40,649 research outputs found
Projected Power Iteration for Network Alignment
The network alignment problem asks for the best correspondence between two
given graphs, so that the largest possible number of edges are matched. This
problem appears in many scientific problems (like the study of protein-protein
interactions) and it is very closely related to the quadratic assignment
problem which has graph isomorphism, traveling salesman and minimum bisection
problems as particular cases. The graph matching problem is NP-hard in general.
However, under some restrictive models for the graphs, algorithms can
approximate the alignment efficiently. In that spirit the recent work by Feizi
and collaborators introduce EigenAlign, a fast spectral method with convergence
guarantees for Erd\H{o}s-Reny\'i graphs. In this work we propose the algorithm
Projected Power Alignment, which is a projected power iteration version of
EigenAlign. We numerically show it improves the recovery rates of EigenAlign
and we describe the theory that may be used to provide performance guarantees
for Projected Power Alignment.Comment: 8 page
Faster subsequence recognition in compressed strings
Computation on compressed strings is one of the key approaches to processing
massive data sets. We consider local subsequence recognition problems on
strings compressed by straight-line programs (SLP), which is closely related to
Lempel--Ziv compression. For an SLP-compressed text of length , and an
uncompressed pattern of length , C{\'e}gielski et al. gave an algorithm for
local subsequence recognition running in time . We improve
the running time to . Our algorithm can also be used to
compute the longest common subsequence between a compressed text and an
uncompressed pattern in time ; the same problem with a
compressed pattern is known to be NP-hard
Multireference Alignment using Semidefinite Programming
The multireference alignment problem consists of estimating a signal from
multiple noisy shifted observations. Inspired by existing Unique-Games
approximation algorithms, we provide a semidefinite program (SDP) based
relaxation which approximates the maximum likelihood estimator (MLE) for the
multireference alignment problem. Although we show that the MLE problem is
Unique-Games hard to approximate within any constant, we observe that our
poly-time approximation algorithm for the MLE appears to perform quite well in
typical instances, outperforming existing methods. In an attempt to explain
this behavior we provide stability guarantees for our SDP under a random noise
model on the observations. This case is more challenging to analyze than
traditional semi-random instances of Unique-Games: the noise model is on
vertices of a graph and translates into dependent noise on the edges.
Interestingly, we show that if certain positivity constraints in the SDP are
dropped, its solution becomes equivalent to performing phase correlation, a
popular method used for pairwise alignment in imaging applications. Finally, we
show how symmetry reduction techniques from matrix representation theory can
simplify the analysis and computation of the SDP, greatly decreasing its
computational cost
Systematics of Aligned Axions
We describe a novel technique that renders theories of axions tractable,
and more generally can be used to efficiently analyze a large class of periodic
potentials of arbitrary dimension. Such potentials are complex energy
landscapes with a number of local minima that scales as , and so for
large appear to be analytically and numerically intractable. Our method is
based on uncovering a set of approximate symmetries that exist in addition to
the periods. These approximate symmetries, which are exponentially close to
exact, allow us to locate the minima very efficiently and accurately and to
analyze other characteristics of the potential. We apply our framework to
evaluate the diameters of flat regions suitable for slow-roll inflation, which
unifies, corrects and extends several forms of "axion alignment" previously
observed in the literature. We find that in a broad class of random theories,
the potential is smooth over diameters enhanced by compared to the
typical scale of the potential. A Mathematica implementation of our framework
is available online.Comment: 68 pages, 17 figure
- …