549 research outputs found
Classification of unions of subspaces with sparse representations
We propose a preliminary investigation on the benefits and limitations of classifiers based on sparse representations. We specifically focus on the union of subspaces data model and examine binary classifiers built on a sparse non linear mapping (in a redundant dictionary) followed by a linear classifier. We study two common sparse non linear mappings (namely \ell_0 and \ell_1) and show that, in both cases, there exists a finite dictionary such that the classifier discriminates the two classes correctly. This result paves the way towards a better understanding of the increasingly popular classifiers based on sparse representatio
Multiple pattern classification by sparse subspace decomposition
A robust classification method is developed on the basis of sparse subspace
decomposition. This method tries to decompose a mixture of subspaces of
unlabeled data (queries) into class subspaces as few as possible. Each query is
classified into the class whose subspace significantly contributes to the
decomposed subspace. Multiple queries from different classes can be
simultaneously classified into their respective classes. A practical greedy
algorithm of the sparse subspace decomposition is designed for the
classification. The present method achieves high recognition rate and robust
performance exploiting joint sparsity.Comment: 8 pages, 3 figures, 2nd IEEE International Workshop on Subspace
Methods, Workshop Proceedings of ICCV 200
Sparse Recovery from Combined Fusion Frame Measurements
Sparse representations have emerged as a powerful tool in signal and
information processing, culminated by the success of new acquisition and
processing techniques such as Compressed Sensing (CS). Fusion frames are very
rich new signal representation methods that use collections of subspaces
instead of vectors to represent signals. This work combines these exciting
fields to introduce a new sparsity model for fusion frames. Signals that are
sparse under the new model can be compressively sampled and uniquely
reconstructed in ways similar to sparse signals using standard CS. The
combination provides a promising new set of mathematical tools and signal
models useful in a variety of applications. With the new model, a sparse signal
has energy in very few of the subspaces of the fusion frame, although it does
not need to be sparse within each of the subspaces it occupies. This sparsity
model is captured using a mixed l1/l2 norm for fusion frames.
A signal sparse in a fusion frame can be sampled using very few random
projections and exactly reconstructed using a convex optimization that
minimizes this mixed l1/l2 norm. The provided sampling conditions generalize
coherence and RIP conditions used in standard CS theory. It is demonstrated
that they are sufficient to guarantee sparse recovery of any signal sparse in
our model. Moreover, a probabilistic analysis is provided using a stochastic
model on the sparse signal that shows that under very mild conditions the
probability of recovery failure decays exponentially with increasing dimension
of the subspaces
Sparse Subspace Clustering: Algorithm, Theory, and Applications
In many real-world problems, we are dealing with collections of
high-dimensional data, such as images, videos, text and web documents, DNA
microarray data, and more. Often, high-dimensional data lie close to
low-dimensional structures corresponding to several classes or categories the
data belongs to. In this paper, we propose and study an algorithm, called
Sparse Subspace Clustering (SSC), to cluster data points that lie in a union of
low-dimensional subspaces. The key idea is that, among infinitely many possible
representations of a data point in terms of other points, a sparse
representation corresponds to selecting a few points from the same subspace.
This motivates solving a sparse optimization program whose solution is used in
a spectral clustering framework to infer the clustering of data into subspaces.
Since solving the sparse optimization program is in general NP-hard, we
consider a convex relaxation and show that, under appropriate conditions on the
arrangement of subspaces and the distribution of data, the proposed
minimization program succeeds in recovering the desired sparse representations.
The proposed algorithm can be solved efficiently and can handle data points
near the intersections of subspaces. Another key advantage of the proposed
algorithm with respect to the state of the art is that it can deal with data
nuisances, such as noise, sparse outlying entries, and missing entries,
directly by incorporating the model of the data into the sparse optimization
program. We demonstrate the effectiveness of the proposed algorithm through
experiments on synthetic data as well as the two real-world problems of motion
segmentation and face clustering
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