In LinkedIn, Phil, Leslie, and me have a productive discussion on the use of BSBL and EM-GM-AMP in practical problems. The whole discussions can be seen at here. For convenience, I copied my words on the use of BSBL and related issues below.
Below are several practical examples where intra-block correlation exists and BSBL can be used.
In fact, there are many examples in practice.
1. Localization of distributed sources (not point-sources). In EEG/MEG
source localization, the sources are generally not just a point. They
have areas, so they are called distributed sources. When modeling the
problem using a sparse linear regression model: y=Ax + v, the
coefficient vector x is expected to contain several nonzero blocks, each
nonzero block corresponding to a distributed source. Entries in the
same nonzero block are highly correlated to each other in amplitude,
since they are all associated with the same source.
(Remark: There are a number of work assuming the sources are
point-sources as well. Thus the problem becomes a traditional DOA
problem. However, in this case, BSBL can be applied as well, which I
will explain later.)
2. In compressed sensing of ECG, an ECG signal has clear block structure
(generally all the blocks are nonzero, but some blocks are very close
to zero), and in each “block” the entries are highly correlated in
amplitude. One can see the Fig.1 in my T-BME paper (
) to get such feeling.
(Remark: Almost all the physiological signals have correlation in the
time domain, although some of them have not clear block-structure.
However, BSBL can also be used for such signals. I will explain later).
3. One should note that the mathematical model used in compressed
sensing is a linear regression model, and such a model has numerous
applications in almost every field. For many applications, the
regression coefficients have block structure. In each block, the entries
are associated with a same physical “force” or a “factor”, and thus
correlated in their amplitude.
In fact, as long as a signal has block structure, then it is highly
possible that intra-block correlation also exists. Even if the
intra-block correlation= 0, it is not harmful to run BSBL, because, as
you can see from my T-SP paper, it also has excellent performance (and
outperforms all the known block-structure-based algorithms) when
intra-block correlation = 0. You do not need to test whether the
correlation exists. You just need to run. That’s it.
Now I want to
emphasize that BSBL can be used in the general-sparse problems (i.e., no
structure in the coefficient vector x). This is due to two factors.
1.One factor is that truly sparse signals do not exist in most practical problems; in fact, they are non-sparse.
As I have mentioned in our personal communication, and also mentioned by
Phil and many other people, the truly sparse signal does not exist in
most practical problems. Those “sparse” signals are compressive; most of
their entries in the time domain (or the coefficients in some
transformed domains) are close to zero but strictly nonzero. In other
words, they are non-sparse ! Using general-sparse recovery algorithms
(such as Lasso, CoSaMP, OMP, FOCUSS, the basic SBL) can recover those
entries with large amplitude. But they always have challenges in
recovery of the entries with small amplitude. So, the quality of the
recovered data has a “glass ceilling”, which is not very high.
However, BSBL has a unique property, i.e., recovering non-sparse signals
(or signals with non-sparse representation coefficients) very well.
As for the recovery of non-sparse signals, I do have a number of work showing this. Please refer to my T-BME papers:
 Compressed Sensing for Energy-Efficient Wireless Telemonitoring of
Noninvasive Fetal ECG via Block Sparse Bayesian Learning
 Compressed Sensing of EEG for Wireless Telemonitoring with Low Energy Consumption and Inexpensive Hardware
In fact, we have achieved much better results than [1-2], which will be release soon.
2.The second factor is that BSBL is a kind of multiscale regression algorithms (although very naïve).
In DOA or similar applications (e.g. earth-quake detection, brain source
localization, or some communication problems), the matrix A (remind the
model: y=Ax + v) is highly coherent (i.e., columns of A are highly
correlated to each other). In this case, even if x is a very sparse
vector, this problem is very difficult, especially in noisy situations.
Using the basic SBL can get better performance than most existing
algorithms. But we found that using BSBL can get much better
performance. This is mainly because that BSBL, using the block
partition, divides the whole search space (i.e. the number of whole
locations in x) into a number of sub search spaces (i.e. the number of
candidate nonzero blocks). This makes the localization problems become
easier. Since x is sparse, generally the nonzero blocks are only a few
and zero blocks are many. During iteration, the zero blocks are deleted
in BSBL gradually. And the problem becomes easier and easier with
iteration. I can dynamically change the block partition in BSBL
according to some criterion. However, experience showed that it is not
necessary to do this. BSBL can eventually find the correct locations of
nonzero entries in x (although the rest entries in a nonzero block have
very small amplitude).
The Fig.4 in my ICASSP 2012 paper (
) may give some feeling on what I said “the rest entries in a
nonzero block have very small amplitude”.
Last, but not least, I
want to say that a comparison between BSBL with other algorithms is
helpful to everybody. I will be very happy to see the result. But I
suggest one performs the comparison in some practical problems, since the
readers generally come from various application fields with questions similar like “which algorithms is the best one for my problems”. Putting the
comparison in specific practical problems is more informative and really
helpful. Computer simulations have several problems, such as the
suitable performance index, the consistency of the simulation model with
the underlying models in practical problems, what’s the criterion to
choose the algorithms (e.g. for an audio compressed sensing problem, does one think it is a general-sparse recovery problem, or a
block-sparse recovery problem, or a non-sparse recovery problem?). Due
to these issues, the conclusions may be not solid, and even more or less
MSE is generally not a good performance index. For example, for image
quality, MSE is not recommended (what’s the suitable performance index
for images is still a hot topic in the image processing field). In my
experience on compressed sensing of EEG, I even found that MSE always
misleading when I compared BSBL with my STSBL algorithms. This is why in
my two T-BME papers (and other papers coming out), I used a
task-oriented performance measure criterion. That is, after recovered
(1) performing task-required signal processing or pattern recognition on the recovered data, obtaining result A;
(2) performing the same task-required signal processing or pattern
recognition on the original data (or the recovered data by another
algorithm), obtaining result B;
(3) comparing result A with result B, which tell me which algorithm has better data recovery ability.
Many people asked me whether my BSBL codes can be used for
complex-valued problems. My answer is YES. But you need to transform
your complex-valued problem into a real-valued problem, as I showed
This transform is very simple. You just need no more than 1 minute to do this.
I will update my BSBL codes in the near future such that no need to do the transform.