1,521 research outputs found
A New Regularity Lemma and Faster Approximation Algorithms for Low Threshold Rank Graphs
Kolla and Tulsiani [KT07,Kolla11} and Arora, Barak and Steurer [ABS10]
introduced the technique of subspace enumeration, which gives approximation
algorithms for graph problems such as unique games and small set expansion; the
running time of such algorithms is exponential in the threshold-rank of the
graph.
Guruswami and Sinop [GS11,GS12], and Barak, Raghavendra, and Steurer [BRS11]
developed an alternative approach to the design of approximation algorithms for
graphs of bounded threshold-rank, based on semidefinite programming relaxations
in the Lassere hierarchy and on novel rounding techniques. These algorithms are
faster than the ones based on subspace enumeration and work on a broad class of
problems.
In this paper we develop a third approach to the design of such algorithms.
We show, constructively, that graphs of bounded threshold-rank satisfy a weak
Szemeredi regularity lemma analogous to the one proved by Frieze and Kannan
[FK99] for dense graphs. The existence of efficient approximation algorithms is
then a consequence of the regularity lemma, as shown by Frieze and Kannan.
Applying our method to the Max Cut problem, we devise an algorithm that is
faster than all previous algorithms, and is easier to describe and analyze
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.
On the Troll-Trust Model for Edge Sign Prediction in Social Networks
In the problem of edge sign prediction, we are given a directed graph
(representing a social network), and our task is to predict the binary labels
of the edges (i.e., the positive or negative nature of the social
relationships). Many successful heuristics for this problem are based on the
troll-trust features, estimating at each node the fraction of outgoing and
incoming positive/negative edges. We show that these heuristics can be
understood, and rigorously analyzed, as approximators to the Bayes optimal
classifier for a simple probabilistic model of the edge labels. We then show
that the maximum likelihood estimator for this model approximately corresponds
to the predictions of a Label Propagation algorithm run on a transformed
version of the original social graph. Extensive experiments on a number of
real-world datasets show that this algorithm is competitive against
state-of-the-art classifiers in terms of both accuracy and scalability.
Finally, we show that troll-trust features can also be used to derive online
learning algorithms which have theoretical guarantees even when edges are
adversarially labeled.Comment: v5: accepted to AISTATS 201
Multivariate Hawkes Processes for Large-scale Inference
In this paper, we present a framework for fitting multivariate Hawkes
processes for large-scale problems both in the number of events in the observed
history and the number of event types (i.e. dimensions). The proposed
Low-Rank Hawkes Process (LRHP) framework introduces a low-rank approximation of
the kernel matrix that allows to perform the nonparametric learning of the
triggering kernels using at most operations, where is the
rank of the approximation (). This comes as a major improvement to
the existing state-of-the-art inference algorithms that are in .
Furthermore, the low-rank approximation allows LRHP to learn representative
patterns of interaction between event types, which may be valuable for the
analysis of such complex processes in real world datasets. The efficiency and
scalability of our approach is illustrated with numerical experiments on
simulated as well as real datasets.Comment: 16 pages, 5 figure
Fast Computation of Fourier Integral Operators
We introduce a general purpose algorithm for rapidly computing certain types
of oscillatory integrals which frequently arise in problems connected to wave
propagation and general hyperbolic equations. The problem is to evaluate
numerically a so-called Fourier integral operator (FIO) of the form at points given on
a Cartesian grid. Here, is a frequency variable, is the
Fourier transform of the input , is an amplitude and
is a phase function, which is typically as large as ;
hence the integral is highly oscillatory at high frequencies. Because an FIO is
a dense matrix, a naive matrix vector product with an input given on a
Cartesian grid of size by would require operations.
This paper develops a new numerical algorithm which requires operations, and as low as in storage space. It operates by
localizing the integral over polar wedges with small angular aperture in the
frequency plane. On each wedge, the algorithm factorizes the kernel into two components: 1) a diffeomorphism which is
handled by means of a nonuniform FFT and 2) a residual factor which is handled
by numerical separation of the spatial and frequency variables. The key to the
complexity and accuracy estimates is that the separation rank of the residual
kernel is \emph{provably independent of the problem size}. Several numerical
examples demonstrate the efficiency and accuracy of the proposed methodology.
We also discuss the potential of our ideas for various applications such as
reflection seismology.Comment: 31 pages, 3 figure
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