Sparse signal reconstruction from a set of underdetermined or ill-conditioned
linear measurements is a fundamental problem in signal processing and statistics.
This problem has over the last few years found application areas from compressive
sensing (Donoho, 2006; Guochang
et al., 2010) to source coding (Goyal et
al., 1998), image restoration (Figueiredo and Nowak,
2003; Yu et al., 2010) and source localization
(Picard and Weiss, 2010).
Suppose A is a measurement matrix in RMxN (M<N) and n is a noise vector in RM, then we are given measurements y ∈RM of the form:
If there are no restrictions on the measurement matrix A and the original signal
x, then the problem of finding an estimate
is NP-hard in general. While there are at least five major classes of computation
techniques for solving sparse reconstruction problem (Tropp
and Wright, 2010), namely greedy pursuit (Mallat and
Zhifeng, 1993), Bayesian framework (Wipf and Rao, 2004),
nonconvex optimization (Chartrand, 2007), brute force
(Miller, 2002) and convex relaxation (Chen
et al., 1998).
Greedy pursuit, one of the most popular ways at present (Chen
et al., 2008), is used in approaches such as Matching Pursuit (MP)
(Mallat and Zhifeng, 1993), Orthogonal Matching Pursuit
(OMP) (Pati et al., 1993), Stagewise Orthogonal
Matching Pursuit (StOMP) (Donoho et al., 2006)
and Stagewise Weak Gradient Pursuit (SWGP) (Blumensath and
Davies, 2009). Greedy pursuit iteratively refines a sparse signal representation
by successively choosing one or more elements on a dictionary that have the
highest correlation with current residual. Bayesian framework employs relevance
vector machine for estimating the underlying signal while nonconvex optimization
procedures develop a related nonconvex problem and attempt to find a stationary
point. Brute force strategies search through all possible support sets to minimize
the number of possibilities. Another very popular approach is to solve the convex
unconstrained optimization problems of the form:
Here, τ is a regularization positive parameter which balances the sparsity
of the data against the data fidelity (Tropp and Wright,
2010). Many optimization algorithms have been proposed to solve the unconstrained
(but nonsmooth) formulation (2), such as Interior-Point (IP) methods (Kim
et al., 2007), Iterative Shrinkage/Thresholding (IST) (Daubechies
et al., 2004) algorithms, GPSR (Figueiredo et
al., 2007) and SpaRSA (Wright et al., 2009).
This study has concentrated on GPSR methods. There are two main problems associated
with the application of GPSR to large scale data sets. On the one hand, the
computational time is easily affected by original signal sparsity and number
of measurements. On the other hand, there is a great deal of interference of
other uselessly searching directions in GPSR so that the computational time
per reconstruction is long if without continuation schemes. Inspired by greedy
pursuit methods, we introduce a variant of GPSR together with iterative weighted
vector estimate and we only update the searching directions of
based on the weighted vector. We refer to the resulting approach as IWGP (Iterative
Weighted Gradient Projection). We show that IWGP retains all the merits of the
GPSR methods and greatly decreases the computational time. We use numerical
experiments to show IWGP can handle lager scale compressive sensing problems
more efficiently than GPSR and IST. We also compare the sensitivity of IWGP
with those of GPSR and IST in terms of change of number of nonzero components
and number of measurements.
GRADIENT PROJECTION FOR SPARSE RECONSTRUCTION
The algorithms discussed and developed in this study are all part of GPSR algorithms.
