4,533 research outputs found
Exponentially Convergent Algorithms for Supervised Matrix Factorization
Supervised matrix factorization (SMF) is a classical machine learning method
that simultaneously seeks feature extraction and classification tasks, which
are not necessarily a priori aligned objectives. Our goal is to use SMF to
learn low-rank latent factors that offer interpretable, data-reconstructive,
and class-discriminative features, addressing challenges posed by
high-dimensional data. Training SMF model involves solving a nonconvex and
possibly constrained optimization with at least three blocks of parameters.
Known algorithms are either heuristic or provide weak convergence guarantees
for special cases. In this paper, we provide a novel framework that 'lifts' SMF
as a low-rank matrix estimation problem in a combined factor space and propose
an efficient algorithm that provably converges exponentially fast to a global
minimizer of the objective with arbitrary initialization under mild
assumptions. Our framework applies to a wide range of SMF-type problems for
multi-class classification with auxiliary features. To showcase an application,
we demonstrate that our algorithm successfully identified well-known
cancer-associated gene groups for various cancers.Comment: 33 pages, 3 figures. arXiv admin note: substantial text overlap with
arXiv:2206.0677
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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