14,709 research outputs found
Block-Sparse Recovery via Convex Optimization
Given a dictionary that consists of multiple blocks and a signal that lives
in the range space of only a few blocks, we study the problem of finding a
block-sparse representation of the signal, i.e., a representation that uses the
minimum number of blocks. Motivated by signal/image processing and computer
vision applications, such as face recognition, we consider the block-sparse
recovery problem in the case where the number of atoms in each block is
arbitrary, possibly much larger than the dimension of the underlying subspace.
To find a block-sparse representation of a signal, we propose two classes of
non-convex optimization programs, which aim to minimize the number of nonzero
coefficient blocks and the number of nonzero reconstructed vectors from the
blocks, respectively. Since both classes of problems are NP-hard, we propose
convex relaxations and derive conditions under which each class of the convex
programs is equivalent to the original non-convex formulation. Our conditions
depend on the notions of mutual and cumulative subspace coherence of a
dictionary, which are natural generalizations of existing notions of mutual and
cumulative coherence. We evaluate the performance of the proposed convex
programs through simulations as well as real experiments on face recognition.
We show that treating the face recognition problem as a block-sparse recovery
problem improves the state-of-the-art results by 10% with only 25% of the
training data.Comment: IEEE Transactions on Signal Processin
Smoothed Analysis of Tensor Decompositions
Low rank tensor decompositions are a powerful tool for learning generative
models, and uniqueness results give them a significant advantage over matrix
decomposition methods. However, tensors pose significant algorithmic challenges
and tensors analogs of much of the matrix algebra toolkit are unlikely to exist
because of hardness results. Efficient decomposition in the overcomplete case
(where rank exceeds dimension) is particularly challenging. We introduce a
smoothed analysis model for studying these questions and develop an efficient
algorithm for tensor decomposition in the highly overcomplete case (rank
polynomial in the dimension). In this setting, we show that our algorithm is
robust to inverse polynomial error -- a crucial property for applications in
learning since we are only allowed a polynomial number of samples. While
algorithms are known for exact tensor decomposition in some overcomplete
settings, our main contribution is in analyzing their stability in the
framework of smoothed analysis.
Our main technical contribution is to show that tensor products of perturbed
vectors are linearly independent in a robust sense (i.e. the associated matrix
has singular values that are at least an inverse polynomial). This key result
paves the way for applying tensor methods to learning problems in the smoothed
setting. In particular, we use it to obtain results for learning multi-view
models and mixtures of axis-aligned Gaussians where there are many more
"components" than dimensions. The assumption here is that the model is not
adversarially chosen, formalized by a perturbation of model parameters. We
believe this an appealing way to analyze realistic instances of learning
problems, since this framework allows us to overcome many of the usual
limitations of using tensor methods.Comment: 32 pages (including appendix
New Guarantees for Blind Compressed Sensing
Blind Compressed Sensing (BCS) is an extension of Compressed Sensing (CS)
where the optimal sparsifying dictionary is assumed to be unknown and subject
to estimation (in addition to the CS sparse coefficients). Since the emergence
of BCS, dictionary learning, a.k.a. sparse coding, has been studied as a matrix
factorization problem where its sample complexity, uniqueness and
identifiability have been addressed thoroughly. However, in spite of the strong
connections between BCS and sparse coding, recent results from the sparse
coding problem area have not been exploited within the context of BCS. In
particular, prior BCS efforts have focused on learning constrained and complete
dictionaries that limit the scope and utility of these efforts. In this paper,
we develop new theoretical bounds for perfect recovery for the general
unconstrained BCS problem. These unconstrained BCS bounds cover the case of
overcomplete dictionaries, and hence, they go well beyond the existing BCS
theory. Our perfect recovery results integrate the combinatorial theories of
sparse coding with some of the recent results from low-rank matrix recovery. In
particular, we propose an efficient CS measurement scheme that results in
practical recovery bounds for BCS. Moreover, we discuss the performance of BCS
under polynomial-time sparse coding algorithms.Comment: To appear in the 53rd Annual Allerton Conference on Communication,
Control and Computing, University of Illinois at Urbana-Champaign, IL, USA,
201
Robust Low-Rank Subspace Segmentation with Semidefinite Guarantees
Recently there is a line of research work proposing to employ Spectral
Clustering (SC) to segment (group){Throughout the paper, we use segmentation,
clustering, and grouping, and their verb forms, interchangeably.}
high-dimensional structural data such as those (approximately) lying on
subspaces {We follow {liu2010robust} and use the term "subspace" to denote both
linear subspaces and affine subspaces. There is a trivial conversion between
linear subspaces and affine subspaces as mentioned therein.} or low-dimensional
manifolds. By learning the affinity matrix in the form of sparse
reconstruction, techniques proposed in this vein often considerably boost the
performance in subspace settings where traditional SC can fail. Despite the
success, there are fundamental problems that have been left unsolved: the
spectrum property of the learned affinity matrix cannot be gauged in advance,
and there is often one ugly symmetrization step that post-processes the
affinity for SC input. Hence we advocate to enforce the symmetric positive
semidefinite constraint explicitly during learning (Low-Rank Representation
with Positive SemiDefinite constraint, or LRR-PSD), and show that factually it
can be solved in an exquisite scheme efficiently instead of general-purpose SDP
solvers that usually scale up poorly. We provide rigorous mathematical
derivations to show that, in its canonical form, LRR-PSD is equivalent to the
recently proposed Low-Rank Representation (LRR) scheme {liu2010robust}, and
hence offer theoretic and practical insights to both LRR-PSD and LRR, inviting
future research. As per the computational cost, our proposal is at most
comparable to that of LRR, if not less. We validate our theoretic analysis and
optimization scheme by experiments on both synthetic and real data sets.Comment: 10 pages, 4 figures. Accepted by ICDM Workshop on Optimization Based
Methods for Emerging Data Mining Problems (OEDM), 2010. Main proof simplified
and typos corrected. Experimental data slightly adde
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