253 research outputs found
Non-smooth Non-convex Bregman Minimization: Unification and new Algorithms
We propose a unifying algorithm for non-smooth non-convex optimization. The
algorithm approximates the objective function by a convex model function and
finds an approximate (Bregman) proximal point of the convex model. This
approximate minimizer of the model function yields a descent direction, along
which the next iterate is found. Complemented with an Armijo-like line search
strategy, we obtain a flexible algorithm for which we prove (subsequential)
convergence to a stationary point under weak assumptions on the growth of the
model function error. Special instances of the algorithm with a Euclidean
distance function are, for example, Gradient Descent, Forward--Backward
Splitting, ProxDescent, without the common requirement of a "Lipschitz
continuous gradient". In addition, we consider a broad class of Bregman
distance functions (generated by Legendre functions) replacing the Euclidean
distance. The algorithm has a wide range of applications including many linear
and non-linear inverse problems in signal/image processing and machine
learning
A Splitting Augmented Lagrangian Method for Low Multilinear-Rank Tensor Recovery
This paper studies a recovery task of finding a low multilinear-rank tensor
that fulfills some linear constraints in the general settings, which has many
applications in computer vision and graphics. This problem is named as the low
multilinear-rank tensor recovery problem. The variable splitting technique and
convex relaxation technique are used to transform this problem into a tractable
constrained optimization problem. Considering the favorable structure of the
problem, we develop a splitting augmented Lagrangian method to solve the
resulting problem. The proposed algorithm is easily implemented and its
convergence can be proved under some conditions. Some preliminary numerical
results on randomly generated and real completion problems show that the
proposed algorithm is very effective and robust for tackling the low
multilinear-rank tensor completion problem
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