923 research outputs found
A Unified View of Graph Regularity via Matrix Decompositions
We prove algorithmic weak and \Szemeredi{} regularity lemmas for several
classes of sparse graphs in the literature, for which only weak regularity
lemmas were previously known. These include core-dense graphs, low threshold
rank graphs, and (a version of) upper regular graphs. More precisely, we
define \emph{cut pseudorandom graphs}, we prove our regularity lemmas for these
graphs, and then we show that cut pseudorandomness captures all of the above
graph classes as special cases.
The core of our approach is an abstracted matrix decomposition, roughly
following Frieze and Kannan [Combinatorica '99] and \Lovasz{} and Szegedy
[Geom.\ Func.\ Anal.\ '07], which can be computed by a simple algorithm by
Charikar [AAC0 '00]. This gives rise to the class of cut pseudorandom graphs,
and using work of Oveis Gharan and Trevisan [TOC '15], it also implies new
PTASes for MAX-CUT, MAX-BISECTION, MIN-BISECTION for a significantly expanded
class of input graphs. (It is NP Hard to get PTASes for these graphs in
general.
Regularity lemmas in a Banach space setting
Szemer\'edi's regularity lemma is a fundamental tool in extremal graph
theory, theoretical computer science and combinatorial number theory. Lov\'asz
and Szegedy [L. Lov\'asz and B. Szegedy: Szemer\'edi's Lemma for the analyst,
Geometric and Functional Analysis 17 (2007), 252-270] gave a Hilbert space
interpretation of the lemma and an interpretation in terms of compact- ness of
the space of graph limits. In this paper we prove several compactness results
in a Banach space setting, generalising results of Lov\'asz and Szegedy as well
as a result of Borgs, Chayes, Cohn and Zhao [C. Borgs, J.T. Chayes, H. Cohn and
Y. Zhao: An Lp theory of sparse graph convergence I: limits, sparse random
graph models, and power law distributions, arXiv preprint arXiv:1401.2906
(2014)].Comment: 15 pages. The topological part has been substantially improved based
on referees comments. To appear in European Journal of Combinatoric
PROMISE: Preconditioned Stochastic Optimization Methods by Incorporating Scalable Curvature Estimates
This paper introduces PROMISE (econditioned Stochastic
ptimization ethods by ncorporating
calable Curvature stimates), a suite of sketching-based
preconditioned stochastic gradient algorithms for solving large-scale convex
optimization problems arising in machine learning. PROMISE includes
preconditioned versions of SVRG, SAGA, and Katyusha; each algorithm comes with
a strong theoretical analysis and effective default hyperparameter values. In
contrast, traditional stochastic gradient methods require careful
hyperparameter tuning to succeed, and degrade in the presence of
ill-conditioning, a ubiquitous phenomenon in machine learning. Empirically, we
verify the superiority of the proposed algorithms by showing that, using
default hyperparameter values, they outperform or match popular tuned
stochastic gradient optimizers on a test bed of ridge and logistic
regression problems assembled from benchmark machine learning repositories. On
the theoretical side, this paper introduces the notion of quadratic regularity
in order to establish linear convergence of all proposed methods even when the
preconditioner is updated infrequently. The speed of linear convergence is
determined by the quadratic regularity ratio, which often provides a tighter
bound on the convergence rate compared to the condition number, both in theory
and in practice, and explains the fast global linear convergence of the
proposed methods.Comment: 127 pages, 31 Figure
On the Complexity of Newman's Community Finding Approach for Biological and Social Networks
Given a graph of interactions, a module (also called a community or cluster)
is a subset of nodes whose fitness is a function of the statistical
significance of the pairwise interactions of nodes in the module. The topic of
this paper is a model-based community finding approach, commonly referred to as
modularity clustering, that was originally proposed by Newman and has
subsequently been extremely popular in practice. Various heuristic methods are
currently employed for finding the optimal solution. However, the exact
computational complexity of this approach is still largely unknown.
To this end, we initiate a systematic study of the computational complexity
of modularity clustering. Due to the specific quadratic nature of the
modularity function, it is necessary to study its value on sparse graphs and
dense graphs separately. Our main results include a (1+\eps)-inapproximability
for dense graphs and a logarithmic approximation for sparse graphs. We make use
of several combinatorial properties of modularity to get these results. These
are the first non-trivial approximability results beyond the previously known
NP-hardness results.Comment: Journal of Computer and System Sciences, 201
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