3,856 research outputs found
Convex Optimization Approaches for Blind Sensor Calibration using Sparsity
We investigate a compressive sensing framework in which the sensors introduce
a distortion to the measurements in the form of unknown gains. We focus on
blind calibration, using measures performed on multiple unknown (but sparse)
signals and formulate the joint recovery of the gains and the sparse signals as
a convex optimization problem. We divide this problem in 3 subproblems with
different conditions on the gains, specifially (i) gains with different
amplitude and the same phase, (ii) gains with the same amplitude and different
phase and (iii) gains with different amplitude and phase. In order to solve the
first case, we propose an extension to the basis pursuit optimization which can
estimate the unknown gains along with the unknown sparse signals. For the
second case, we formulate a quadratic approach that eliminates the unknown
phase shifts and retrieves the unknown sparse signals. An alternative form of
this approach is also formulated to reduce complexity and memory requirements
and provide scalability with respect to the number of input signals. Finally
for the third case, we propose a formulation that combines the earlier two
approaches to solve the problem. The performance of the proposed algorithms is
investigated extensively through numerical simulations, which demonstrates that
simultaneous signal recovery and calibration is possible with convex methods
when sufficiently many (unknown, but sparse) calibrating signals are provided
TV-min and Greedy Pursuit for Constrained Joint Sparsity and Application to Inverse Scattering
This paper proposes a general framework for compressed sensing of constrained
joint sparsity (CJS) which includes total variation minimization (TV-min) as an
example. TV- and 2-norm error bounds, independent of the ambient dimension, are
derived for the CJS version of Basis Pursuit and Orthogonal Matching Pursuit.
As an application the results extend Cand`es, Romberg and Tao's proof of exact
recovery of piecewise constant objects with noiseless incomplete Fourier data
to the case of noisy data.Comment: Mathematics and Mechanics of Complex Systems (2013
Hybrid approximate message passing
Gaussian and quadratic approximations of message passing algorithms on graphs have attracted considerable recent attention due to their computational simplicity, analytic tractability, and wide applicability in optimization and statistical inference problems. This paper presents a systematic framework for incorporating such approximate message passing (AMP) methods in general graphical models. The key concept is a partition of dependencies of a general graphical model into strong and weak edges, with the weak edges representing interactions through aggregates of small, linearizable couplings of variables. AMP approximations based on the Central Limit Theorem can be readily applied to aggregates of many weak edges and integrated with standard message passing updates on the strong edges. The resulting algorithm, which we call hybrid generalized approximate message passing (HyGAMP), can yield significantly simpler implementations of sum-product and max-sum loopy belief propagation. By varying the partition of strong and weak edges, a performance--complexity trade-off can be achieved. Group sparsity and multinomial logistic regression problems are studied as examples of the proposed methodology.The work of S. Rangan was supported in part by the National Science Foundation under Grants 1116589, 1302336, and 1547332, and in part by the industrial affiliates of NYU WIRELESS. The work of A. K. Fletcher was supported in part by the National Science Foundation under Grants 1254204 and 1738286 and in part by the Office of Naval Research under Grant N00014-15-1-2677. The work of V. K. Goyal was supported in part by the National Science Foundation under Grant 1422034. The work of E. Byrne and P. Schniter was supported in part by the National Science Foundation under Grant CCF-1527162. (1116589 - National Science Foundation; 1302336 - National Science Foundation; 1547332 - National Science Foundation; 1254204 - National Science Foundation; 1738286 - National Science Foundation; 1422034 - National Science Foundation; CCF-1527162 - National Science Foundation; NYU WIRELESS; N00014-15-1-2677 - Office of Naval Research
Constrained Overcomplete Analysis Operator Learning for Cosparse Signal Modelling
We consider the problem of learning a low-dimensional signal model from a
collection of training samples. The mainstream approach would be to learn an
overcomplete dictionary to provide good approximations of the training samples
using sparse synthesis coefficients. This famous sparse model has a less well
known counterpart, in analysis form, called the cosparse analysis model. In
this new model, signals are characterised by their parsimony in a transformed
domain using an overcomplete (linear) analysis operator. We propose to learn an
analysis operator from a training corpus using a constrained optimisation
framework based on L1 optimisation. The reason for introducing a constraint in
the optimisation framework is to exclude trivial solutions. Although there is
no final answer here for which constraint is the most relevant constraint, we
investigate some conventional constraints in the model adaptation field and use
the uniformly normalised tight frame (UNTF) for this purpose. We then derive a
practical learning algorithm, based on projected subgradients and
Douglas-Rachford splitting technique, and demonstrate its ability to robustly
recover a ground truth analysis operator, when provided with a clean training
set, of sufficient size. We also find an analysis operator for images, using
some noisy cosparse signals, which is indeed a more realistic experiment. As
the derived optimisation problem is not a convex program, we often find a local
minimum using such variational methods. Some local optimality conditions are
derived for two different settings, providing preliminary theoretical support
for the well-posedness of the learning problem under appropriate conditions.Comment: 29 pages, 13 figures, accepted to be published in TS
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
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