CS: Bob's POD's, An integral computed, a UWB hardware, Mixed Operators in Compressed Sensing

You might be interested in reading these reviews by Bob Sturm:

• Paper of the Day (Po'D): Content-adaptive Dictionaries Edition
• Paper of the Day (Po'D): Even More Sparse Separation Edition
• Paper of the Day (Po'D): Demixing Pursuit Edition
• Paper of the Day (Po'D): Dual Matching Pursuit Edition
• Paper of the Day (Po'D): Stereo Matching Pursuit Edition

In a different direction, you probably recalled this entry: Challenge: Compute a high dimensional integral, Win a 1,000 Euros where the integral to be evaluated is required to help evaluate low probability events as featured in The information associated with a sample, by Bernard Beauzamy The prize was won by Peter Robinson of Quintessa Ltd. Here is his entry.

It looks like I missed the following two papers that features some hardware component:

Compressed Sensing Based UWB Receiver: Hardware Compressing and FPGA Reconstruction by Depeng Yang, Husheng Li, Gregory D. Peterson and Aly Fathy. The abstract reads:

A low sampling rate approach for recovering ultra wide band (UWB) signals is proposed, using Distributed Amplifiers (DAs) and low speed Analog-to-Digital Converters (ADCs) and based on the theory of compressed sensing. A microwave circuit consisting of a bank of DAs, followed by a bank of ADCs, is designed to implement analog compressing, where the elements of measurement matrix are realized by picosecond delay tap and flexible gain coefficients in DAs. Numerical simulation shows that a bank of eight DAs and ADCs with 500MHz sampling rate can almost perfectly recover a 100ps-resolution UWB echo signal in the noiseless case. For recovering the UWB signals in a real-time way, issues in field programmable gate array (FPGA) implementation are discussed.

I just added it to the Compressive Sensing hardware page. And here is the second one: UWB Signal Acquisition in Positioning Systems: Bayesian Compressed Sensing with Redundancy by Depeng Yang, Husheng Li, Gregory D. Peterson and Aly E. Fathy. The abstract reads:

In ultra wide band (UWB) positioning systems, the key problem is to detect the timing of transmitted UWB pulses. Narrow pulses are required for high precision of positioning, thus demanding high sampling rate. To alleviate the difficulty for analog-to-digital converters (ADC) and utilize the feature of time sparsity of UWB pulses, a compressed sensing based scheme is proposed, in which the received signal is mixed using distributed amplifiers, sampled using a bank of low rate ADCs and then reconstructed. To combat the detrimental effect of Gaussian noise, Bayesian compressed sensing is applied, using a structure similar to the iterative decoder for turbo codes to exploit the redundancy in time, history and space. Numerical simulation results demonstrate that the Bayesian compressed sensing using redundancy achieves substantial performance gain (UWB pulse recovery rate and positioning precision), compared with traditional compressed sensing approaches.

Finally, here is the latest entry on arxiv: Mixed Operators in Compressed Sensing by Matthew Herman and Deanna Needell. The abstract reads:

Applications of compressed sensing motivate the possibility of using different operators to encode and decode a signal of interest. Since it is clear that the operators cannot be too different, we can view the discrepancy between the two matrices as a perturbation. The stability of l_1-minimization and greedy algorithms to recover the signal in the presence of additive noise is by now well-known. Recently however, work has been done to analyze these methods with noise in the measurement matrix, which generates a multiplicative noise term. This new framework of generalized perturbations (i.e., both additive and multiplicative noise) extends the prior work on stable signal recovery from incomplete and inaccurate measurements of Cand`es, Romberg and Tao using Basis Pursuit (BP), and of Needell and Tropp using Compressive Sampling Matching Pursuit (CoSaMP). We show, under reasonable assumptions, that the stability of the reconstructed signal by both BP and CoSaMP is limited by the noise level in the observation. Our analysis extends easily to arbitrary greedy methods.

When I am reading this about this multiplicative noise, I am also reminded of how one computes the pseudo-spectrum (an issue debated with Anders Hansen). But since there is no obvious connection between pseudo-spectra and the ability for matrices to be a good measurement matrices in the framework of compressive sensing. well ... it doesn't matter, does it ?

Credit: NASA, One of the last photos of Spirit.