1,057 research outputs found
Convex Relaxations for Permutation Problems
Seriation seeks to reconstruct a linear order between variables using
unsorted, pairwise similarity information. It has direct applications in
archeology and shotgun gene sequencing for example. We write seriation as an
optimization problem by proving the equivalence between the seriation and
combinatorial 2-SUM problems on similarity matrices (2-SUM is a quadratic
minimization problem over permutations). The seriation problem can be solved
exactly by a spectral algorithm in the noiseless case and we derive several
convex relaxations for 2-SUM to improve the robustness of seriation solutions
in noisy settings. These convex relaxations also allow us to impose structural
constraints on the solution, hence solve semi-supervised seriation problems. We
derive new approximation bounds for some of these relaxations and present
numerical experiments on archeological data, Markov chains and DNA assembly
from shotgun gene sequencing data.Comment: Final journal version, a few typos and references fixe
Tightness of the maximum likelihood semidefinite relaxation for angular synchronization
Maximum likelihood estimation problems are, in general, intractable
optimization problems. As a result, it is common to approximate the maximum
likelihood estimator (MLE) using convex relaxations. In some cases, the
relaxation is tight: it recovers the true MLE. Most tightness proofs only apply
to situations where the MLE exactly recovers a planted solution (known to the
analyst). It is then sufficient to establish that the optimality conditions
hold at the planted signal. In this paper, we study an estimation problem
(angular synchronization) for which the MLE is not a simple function of the
planted solution, yet for which the convex relaxation is tight. To establish
tightness in this context, the proof is less direct because the point at which
to verify optimality conditions is not known explicitly.
Angular synchronization consists in estimating a collection of phases,
given noisy measurements of the pairwise relative phases. The MLE for angular
synchronization is the solution of a (hard) non-bipartite Grothendieck problem
over the complex numbers. We consider a stochastic model for the data: a
planted signal (that is, a ground truth set of phases) is corrupted with
non-adversarial random noise. Even though the MLE does not coincide with the
planted signal, we show that the classical semidefinite relaxation for it is
tight, with high probability. This holds even for high levels of noise.Comment: 2 figure
Faster SDP hierarchy solvers for local rounding algorithms
Convex relaxations based on different hierarchies of linear/semi-definite
programs have been used recently to devise approximation algorithms for various
optimization problems. The approximation guarantee of these algorithms improves
with the number of {\em rounds} in the hierarchy, though the complexity of
solving (or even writing down the solution for) the 'th level program grows
as where is the input size.
In this work, we observe that many of these algorithms are based on {\em
local} rounding procedures that only use a small part of the SDP solution (of
size instead of ). We give an algorithm to
find the requisite portion in time polynomial in its size. The challenge in
achieving this is that the required portion of the solution is not fixed a
priori but depends on other parts of the solution, sometimes in a complicated
iterative manner.
Our solver leads to time algorithms to obtain the same
guarantees in many cases as the earlier time algorithms based on
rounds of the Lasserre hierarchy. In particular, guarantees based on rounds can be realized in polynomial time.
We develop and describe our algorithm in a fairly general abstract framework.
The main technical tool in our work, which might be of independent interest in
convex optimization, is an efficient ellipsoid algorithm based separation
oracle for convex programs that can output a {\em certificate of infeasibility
with restricted support}. This is used in a recursive manner to find a sequence
of consistent points in nested convex bodies that "fools" local rounding
algorithms.Comment: 30 pages, 8 figure
Probabilistic Clustering Using Maximal Matrix Norm Couplings
In this paper, we present a local information theoretic approach to
explicitly learn probabilistic clustering of a discrete random variable. Our
formulation yields a convex maximization problem for which it is NP-hard to
find the global optimum. In order to algorithmically solve this optimization
problem, we propose two relaxations that are solved via gradient ascent and
alternating maximization. Experiments on the MSR Sentence Completion Challenge,
MovieLens 100K, and Reuters21578 datasets demonstrate that our approach is
competitive with existing techniques and worthy of further investigation.Comment: Presented at 56th Annual Allerton Conference on Communication,
Control, and Computing, 201
Multiclass Total Variation Clustering
Ideas from the image processing literature have recently motivated a new set
of clustering algorithms that rely on the concept of total variation. While
these algorithms perform well for bi-partitioning tasks, their recursive
extensions yield unimpressive results for multiclass clustering tasks. This
paper presents a general framework for multiclass total variation clustering
that does not rely on recursion. The results greatly outperform previous total
variation algorithms and compare well with state-of-the-art NMF approaches
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