walks on graphs to high dimensional simplicial
complexes, which are the high dimensional analogues of graphs. A simplicial
complex of dimension has vertices, edges, triangles, pyramids, up to
-dimensional cells. For any , a high order random walk on
dimension moves between neighboring -faces (e.g., edges) of the complex,
where two -faces are considered neighbors if they share a common
-face (e.g., a triangle). The case of recovers the well studied
random walk on graphs.
We provide a {\em local-to-global criterion} on a complex which implies {\em
rapid convergence of all high order random walks} on it. Specifically, we prove
that if the -dimensional skeletons of all the links of a complex are
spectral expanders, then for {\em all} the high order random walk
on dimension converges rapidly to its stationary distribution.
We derive our result through a new notion of high dimensional combinatorial
expansion of complexes which we term {\em colorful expansion}. This notion is a
natural generalization of combinatorial expansion of graphs and is strongly
related to the convergence rate of the high order random walks.
We further show an explicit family of {\em bounded degree} complexes which
satisfy this criterion. Specifically, we show that Ramanujan complexes meet
this criterion, and thus form an explicit family of bounded degree high
dimensional simplicial complexes in which all of the high order random walks
converge rapidly to their stationary distribution.Comment: 27 page
Ramanujan Complexes and bounded degree topological expanders
Expander graphs have been a focus of attention in computer science in the
last four decades. In recent years a high dimensional theory of expanders is
emerging. There are several possible generalizations of the theory of expansion
to simplicial complexes, among them stand out coboundary expansion and
topological expanders. It is known that for every d there are unbounded degree
simplicial complexes of dimension d with these properties. However, a major
open problem, formulated by Gromov, is whether bounded degree high dimensional
expanders, according to these definitions, exist for d >= 2. We present an
explicit construction of bounded degree complexes of dimension d = 2 which are
high dimensional expanders. More precisely, our main result says that the
2-skeletons of the 3-dimensional Ramanujan complexes are topological expanders.
Assuming a conjecture of Serre on the congruence subgroup property, infinitely
many of them are also coboundary expanders.Comment: To appear in FOCS 201
The implementation of a one-to-one iPAD program in an urban high school
The purpose of this qualitative study was to apply the lessons learned from the Apple
Classrooms of Tomorrow studies, the SAMR model, and Diffusion of Innovations theory
to explore stakeholder perceptions of iPad integration at an urban high school in
Massachusetts. The implementation was viewed through the lenses of the Apple
Classrooms of Tomorrow (ACOT) studies (Baker, Gearhart, & Herman, 1990; Dwyer,
Ringstaff, & Haymore Sandholtz, 1990a; Dwyer, Ringstaff, & Haymore Sandholtz,
1990b), Rogersâ (2003) Diffusion of Innovation (DOI) Model, and Puenteduraâs (2012)
Substitution, Augmentation, Modification, and Redefinition (SAMR) Model. The
researcher used qualitative analysis to code the data. Through data analysis, five themes emerged: communication, control, division, distraction, and workflow. The iPads
changed how and when students and teachers communicated. Teachers sought more
control over the iPads in the classroom. Control over learning shifted toward the students
with the introduction of the iPads. Divisions became apparent with iPad use: new
teachers versus veteran teachers and upperclassman versus underclassman. Distractions were rampant. The iPads influenced the workflow of how teachers taught and how students accessed the curriculum
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