7,802 research outputs found
The MM Alternative to EM
The EM algorithm is a special case of a more general algorithm called the MM
algorithm. Specific MM algorithms often have nothing to do with missing data.
The first M step of an MM algorithm creates a surrogate function that is
optimized in the second M step. In minimization, MM stands for
majorize--minimize; in maximization, it stands for minorize--maximize. This
two-step process always drives the objective function in the right direction.
Construction of MM algorithms relies on recognizing and manipulating
inequalities rather than calculating conditional expectations. This survey
walks the reader through the construction of several specific MM algorithms.
The potential of the MM algorithm in solving high-dimensional optimization and
estimation problems is its most attractive feature. Our applications to random
graph models, discriminant analysis and image restoration showcase this
ability.Comment: Published in at http://dx.doi.org/10.1214/08-STS264 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Stochastic Majorize-Minimize Subspace Algorithm for Online Penalized Least Squares Estimation
Stochastic approximation techniques play an important role in solving many
problems encountered in machine learning or adaptive signal processing. In
these contexts, the statistics of the data are often unknown a priori or their
direct computation is too intensive, and they have thus to be estimated online
from the observed signals. For batch optimization of an objective function
being the sum of a data fidelity term and a penalization (e.g. a sparsity
promoting function), Majorize-Minimize (MM) methods have recently attracted
much interest since they are fast, highly flexible, and effective in ensuring
convergence. The goal of this paper is to show how these methods can be
successfully extended to the case when the data fidelity term corresponds to a
least squares criterion and the cost function is replaced by a sequence of
stochastic approximations of it. In this context, we propose an online version
of an MM subspace algorithm and we study its convergence by using suitable
probabilistic tools. Simulation results illustrate the good practical
performance of the proposed algorithm associated with a memory gradient
subspace, when applied to both non-adaptive and adaptive filter identification
problems
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