CS: Adaptive Compressed Sensing, Gradient Based Recovery Algos, Modified Frame Reconstruction Algo, On Sparse Channel Estimation, a presentation, CVX

Today we have four new papers on arxiv that I have not covered yet:

Echoing some of the babbling written here, here is a serious attempt at mapping cortical information and compressed sensing. As I have written before, the robustness of compressed sensing enabled by random sampling is just too nice of a property to not think about mapping it to a biological system. This is great.

Adaptive compressed sensing - a new class of self-organizing coding models for neuroscience by William Coulter, Christopher Hillar, Friedrich Sommer. The abstract reads:

Sparse coding networks, which utilize unsupervised learning to maximize coding efficiency, have successfully reproduced response properties found in primary visual cortex \cite{AN:OlshausenField96}. However, conventional sparse coding models require that the coding circuit can fully sample the sensory data in a one-to-one fashion, a requirement not supported by experimental data from the thalamo-cortical projection. To relieve these strict wiring requirements, we propose a sparse coding network constructed by introducing synaptic learning in the framework of compressed sensing. We demonstrate that the new model evolves biologically realistic spatially smooth receptive fields despite the fact that the feedforward connectivity subsamples the input and thus the learning has to rely on an impoverished and distorted account of the original visual data. Further, we demonstrate that the model could form a general scheme of cortical communication: it can form meaningful representations in a secondary sensory area, which receives input from the primary sensory area through a "compressing" cortico-cortical projection. Finally, we prove that our model belongs to a new class of sparse coding algorithms in which recurrent connections are essential in forming the spatial receptive fields.

And here are the three remaining ones (by the way I could not find the webpages of any of the authors, can somebody help ?)

The Physics of Compressive Sensing and the Gradient-Based Recovery Algorithms by Qi Dai, Wei Sha. The abstract reads:

The physics of compressive sensing (CS) and the gradient-based recovery algorithms are presented. First, the different forms for CS are summarized. Second, the physical meanings of coherence and measurement are given. Third, the gradient-based recovery algorithms and their geometry explanations are provided. Finally, we conclude the report and give some suggestion for future work.

Modified Frame Reconstruction Algorithm for Compressive Sensing by Graeme Pope. The abstract reads:

Compressive sensing is a technique to sample signals well below the Nyquist rate using linear measurement operators. In this paper we present an algorithm for signal reconstruction given such a set of measurements. This algorithm generalises and extends previous iterative hard thresholding algorithms and we give sufficient conditions for successful reconstruction of the original data signal. In addition we show that by underestimating the sparsity of the data signal we can increase the success rate of the algorithm.
We also present a number of modifications to this algorithm: the incorporation of a least squares step, polynomial acceleration and an adaptive method for choosing the step-length. These modified algorithms converge to the correct solution under similar conditions to the original un-modified algorithm. Empirical evidence show that these modifications dramatically increase both the success rate and the rate of convergence, and can outperform other algorithms previously used for signal reconstruction in compressive sensing.

and finally, On Sparse Channel Estimation by Brian Carroll. The abstract reads:

The abstract reads:

Channel Estimation is an essential component in applications such as radar and data communication. In multi path time varying environments, it is necessary to estimate time-shifts, scale-shifts (the wideband equivalent of Doppler-shifts), and the gains/phases of each of the multiple paths. With recent advances in sparse estimation (or "compressive sensing"), new estimation techniques have emerged which yield more accurate estimates of these channel parameters than traditional strategies. These estimation strategies, however, restrict potential estimates of time-shifts and scale-shifts to a finite set of values separated by a choice of grid spacing. A small grid spacing increases the number of potential estimates, thus lowering the quantization error, but also increases complexity and estimation time. Conversely, a large grid spacing lowers the number of potential estimates, thus lowering the complexity and estimation time, but increases the quantization error. In this thesis, we derive an expression which relates the choice of grid spacing to the mean-squared quantization error. Furthermore, we consider the case when scale-shifts are approximated by Doppler-shifts, and derive a similar expression relating the choice of the grid spacing and the quantization error. Using insights gained from these expressions, we further explore the effects of the choice and grid spacing, and examine when a wideband model can be well approximated by a narrowband model.

We also have a presentation on Minimum Sum of Distances Estimator: Robustness and Stability by Yariv Sharon, John Wright and Yi Ma. I talked about this here.

In a different direction, Michael Grant and Stephen Boyd let us know that CVX (Matlab Software for Disciplined Convex Programming) has problems:

Reports of problems with 64-bit Windows and Matlab R2009a

Apparently, the MEX files that ship with CVX do not work with the latest and greatest version of the 64-bit Windows version of Matlab. I do not recommend upgrading to R2009a if this is your operating system of choice and you rely on CVX for your work.

Fuller support for 64-bit platforms

Image Credit: NASA/JPL/Space Science Institut, Titan as seen by Cassini three days ago.