**T-MSBL for Sparse Signal Recovery and Compressed
Sensing**

Zhilin Zhang (z4zhang@ucsd.edu)

University of California, San Diego

Updated: Nov. 11, 2011

**For latest
updates, please check the new page:**

**https://sites.google.com/site/researchbyzhang/t-msbl**

**0. What is T-MSBL?**

The algorithms were developed for the multiple measurement vector (MMV) model, i.e.

*Y* = *A* *X*
+ *V*,

where
*A* is a known N by M dictionary matrix with N < M, *Y* is an
available data matrix, and *V* is an unknown noise matrix. The goal is to
recover the source matrix *X*. T-MSBL exploits the correlation exists in
each nonzero row of *X* (namely, temporal correlation). For details, see
the reference [1] below.

**1. Download Link**

The
codes of T-MSBL (and also T-SBL) can be downloaded at: **http://sccn.ucsd.edu/~zhang/TMSBL_code.zip****.**

Current version: 2.4 (Updated: Nov.11, 2011)

Here is the User Guide, which is included in the code package.

(Latest updates to the codes can be found at: https://sites.google.com/site/researchbyzhang/t-msbl. Or, send email to me)

**2. When Shall I use T-MSBL/T-SBL?**

Ø **When Phi is highly
coherent**

It performs well when the matrix *A*
is highly coherent both in MMV model and in the
standard compressed sensing model (i.e., the single measurement vector model).
See the research note for a
comparison with other 11 typical algorithms in the single measurement vector
model, and click here to
download a demo on an MMV model simulating a source localization task(notice the
input arguments in this case).

** **

Ø **Multiple
measurement vector (MMV) model**

This scenario is the main application of T-MSBL/T-SBL. By extensive experiments (compared with 17 state-of-the-art algorithms under various experiment settings), we found T-MSBL/T-SBL has top recovery performance among existing algorithms for this model, especially when the inverse problem is more difficult, such as highly under-determinacy ratio, more nonzero rows, and/or low SNR (6-15dB).

Ø **Single measurement
vector (SMV) model**

T-MSBL/T-SBL can be used for SMV models as well. Of course, in this case there is no temporal information can be exploited. However, due to an effective learning rule to automatically choose the optimal regularization value, it still has better performance than many algorithms for this model, especially in noisy environment.

Ø **Time-varying
sparsity model /dynamic compressed sensing model**

The support of each source vector X(:,i)
is slowly time-varying, we can use the concatenate of MMV models to approximate
this scenario. See the reference [2] below and the demo file in the code
package: **demo_time_varying.m**. In
extensive experiments, it shows good performance for the time-varying sparsity
model.

Ø **MMV model with
identical source vectors**

This rare model sometimes appears in
theoretical works or in some random walk models. See the demo file **demo_identicalVector.m** for the use guide.

**3. The most convenient ways to use T-MSBL**

Ø You can simply use the command to call T-MSBL for most cases (MMV model with temporal correlated source vectors, when SNR varies from 6 dB to 22 dB)

**X_est
= TMSBL(A, Y);**

Ø In most cases you can roughly know or estimate the noise level (of course you need not to know the exact value). For example, when solving practical problems, you may have some idea that the noise is large/mild/small. In this case, you can choose these commands for a better performance:

o When noise is large (e.g. SNR <=6 dB)

**X_est = TMSBL(A, Y, 'noise', 'large')**

o When noise is mild (e.g. 7 dB <= SNR <=22 dB)

**X_est = TMSBL(A, Y, 'noise', 'mild')**

o When noise is small (e.g. SNR >22 dB)

**X_est = TMSBL(A, Y, 'noise', 'small')**

o When no noise

**X_est = TMSBL(A, Y, 'noise', 'no')**

**Note 1: ****The above five commands uses a set of pre-defined
parameter values, which work well for most standard
compressed sensing experiment settings****.**

**Note 2****:
For better understand on SBL, see the Reference [5] at the bottom.**

**4. References**

[1] Zhilin Zhang, Bhaskar D. Rao, Sparse Signal Recovery with Temporally Correlated Source Vectors Using Sparse Bayesian Learning, IEEE Journal of Selected Topics in Signal Processing, Special Issue on Adaptive Sparse Representation of Data and Applications in Signal and Image Processing, vol. 5, no. 5, pp. 912-926, 2011

*(Comment: Main paper on T-MSBL/T-SBL;
The first paper that systematically studies the effects of temporal correlation
on the performance in the MMV model)*

[2] Zhilin Zhang, Bhaskar D. Rao, Exploiting Correlation in Sparse Signal Recovery Problems: Multiple Measurement Vectors, Block Sparsity, and Time-Varying Sparsity, ICML 2011 Workshop on Structured Sparsity: Learning and Inference, July, 2011

*(Comment: advises the use of MMV models
to approximate time-varying sparsity model)*

[3] Zhilin Zhang, Bhaskar D. Rao, Iterative Reweighted Algorithms for Sparse Signal Recovery with Temporally Correlated Source Vectors, Proc. of the 36th International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2011), Prague, the Czech Republic, 2011

*(Comment: Proposes a strategy to improve
existing iterative reweighted L2 algorithms)*

[4] Zhilin Zhang, Bhaskar D. Rao, Sparse Signal Recovery in the Presence of Correlated Multiple Measurement Vectors, Proc. of the 35th International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2010), Texas, USA, 2010

*(Comment: The first paper to exploit the
temporal correlation in the MMV model)*

[5] Zhilin Zhang, Bhaskar D. Rao, Clarify Some Issues on the Sparse Bayesian Learning for Sparse Signal Recovery, Technical Report, 2011

*(Comment: Clarifies some
misunderstandings on SBL and serves as guidance for correctly using SBL.)*