22,671 research outputs found
1-Bit Matrix Completion under Exact Low-Rank Constraint
We consider the problem of noisy 1-bit matrix completion under an exact rank
constraint on the true underlying matrix . Instead of observing a subset
of the noisy continuous-valued entries of a matrix , we observe a subset
of noisy 1-bit (or binary) measurements generated according to a probabilistic
model. We consider constrained maximum likelihood estimation of , under a
constraint on the entry-wise infinity-norm of and an exact rank
constraint. This is in contrast to previous work which has used convex
relaxations for the rank. We provide an upper bound on the matrix estimation
error under this model. Compared to the existing results, our bound has faster
convergence rate with matrix dimensions when the fraction of revealed 1-bit
observations is fixed, independent of the matrix dimensions. We also propose an
iterative algorithm for solving our nonconvex optimization with a certificate
of global optimality of the limiting point. This algorithm is based on low rank
factorization of . We validate the method on synthetic and real data with
improved performance over existing methods.Comment: 6 pages, 3 figures, to appear in CISS 201
Towards Fast-Convergence, Low-Delay and Low-Complexity Network Optimization
Distributed network optimization has been studied for well over a decade.
However, we still do not have a good idea of how to design schemes that can
simultaneously provide good performance across the dimensions of utility
optimality, convergence speed, and delay. To address these challenges, in this
paper, we propose a new algorithmic framework with all these metrics
approaching optimality. The salient features of our new algorithm are
three-fold: (i) fast convergence: it converges with only
iterations that is the fastest speed among all the existing algorithms; (ii)
low delay: it guarantees optimal utility with finite queue length; (iii) simple
implementation: the control variables of this algorithm are based on virtual
queues that do not require maintaining per-flow information. The new technique
builds on a kind of inexact Uzawa method in the Alternating Directional Method
of Multiplier, and provides a new theoretical path to prove global and linear
convergence rate of such a method without requiring the full rank assumption of
the constraint matrix
Smoothed analysis of the low-rank approach for smooth semidefinite programs
We consider semidefinite programs (SDPs) of size n with equality constraints.
In order to overcome scalability issues, Burer and Monteiro proposed a
factorized approach based on optimizing over a matrix Y of size by such
that is the SDP variable. The advantages of such formulation are
twofold: the dimension of the optimization variable is reduced and positive
semidefiniteness is naturally enforced. However, the problem in Y is
non-convex. In prior work, it has been shown that, when the constraints on the
factorized variable regularly define a smooth manifold, provided k is large
enough, for almost all cost matrices, all second-order stationary points
(SOSPs) are optimal. Importantly, in practice, one can only compute points
which approximately satisfy necessary optimality conditions, leading to the
question: are such points also approximately optimal? To this end, and under
similar assumptions, we use smoothed analysis to show that approximate SOSPs
for a randomly perturbed objective function are approximate global optima, with
k scaling like the square root of the number of constraints (up to log
factors). Moreover, we bound the optimality gap at the approximate solution of
the perturbed problem with respect to the original problem. We particularize
our results to an SDP relaxation of phase retrieval
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