4 research outputs found
Inertial Stochastic PALM (iSPALM) and Applications in Machine Learning
Inertial algorithms for minimizing nonsmooth and nonconvex functions as the
inertial proximal alternating linearized minimization algorithm (iPALM) have
demonstrated their superiority with respect to computation time over their non
inertial variants. In many problems in imaging and machine learning, the
objective functions have a special form involving huge data which encourage the
application of stochastic algorithms. While algorithms based on stochastic
gradient descent are still used in the majority of applications, recently also
stochastic algorithms for minimizing nonsmooth and nonconvex functions were
proposed. In this paper, we derive an inertial variant of a stochastic PALM
algorithm with variance-reduced gradient estimator, called iSPALM, and prove
linear convergence of the algorithm under certain assumptions. Our inertial
approach can be seen as generalization of momentum methods widely used to speed
up and stabilize optimization algorithms, in particular in machine learning, to
nonsmooth problems. Numerical experiments for learning the weights of a
so-called proximal neural network and the parameters of Student-t mixture
models show that our new algorithm outperforms both stochastic PALM and its
deterministic counterparts
Alternatives to the EM Algorithm for ML-Estimation of Location, Scatter Matrix and Degree of Freedom of the Student- Distribution
In this paper, we consider maximum likelihood estimations of the degree of
freedom parameter , the location parameter and the scatter matrix
of the multivariate Student- distribution. In particular, we are
interested in estimating the degree of freedom parameter that determines
the tails of the corresponding probability density function and was rarely
considered in detail in the literature so far. We prove that under certain
assumptions a minimizer of the negative log-likelihood function exists, where
we have to take special care of the case , for which
the Student- distribution approaches the Gaussian distribution. As
alternatives to the classical EM algorithm we propose three other algorithms
which cannot be interpreted as EM algorithm. For fixed , the first
algorithm is an accelerated EM algorithm known from the literature. However,
since we do not fix , we cannot apply standard convergence results for the
EM algorithm. The other two algorithms differ from this algorithm in the
iteration step for . We show how the objective function behaves for the
different updates of and prove for all three algorithms that it decreases
in each iteration step. We compare the algorithms as well as some accelerated
versions by numerical simulation and apply one of them for estimating the
degree of freedom parameter in images corrupted by Student- noise
A Convex Constraint Variational Method for Restoring Blurred Images in the Presence of Alpha-Stable Noises
Blurred image restoration poses a great challenge under the non-Gaussian noise environments in various communication systems. In order to restore images from blur and alpha-stable noise while also preserving their edges, this paper proposes a variational method to restore the blurred images with alpha-stable noises based on the property of the meridian distribution and the total variation (TV). Since the variational model is non-convex, it cannot guarantee a global optimal solution. To overcome this drawback, we also incorporate an additional penalty term into the deblurring and denoising model and propose a strictly convex variational method. Due to the convexity of our model, the primal-dual algorithm is adopted to solve this convex variational problem. Our simulation results validate the proposed method