25 research outputs found
Nonconvex Nonsmooth Low-Rank Minimization via Iteratively Reweighted Nuclear Norm
The nuclear norm is widely used as a convex surrogate of the rank function in
compressive sensing for low rank matrix recovery with its applications in image
recovery and signal processing. However, solving the nuclear norm based relaxed
convex problem usually leads to a suboptimal solution of the original rank
minimization problem. In this paper, we propose to perform a family of
nonconvex surrogates of -norm on the singular values of a matrix to
approximate the rank function. This leads to a nonconvex nonsmooth minimization
problem. Then we propose to solve the problem by Iteratively Reweighted Nuclear
Norm (IRNN) algorithm. IRNN iteratively solves a Weighted Singular Value
Thresholding (WSVT) problem, which has a closed form solution due to the
special properties of the nonconvex surrogate functions. We also extend IRNN to
solve the nonconvex problem with two or more blocks of variables. In theory, we
prove that IRNN decreases the objective function value monotonically, and any
limit point is a stationary point. Extensive experiments on both synthesized
data and real images demonstrate that IRNN enhances the low-rank matrix
recovery compared with state-of-the-art convex algorithms
Nonconvex Sparse Spectral Clustering by Alternating Direction Method of Multipliers and Its Convergence Analysis
Spectral Clustering (SC) is a widely used data clustering method which first
learns a low-dimensional embedding of data by computing the eigenvectors of
the normalized Laplacian matrix, and then performs k-means on to get
the final clustering result. The Sparse Spectral Clustering (SSC) method
extends SC with a sparse regularization on by using the block
diagonal structure prior of in the ideal case. However, encouraging
to be sparse leads to a heavily nonconvex problem which is
challenging to solve and the work (Lu, Yan, and Lin 2016) proposes a convex
relaxation in the pursuit of this aim indirectly. However, the convex
relaxation generally leads to a loose approximation and the quality of the
solution is not clear. This work instead considers to solve the nonconvex
formulation of SSC which directly encourages to be sparse. We propose
an efficient Alternating Direction Method of Multipliers (ADMM) to solve the
nonconvex SSC and provide the convergence guarantee. In particular, we prove
that the sequences generated by ADMM always exist a limit point and any limit
point is a stationary point. Our analysis does not impose any assumptions on
the iterates and thus is practical. Our proposed ADMM for nonconvex problems
allows the stepsize to be increasing but upper bounded, and this makes it very
efficient in practice. Experimental analysis on several real data sets verifies
the effectiveness of our method.Comment: Proceedings of the AAAI Conference on Artificial Intelligence (AAAI).
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