Given a vector x and a measurement matrix A, the approach of GPSR is to express
Eq. 2 as a Bound-Constrained Quadratic Program (BCQP) (Wu
et al., 2010) via splitting the variable x into positive and negative
parts (Figueiredo et al., 2007) and can be rewritten
Eq. 2 as:
Equation 3 can be more properly rewritten in a standard BCQP form as:
The gradient of the objective function G ()
is defined as:
GPSR chooses the negative gradient -Δ G ()
to search for each iterate :
where, α(k) is the step size. This can be solved by two strategies in GPSR, called GRSR-Basic algorithm and GPSR-BB algorithm respectively. In GPSR-Basic, the step size α(k) is defined initially as:
Then it is achieved the proper step size until a sufficient decrease is attained
in G by running a backtracking line search. Alternatively, in order to avoid
the fussily inner searching loop, the Barizilai and Borwein equation (Figueiredo
et al., 2007) is applied to approximate to the Hessian of G at
in GPSR-BB. So the stp size is calculated simply as:
Note that GPSR-Basic ensures objective function decrease per iteration and
GPSR-BB makes the iteration easier. They are easy to implement and do not appear
to require application-specific tuning. In reality, GPSR algorithms tend not
to work very well for large-scale settings because they spend much computational
time. Instead, it is necessary to apply specialized techniques, such as warm-start
and continuation. A more detailed discussion of GPSR algorithms can be found
by Figueiredo et al. (2007).
ITERATIVE WEIGHTED GRADIENT PROJECTION
In this study, we propose a novel approach, called IWGP which combines a weighted
vector with GPSR to achieve a spare approximation solution to Eq.
1. More precisely, we mean that it is more efficient to compute the gradient
and to solve Eq. 2 by GPSR. In this section, we discuss how
the algorithm works efficiently as well as the different ways to choose threshold,
describing the IWGP framework formally.
In fact the gradient ΔG ()
contain more information than its form suggests, by further analyzing its particular
Eq. 5. For a given ,
We define a residual vector (Liejun, 2011):
So Eq. 9 can be rewritten as:
It may be worth noting that the gradient of the objective function in Eq.
11 can be constitution of a constant vector τ1 and a vector ATr
which means that the gradient ΔG ()
includes the information of residual correlation between the current residual
r and measurement matrix A.
Now we may concentrate on showing that the proposed algorithm makes full use of the residual correlation to build up a weighted vector to filter out unwanted gradient value. We define a vector J in RN by:
where, c = Atr, Th is a threshold parameter, hard (x, th) = 2xsign (|x|-th1) +1 is the well-known hard-threshold function. We merge the newly identified vector with the previous weighted vector estimate, thereby updating the estimate:
Since the original sparse signal x is a signal populated primarily with zeros,
we can be only interested in the gradient directions of the locations of the
nonzeros in .
Specifically, we define a new gradient vector:
With the direction to move given by Eq. 15, we can simply
compute the step size as Eq. 9 and update the next estimate
by Eq. 6.
Threshold selection: To recover the original signal x, we hope to know which columns of A participate in the measurement vector y. The idea behind the algorithm is to identify columns in a thresholding fashion. At each iteration, we firstly wish to determine a weighted vector which represents the column of A that is most strongly correlated with the remaining part of y. Then we start to search from a new approximation along the negative gradient filtered by the weighted vector. After proper iterations, we wish that the algorithm will have identified the correct vector of columns.
These considerations motivate a few number of possible threshold selection. One simple selection is:
where, t typically takes values in the range 2≤t≤3. The selection strategy
is inspired by Gaussian behavior of residual Multiple-Access Interference (MAI)
(Donoho et al., 2006).
A similar strategy is called stagewise weak element selection (Blumensath
and Davies, 2009). This selection takes advantage of a threshold based on
the maximum of |ri|. Based on this approach, the threshold is calculated
where, β∈ (0, 1) is a weakness parameter.
Both these strategies are well suited to the class of problems addressed in this study. In the experiments, unless otherwise noted, we choose (18), with β = 0.75 as the threshold.
|| Schematic Representation of the IWGP algorithm
The IWGP framework: IWGP operates in k iterations, building up a sequence
by searching along a sequence of negative weighted gradient vectors
Moreover, the algorithm also maintains a sequence of estimates W(0),
, of weighted coefficients. Figure 1
gives a diagrammatic representation.
