1,190 research outputs found
Compressive PCA for Low-Rank Matrices on Graphs
We introduce a novel framework for an approxi- mate recovery of data matrices
which are low-rank on graphs, from sampled measurements. The rows and columns
of such matrices belong to the span of the first few eigenvectors of the graphs
constructed between their rows and columns. We leverage this property to
recover the non-linear low-rank structures efficiently from sampled data
measurements, with a low cost (linear in n). First, a Resrtricted Isometry
Property (RIP) condition is introduced for efficient uniform sampling of the
rows and columns of such matrices based on the cumulative coherence of graph
eigenvectors. Secondly, a state-of-the-art fast low-rank recovery method is
suggested for the sampled data. Finally, several efficient, parallel and
parameter-free decoders are presented along with their theoretical analysis for
decoding the low-rank and cluster indicators for the full data matrix. Thus, we
overcome the computational limitations of the standard linear low-rank recovery
methods for big datasets. Our method can also be seen as a major step towards
efficient recovery of non- linear low-rank structures. For a matrix of size n X
p, on a single core machine, our method gains a speed up of over Robust
Principal Component Analysis (RPCA), where k << p is the subspace dimension.
Numerically, we can recover a low-rank matrix of size 10304 X 1000, 100 times
faster than Robust PCA
Robust PCA as Bilinear Decomposition with Outlier-Sparsity Regularization
Principal component analysis (PCA) is widely used for dimensionality
reduction, with well-documented merits in various applications involving
high-dimensional data, including computer vision, preference measurement, and
bioinformatics. In this context, the fresh look advocated here permeates
benefits from variable selection and compressive sampling, to robustify PCA
against outliers. A least-trimmed squares estimator of a low-rank bilinear
factor analysis model is shown closely related to that obtained from an
-(pseudo)norm-regularized criterion encouraging sparsity in a matrix
explicitly modeling the outliers. This connection suggests robust PCA schemes
based on convex relaxation, which lead naturally to a family of robust
estimators encompassing Huber's optimal M-class as a special case. Outliers are
identified by tuning a regularization parameter, which amounts to controlling
sparsity of the outlier matrix along the whole robustification path of (group)
least-absolute shrinkage and selection operator (Lasso) solutions. Beyond its
neat ties to robust statistics, the developed outlier-aware PCA framework is
versatile to accommodate novel and scalable algorithms to: i) track the
low-rank signal subspace robustly, as new data are acquired in real time; and
ii) determine principal components robustly in (possibly) infinite-dimensional
feature spaces. Synthetic and real data tests corroborate the effectiveness of
the proposed robust PCA schemes, when used to identify aberrant responses in
personality assessment surveys, as well as unveil communities in social
networks, and intruders from video surveillance data.Comment: 30 pages, submitted to IEEE Transactions on Signal Processin
Subspace Methods for Joint Sparse Recovery
We propose robust and efficient algorithms for the joint sparse recovery
problem in compressed sensing, which simultaneously recover the supports of
jointly sparse signals from their multiple measurement vectors obtained through
a common sensing matrix. In a favorable situation, the unknown matrix, which
consists of the jointly sparse signals, has linearly independent nonzero rows.
In this case, the MUSIC (MUltiple SIgnal Classification) algorithm, originally
proposed by Schmidt for the direction of arrival problem in sensor array
processing and later proposed and analyzed for joint sparse recovery by Feng
and Bresler, provides a guarantee with the minimum number of measurements. We
focus instead on the unfavorable but practically significant case of
rank-defect or ill-conditioning. This situation arises with limited number of
measurement vectors, or with highly correlated signal components. In this case
MUSIC fails, and in practice none of the existing methods can consistently
approach the fundamental limit. We propose subspace-augmented MUSIC (SA-MUSIC),
which improves on MUSIC so that the support is reliably recovered under such
unfavorable conditions. Combined with subspace-based greedy algorithms also
proposed and analyzed in this paper, SA-MUSIC provides a computationally
efficient algorithm with a performance guarantee. The performance guarantees
are given in terms of a version of restricted isometry property. In particular,
we also present a non-asymptotic perturbation analysis of the signal subspace
estimation that has been missing in the previous study of MUSIC.Comment: submitted to IEEE transactions on Information Theory, revised versio
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