Sensing matrix for compressed sensing

Question

• What are the differences between random binary sensing matrix  and random Gaussian sensing matrix?
• What the advantages and disadvantages of each matrix?

How can I choose the suitable matrix for a certain signal?

Answer 1

A sensing matrix maps input vector to measurement vector through linear wighted summation of input. What makes a specific matrix good, is application dependent. Now, both distributions more or less satisfy RIP. However hardware implementation of the Bernoulli matrix (binary or bipolar) is much much easier especially in analog domain. A Bernoulli wight is either 0 or 1 (or -1/1 in case of polar Bernoulli), but a Gaussian wight is a floating point figure. Multiplication of a flouting point number either in digital or analog, is resource consuming, while multiplication of a Bernoulli wight is feasible through implementation of a simple switch in analog domain or an and instruction in digital. As an example consider RMPI analog to information devices which compressively sample the signal in analog and then reconstruct the signal in digital. Prior to quantization the sensing matrix should be applied, were through incorporating a Bernoulli matrix, the multipliers are implemented as simples switches.

I also think, Gaussian Matrix has tighter RIP than Bernoulli (especially binary Bernoulli). As an example assume we have an sparse matrix $X=[0,1,0,2,0]^T$ , and assume we are measuring it using a random matrix with two rows and 5 columns. A binary Bernoulli, is more probable to multiply one of the non sparse entries by zero and hence miss its contribution in the measurements vector. Also, having only 1,0 and -1, the matrix rows are more probable to be similar and hence more coherency occurs.

Although it seems not very practical, in this paper the authors presented an analog circuit for application of Gaussian Matrix.