713 research outputs found
Augmented L1 and Nuclear-Norm Models with a Globally Linearly Convergent Algorithm
This paper studies the long-existing idea of adding a nice smooth function to
"smooth" a non-differentiable objective function in the context of sparse
optimization, in particular, the minimization of
, where is a vector, as well as the
minimization of , where is a matrix and
and are the nuclear and Frobenius norms of ,
respectively. We show that they can efficiently recover sparse vectors and
low-rank matrices. In particular, they enjoy exact and stable recovery
guarantees similar to those known for minimizing and under
the conditions on the sensing operator such as its null-space property,
restricted isometry property, spherical section property, or RIPless property.
To recover a (nearly) sparse vector , minimizing
returns (nearly) the same solution as minimizing
almost whenever . The same relation also
holds between minimizing and minimizing
for recovering a (nearly) low-rank matrix , if . Furthermore, we show that the linearized Bregman algorithm for
minimizing subject to enjoys global
linear convergence as long as a nonzero solution exists, and we give an
explicit rate of convergence. The convergence property does not require a
solution solution or any properties on . To our knowledge, this is the best
known global convergence result for first-order sparse optimization algorithms.Comment: arXiv admin note: text overlap with arXiv:1207.5326 by other author
Relaxed Majorization-Minimization for Non-smooth and Non-convex Optimization
We propose a new majorization-minimization (MM) method for non-smooth and
non-convex programs, which is general enough to include the existing MM
methods. Besides the local majorization condition, we only require that the
difference between the directional derivatives of the objective function and
its surrogate function vanishes when the number of iterations approaches
infinity, which is a very weak condition. So our method can use a surrogate
function that directly approximates the non-smooth objective function. In
comparison, all the existing MM methods construct the surrogate function by
approximating the smooth component of the objective function. We apply our
relaxed MM methods to the robust matrix factorization (RMF) problem with
different regularizations, where our locally majorant algorithm shows
advantages over the state-of-the-art approaches for RMF. This is the first
algorithm for RMF ensuring, without extra assumptions, that any limit point of
the iterates is a stationary point.Comment: AAAI1
Augmented ℓ1 and nuclear-norm models with a globally linearly convergent algorithm
Abstract. This paper studies the long-existing idea of adding a nice smooth function to “smooth ” a nondifferentiable objective function in the context of sparse optimization, in particular, the minimization of ‖x‖1 + 1 2α ‖x‖22,wherexis a vector, as well as the minimization of ‖X‖ ∗ + 1 2α ‖X‖2F,whereX is a matrix and ‖X‖ ∗ and ‖X‖F are the nuclear and Frobenius norms of X, respectively. We show that they let sparse vectors and low-rank matrices be efficiently recovered. In particular, they enjoy exact and stable recovery guarantees similar to those known for the minimization of ‖x‖1 and ‖X‖∗ under the conditions on the sensing operator such as its null-space property, restricted isometry property (RIP), spherical section property, or “RIPless ” property. To recover a (nearly) sparse vector x 0, minimizing ‖x‖1 + 1 2α ‖x‖22 returns (nearly) the same solution as minimizing ‖x‖1 whenever α ≥ 10‖x 0 ‖∞. The same relation also holds between minimizing ‖X‖ ∗ + 1 2α ‖X‖2F and minimizing ‖X‖ ∗ for recovering a (nearly) low-rank matrix X 0 if α ≥ 10‖X 0 ‖2. Furthermore, we show that the linearized Bregman algorithm, as well as its two fast variants, for minimizing ‖x‖1 + 1 2α ‖x‖2 2 subject to Ax = b enjoys global linear convergence as long as a nonzero solution exists, and we give an explicit rate of convergence. The convergence property does not require a sparse solution or any properties on A. To the best of our knowledge, this is the best known global convergence result for first-order sparse optimization algorithms
Successive Convex Approximation Algorithms for Sparse Signal Estimation with Nonconvex Regularizations
In this paper, we propose a successive convex approximation framework for
sparse optimization where the nonsmooth regularization function in the
objective function is nonconvex and it can be written as the difference of two
convex functions. The proposed framework is based on a nontrivial combination
of the majorization-minimization framework and the successive convex
approximation framework proposed in literature for a convex regularization
function. The proposed framework has several attractive features, namely, i)
flexibility, as different choices of the approximate function lead to different
type of algorithms; ii) fast convergence, as the problem structure can be
better exploited by a proper choice of the approximate function and the
stepsize is calculated by the line search; iii) low complexity, as the
approximate function is convex and the line search scheme is carried out over a
differentiable function; iv) guaranteed convergence to a stationary point. We
demonstrate these features by two example applications in subspace learning,
namely, the network anomaly detection problem and the sparse subspace
clustering problem. Customizing the proposed framework by adopting the
best-response type approximation, we obtain soft-thresholding with exact line
search algorithms for which all elements of the unknown parameter are updated
in parallel according to closed-form expressions. The attractive features of
the proposed algorithms are illustrated numerically.Comment: submitted to IEEE Journal of Selected Topics in Signal Processing,
special issue in Robust Subspace Learnin
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