233 research outputs found
Polar Polytopes and Recovery of Sparse Representations
Suppose we have a signal y which we wish to represent using a linear
combination of a number of basis atoms a_i, y=sum_i x_i a_i = Ax. The problem
of finding the minimum L0 norm representation for y is a hard problem. The
Basis Pursuit (BP) approach proposes to find the minimum L1 norm representation
instead, which corresponds to a linear program (LP) that can be solved using
modern LP techniques, and several recent authors have given conditions for the
BP (minimum L1 norm) and sparse (minimum L0 solutions) representations to be
identical. In this paper, we explore this sparse representation problem} using
the geometry of convex polytopes, as recently introduced into the field by
Donoho. By considering the dual LP we find that the so-called polar polytope P
of the centrally-symmetric polytope P whose vertices are the atom pairs +-a_i
is particularly helpful in providing us with geometrical insight into
optimality conditions given by Fuchs and Tropp for non-unit-norm atom sets. In
exploring this geometry we are able to tighten some of these earlier results,
showing for example that the Fuchs condition is both necessary and sufficient
for L1-unique-optimality, and that there are situations where Orthogonal
Matching Pursuit (OMP) can eventually find all L1-unique-optimal solutions with
m nonzeros even if ERC fails for m, if allowed to run for more than m steps
Group polytope faces pursuit for recovery of block-sparse signals
This is the accepted version of the article. The final publication is available at link.springer.com. http://www.springerlink.com/content/e0r61416446277w0
The Geometry of Uniqueness, Sparsity and Clustering in Penalized Estimation
We provide a necessary and sufficient condition for the uniqueness of
penalized least-squares estimators whose penalty term is given by a norm with a
polytope unit ball, covering a wide range of methods including SLOPE and LASSO,
as well as the related method of basis pursuit. We consider a strong type of
uniqueness that is relevant for statistical problems. The uniqueness condition
is geometric and involves how the row span of the design matrix intersects the
faces of the dual norm unit ball, which for SLOPE is given by the sign
permutahedron. Further considerations based this condition also allow to derive
results on sparsity and clustering features. In particular, we define the
notion of a SLOPE model to describe both sparsity and clustering properties of
this method and also provide a geometric characterization of accessible SLOPE
models.Comment: new title, minor change
On Modified l_1-Minimization Problems in Compressed Sensing
Sparse signal modeling has received much attention recently because of its application in medical imaging, group testing and radar technology, among others. Compressed sensing, a recently coined term, has showed us, both in theory and practice, that various
signals of interest which are sparse or approximately sparse can be efficiently recovered by using far fewer samples than suggested by Shannon sampling theorem.
Sparsity is the only prior information about an unknown signal assumed in traditional compressed sensing techniques. But in many applications, other kinds of prior information are also available, such as partial knowledge of the support, tree structure
of signal and clustering of large coefficients around a small set of coefficients.
In this thesis, we consider compressed sensing problems with prior information on the support of the signal, together with sparsity. We modify regular l_1 -minimization problems considered in compressed sensing, using this extra information. We call these
modified l_1 -minimization problems.
We show that partial knowledge of the support helps us to weaken sufficient conditions for the recovery of sparse signals using modified ` 1 minimization problems. In case
of deterministic compressed sensing, we show that a sharp condition for sparse recovery
can be improved using modified ` 1 minimization problems. We also derive algebraic necessary and sufficient condition for modified basis pursuit problem and use an open source algorithm known as l_1 -homotopy algorithm to perform some numerical experiments and compare the performance of modified Basis Pursuit Denoising with the regular Basis Pursuit Denoising
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