3,881 research outputs found
Stable Principal Component Pursuit
In this paper, we study the problem of recovering a low-rank matrix (the
principal components) from a high-dimensional data matrix despite both small
entry-wise noise and gross sparse errors. Recently, it has been shown that a
convex program, named Principal Component Pursuit (PCP), can recover the
low-rank matrix when the data matrix is corrupted by gross sparse errors. We
further prove that the solution to a related convex program (a relaxed PCP)
gives an estimate of the low-rank matrix that is simultaneously stable to small
entrywise noise and robust to gross sparse errors. More precisely, our result
shows that the proposed convex program recovers the low-rank matrix even though
a positive fraction of its entries are arbitrarily corrupted, with an error
bound proportional to the noise level. We present simulation results to support
our result and demonstrate that the new convex program accurately recovers the
principal components (the low-rank matrix) under quite broad conditions. To our
knowledge, this is the first result that shows the classical Principal
Component Analysis (PCA), optimal for small i.i.d. noise, can be made robust to
gross sparse errors; or the first that shows the newly proposed PCP can be made
stable to small entry-wise perturbations.Comment: 5-page paper submitted to ISIT 201
Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)
The implicit objective of the biennial "international - Traveling Workshop on
Interactions between Sparse models and Technology" (iTWIST) is to foster
collaboration between international scientific teams by disseminating ideas
through both specific oral/poster presentations and free discussions. For its
second edition, the iTWIST workshop took place in the medieval and picturesque
town of Namur in Belgium, from Wednesday August 27th till Friday August 29th,
2014. The workshop was conveniently located in "The Arsenal" building within
walking distance of both hotels and town center. iTWIST'14 has gathered about
70 international participants and has featured 9 invited talks, 10 oral
presentations, and 14 posters on the following themes, all related to the
theory, application and generalization of the "sparsity paradigm":
Sparsity-driven data sensing and processing; Union of low dimensional
subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph
sensing/processing; Blind inverse problems and dictionary learning; Sparsity
and computational neuroscience; Information theory, geometry and randomness;
Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?;
Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website:
http://sites.google.com/site/itwist1
Completing Low-Rank Matrices with Corrupted Samples from Few Coefficients in General Basis
Subspace recovery from corrupted and missing data is crucial for various
applications in signal processing and information theory. To complete missing
values and detect column corruptions, existing robust Matrix Completion (MC)
methods mostly concentrate on recovering a low-rank matrix from few corrupted
coefficients w.r.t. standard basis, which, however, does not apply to more
general basis, e.g., Fourier basis. In this paper, we prove that the range
space of an matrix with rank can be exactly recovered from few
coefficients w.r.t. general basis, though and the number of corrupted
samples are both as high as . Our model covers
previous ones as special cases, and robust MC can recover the intrinsic matrix
with a higher rank. Moreover, we suggest a universal choice of the
regularization parameter, which is . By our
filtering algorithm, which has theoretical guarantees, we can
further reduce the computational cost of our model. As an application, we also
find that the solutions to extended robust Low-Rank Representation and to our
extended robust MC are mutually expressible, so both our theory and algorithm
can be applied to the subspace clustering problem with missing values under
certain conditions. Experiments verify our theories.Comment: To appear in IEEE Transactions on Information Theor
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