54 research outputs found
A Non-monotone Alternating Updating Method for A Class of Matrix Factorization Problems
In this paper we consider a general matrix factorization model which covers a
large class of existing models with many applications in areas such as machine
learning and imaging sciences. To solve this possibly nonconvex, nonsmooth and
non-Lipschitz problem, we develop a non-monotone alternating updating method
based on a potential function. Our method essentially updates two blocks of
variables in turn by inexactly minimizing this potential function, and updates
another auxiliary block of variables using an explicit formula. The special
structure of our potential function allows us to take advantage of efficient
computational strategies for non-negative matrix factorization to perform the
alternating minimization over the two blocks of variables. A suitable line
search criterion is also incorporated to improve the numerical performance.
Under some mild conditions, we show that the line search criterion is well
defined, and establish that the sequence generated is bounded and any cluster
point of the sequence is a stationary point. Finally, we conduct some numerical
experiments using real datasets to compare our method with some existing
efficient methods for non-negative matrix factorization and matrix completion.
The numerical results show that our method can outperform these methods for
these specific applications
Low-Rank Tensor Recovery with Euclidean-Norm-Induced Schatten-p Quasi-Norm Regularization
The nuclear norm and Schatten- quasi-norm of a matrix are popular rank
proxies in low-rank matrix recovery. Unfortunately, computing the nuclear norm
or Schatten- quasi-norm of a tensor is NP-hard, which is a pity for low-rank
tensor completion (LRTC) and tensor robust principal component analysis
(TRPCA). In this paper, we propose a new class of rank regularizers based on
the Euclidean norms of the CP component vectors of a tensor and show that these
regularizers are monotonic transformations of tensor Schatten- quasi-norm.
This connection enables us to minimize the Schatten- quasi-norm in LRTC and
TRPCA implicitly. The methods do not use the singular value decomposition and
hence scale to big tensors. Moreover, the methods are not sensitive to the
choice of initial rank and provide an arbitrarily sharper rank proxy for
low-rank tensor recovery compared to nuclear norm. We provide theoretical
guarantees in terms of recovery error for LRTC and TRPCA, which show relatively
smaller of Schatten- quasi-norm leads to tighter error bounds.
Experiments using LRTC and TRPCA on synthetic data and natural images verify
the effectiveness and superiority of our methods compared to baseline methods
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