1,462 research outputs found
Low-Rank Inducing Norms with Optimality Interpretations
Optimization problems with rank constraints appear in many diverse fields
such as control, machine learning and image analysis. Since the rank constraint
is non-convex, these problems are often approximately solved via convex
relaxations. Nuclear norm regularization is the prevailing convexifying
technique for dealing with these types of problem. This paper introduces a
family of low-rank inducing norms and regularizers which includes the nuclear
norm as a special case. A posteriori guarantees on solving an underlying rank
constrained optimization problem with these convex relaxations are provided. We
evaluate the performance of the low-rank inducing norms on three matrix
completion problems. In all examples, the nuclear norm heuristic is
outperformed by convex relaxations based on other low-rank inducing norms. For
two of the problems there exist low-rank inducing norms that succeed in
recovering the partially unknown matrix, while the nuclear norm fails. These
low-rank inducing norms are shown to be representable as semi-definite
programs. Moreover, these norms have cheaply computable proximal mappings,
which makes it possible to also solve problems of large size using first-order
methods
PhasePack: A Phase Retrieval Library
Phase retrieval deals with the estimation of complex-valued signals solely
from the magnitudes of linear measurements. While there has been a recent
explosion in the development of phase retrieval algorithms, the lack of a
common interface has made it difficult to compare new methods against the
state-of-the-art. The purpose of PhasePack is to create a common software
interface for a wide range of phase retrieval algorithms and to provide a
common testbed using both synthetic data and empirical imaging datasets.
PhasePack is able to benchmark a large number of recent phase retrieval methods
against one another to generate comparisons using a range of different
performance metrics. The software package handles single method testing as well
as multiple method comparisons.
The algorithm implementations in PhasePack differ slightly from their
original descriptions in the literature in order to achieve faster speed and
improved robustness. In particular, PhasePack uses adaptive stepsizes,
line-search methods, and fast eigensolvers to speed up and automate
convergence
Robust Low-Rank Subspace Segmentation with Semidefinite Guarantees
Recently there is a line of research work proposing to employ Spectral
Clustering (SC) to segment (group){Throughout the paper, we use segmentation,
clustering, and grouping, and their verb forms, interchangeably.}
high-dimensional structural data such as those (approximately) lying on
subspaces {We follow {liu2010robust} and use the term "subspace" to denote both
linear subspaces and affine subspaces. There is a trivial conversion between
linear subspaces and affine subspaces as mentioned therein.} or low-dimensional
manifolds. By learning the affinity matrix in the form of sparse
reconstruction, techniques proposed in this vein often considerably boost the
performance in subspace settings where traditional SC can fail. Despite the
success, there are fundamental problems that have been left unsolved: the
spectrum property of the learned affinity matrix cannot be gauged in advance,
and there is often one ugly symmetrization step that post-processes the
affinity for SC input. Hence we advocate to enforce the symmetric positive
semidefinite constraint explicitly during learning (Low-Rank Representation
with Positive SemiDefinite constraint, or LRR-PSD), and show that factually it
can be solved in an exquisite scheme efficiently instead of general-purpose SDP
solvers that usually scale up poorly. We provide rigorous mathematical
derivations to show that, in its canonical form, LRR-PSD is equivalent to the
recently proposed Low-Rank Representation (LRR) scheme {liu2010robust}, and
hence offer theoretic and practical insights to both LRR-PSD and LRR, inviting
future research. As per the computational cost, our proposal is at most
comparable to that of LRR, if not less. We validate our theoretic analysis and
optimization scheme by experiments on both synthetic and real data sets.Comment: 10 pages, 4 figures. Accepted by ICDM Workshop on Optimization Based
Methods for Emerging Data Mining Problems (OEDM), 2010. Main proof simplified
and typos corrected. Experimental data slightly adde
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