321 research outputs found
Online Matrix Completion Through Nuclear Norm Regularisation
It is the main goal of this paper to propose a novel method to perform matrix
completion on-line. Motivated by a wide variety of applications, ranging from
the design of recommender systems to sensor network localization through
seismic data reconstruction, we consider the matrix completion problem when
entries of the matrix of interest are observed gradually. Precisely, we place
ourselves in the situation where the predictive rule should be refined
incrementally, rather than recomputed from scratch each time the sample of
observed entries increases. The extension of existing matrix completion methods
to the sequential prediction context is indeed a major issue in the Big Data
era, and yet little addressed in the literature. The algorithm promoted in this
article builds upon the Soft Impute approach introduced in Mazumder et al.
(2010). The major novelty essentially arises from the use of a randomised
technique for both computing and updating the Singular Value Decomposition
(SVD) involved in the algorithm. Though of disarming simplicity, the method
proposed turns out to be very efficient, while requiring reduced computations.
Several numerical experiments based on real datasets illustrating its
performance are displayed, together with preliminary results giving it a
theoretical basis.Comment: Corrected a typo in the affiliatio
Fast Low-Rank Matrix Learning with Nonconvex Regularization
Low-rank modeling has a lot of important applications in machine learning,
computer vision and social network analysis. While the matrix rank is often
approximated by the convex nuclear norm, the use of nonconvex low-rank
regularizers has demonstrated better recovery performance. However, the
resultant optimization problem is much more challenging. A very recent
state-of-the-art is based on the proximal gradient algorithm. However, it
requires an expensive full SVD in each proximal step. In this paper, we show
that for many commonly-used nonconvex low-rank regularizers, a cutoff can be
derived to automatically threshold the singular values obtained from the
proximal operator. This allows the use of power method to approximate the SVD
efficiently. Besides, the proximal operator can be reduced to that of a much
smaller matrix projected onto this leading subspace. Convergence, with a rate
of O(1/T) where T is the number of iterations, can be guaranteed. Extensive
experiments are performed on matrix completion and robust principal component
analysis. The proposed method achieves significant speedup over the
state-of-the-art. Moreover, the matrix solution obtained is more accurate and
has a lower rank than that of the traditional nuclear norm regularizer.Comment: Long version of conference paper appeared ICDM 201
PEAR: PEriodic And fixed Rank separation for fast fMRI
In functional MRI (fMRI), faster acquisition via undersampling of data can
improve the spatial-temporal resolution trade-off and increase statistical
robustness through increased degrees-of-freedom. High quality reconstruction of
fMRI data from undersampled measurements requires proper modeling of the data.
We present an fMRI reconstruction approach based on modeling the fMRI signal as
a sum of periodic and fixed rank components, for improved reconstruction from
undersampled measurements. We decompose the fMRI signal into a component which
a has fixed rank and a component consisting of a sum of periodic signals which
is sparse in the temporal Fourier domain. Data reconstruction is performed by
solving a constrained problem that enforces a fixed, moderate rank on one of
the components, and a limited number of temporal frequencies on the other. Our
approach is coined PEAR - PEriodic And fixed Rank separation for fast fMRI.
Experimental results include purely synthetic simulation, a simulation with
real timecourses and retrospective undersampling of a real fMRI dataset.
Evaluation was performed both quantitatively and visually versus ground truth,
comparing PEAR to two additional recent methods for fMRI reconstruction from
undersampled measurements. Results demonstrate PEAR's improvement in estimating
the timecourses and activation maps versus the methods compared against at
acceleration ratios of R=8,16 (for simulated data) and R=6.66,10 (for real
data). PEAR results in reconstruction with higher fidelity than when using a
fixed-rank based model or a conventional Low-rank+Sparse algorithm. We have
shown that splitting the functional information between the components leads to
better modeling of fMRI, over state-of-the-art methods
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