1,346 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
Similarity Learning via Kernel Preserving Embedding
Data similarity is a key concept in many data-driven applications. Many
algorithms are sensitive to similarity measures. To tackle this fundamental
problem, automatically learning of similarity information from data via
self-expression has been developed and successfully applied in various models,
such as low-rank representation, sparse subspace learning, semi-supervised
learning. However, it just tries to reconstruct the original data and some
valuable information, e.g., the manifold structure, is largely ignored. In this
paper, we argue that it is beneficial to preserve the overall relations when we
extract similarity information. Specifically, we propose a novel similarity
learning framework by minimizing the reconstruction error of kernel matrices,
rather than the reconstruction error of original data adopted by existing work.
Taking the clustering task as an example to evaluate our method, we observe
considerable improvements compared to other state-of-the-art methods. More
importantly, our proposed framework is very general and provides a novel and
fundamental building block for many other similarity-based tasks. Besides, our
proposed kernel preserving opens up a large number of possibilities to embed
high-dimensional data into low-dimensional space.Comment: Published in AAAI 201
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
Joint Tensor Factorization and Outlying Slab Suppression with Applications
We consider factoring low-rank tensors in the presence of outlying slabs.
This problem is important in practice, because data collected in many
real-world applications, such as speech, fluorescence, and some social network
data, fit this paradigm. Prior work tackles this problem by iteratively
selecting a fixed number of slabs and fitting, a procedure which may not
converge. We formulate this problem from a group-sparsity promoting point of
view, and propose an alternating optimization framework to handle the
corresponding () minimization-based low-rank tensor
factorization problem. The proposed algorithm features a similar per-iteration
complexity as the plain trilinear alternating least squares (TALS) algorithm.
Convergence of the proposed algorithm is also easy to analyze under the
framework of alternating optimization and its variants. In addition,
regularization and constraints can be easily incorporated to make use of
\emph{a priori} information on the latent loading factors. Simulations and real
data experiments on blind speech separation, fluorescence data analysis, and
social network mining are used to showcase the effectiveness of the proposed
algorithm
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