Meantime a precise framework of the algorithm is defined as follows.
||Start with initial solution =
0, initial residual r(0) = y and the weighted vector W(0)
= 0. The iteration counter k starts at k = 1
||Apply matched filtering to the current residual, achieving the vector
of residual correlations:
||Compute the threshold Th(k) from Eq.
18, the identified vector J(k) as in Eq.
13 and then yield the weighted vector W(k)as in Eq.
||Compute the weighted gradient
as in Eq. 16
||Calculate the new estimate and the new residual:
||Compute the step size α(k-1) from Eq.
||Check the stopping criterion which is chosen as suggested by Figueiredo
et al. (2007) ||min (z, ΔG (z))||≤tolP and if it is
as the final output of the iteration; otherwise, set k7k + 1 and return
to step 2
This section illustrates experimentally that IWGP is a powerful algorithm for several types of sparse reconstruction problems of the form Eq. 2. All the experiments were carried out on a personal computer with an Intel Pentium (R) Dual E2200 processor and 1.99GB of memory.
Let us describe the experimental setup. In each trial, we start with a d-sparse signal x and its measurements y according to the model in Eq. 1. In all of the experiments below, we have used an MxN Gaussian matrix as our measurement matrix A, with all entries independently distributed Normal (0, 1) and then orthonormalizing the rows. We used parameter of stopping criterion tolP = 10-5.
In our first experiment, we consider a length N = 4096 signal x that contains d = 100 randomly placed±1 spikes. To simulate measurement noise, zero-mean Gaussian white noise with variance 10-4 is added to each of the M measurements that define the observation y. In the example M = 1024, τ = 0.005 ||ATy||∞ and the reconstructions are implemented by GPSR and IWGP.
Figure 2 shows that the original signal and the reconstruction results with GPSR-BB and IWGP, respectively. Because of noisy reconstruction, GPSR-BB cannot recover the underlying sparse signal exactly, nor can IWGP. However, the IWGP reconstruction exhibits a bit lower Mean Squared Error (MSE) with respect to the original signal.
In the second experiment, we compare the speed of IWGP with that of other recently
proposed algorithms, namely, variants of GPSR and IST (Daubechies
et al., 2004). All the parameters are similar in the first experiment
except that we perform 100 independent trials. Table 1 reports
the average CPU times needed by the several algorithms.
||From top to bottom: original signal, reconstruction via GPSR-BB
|| CPU time and number of nonzero components
|| Average CPU times
The finding in this table presents IWGP is significantly faster than the others.
More in detail, IWGP is approximate 12 times faster than IST and about 5 times
faster than GPSR.
The plot in Fig. 3 presents how the CPU time of IWGP changes when the number of nonzero components increases. For comparison, also shown are the results obtained with GPSR and IST. And the noises variances in this example are fixed the same and the results are averaged over 100 runs. In general the CPU time of the algorithms is in line with the increase of the number of nonzero components. However, IWGP keeps this growth very mild. In other words, IWGP is insensitive to the change in the sparsity level.
Then let us see how much CPU time of the algorithms is required to recover
a fixed sparsity level of signal with change in the number of measurements.
|| CPU time and number of measurements
Results in Fig. 4 display that CPU time of IWGP (as well
as GPSR and IST) is affected by increase in the number of measurements. Not
surprisingly, IWGP costs the least CPU time solving spare reconstruction problem
although the others also decrease gradually as the number of measurements grows.
We have presented a novel gradient projection algorithm to more effectively solve the solution to sparse signal reconstruction problems arising in compressive sensing and other inverse problems. We have employed a weighted technique to filter out unwanted gradient components and the resulting algorithm is well suited to handling large-scale settings. In the numerical experiments, we have tested IWGP by utilizing a large-scale example. The novel algorithm is significantly faster compared to state-of-the-art algorithms. Moreover, the decrease of sparsity level can hardly interfere the computational time of the algorithm derived in this paper. The numerical results further show that IWGP is superior to GPSR algorithms and IST.