It may come under different heading: manifold signal processing, structured sparsity and norms, model based sensing, dictionary learning and even cosparsity: The idea is that most signals are really low dimensional without having to be sparse.
What can we learn from Zhilin's Block Sparse Bayesian Learning algorithm and results on a non sparse but structured signal ?
- Yes, it uses a random Bernouilli like measurement matrix for signal acquisition and that is compressive sensing
- Yes, it does a reconstruction of the original signal using structured sparsity, as opposed to vanilla sparsity
But does this approach help in performing classification or estimation on the compressed data ? (which is really what we are after) as we want lightweight sensors to not only gather data optimally but we also want (rough) classification near the sensor ... at very low cost.
The answer is yes because the reconstruction step is proof that all the information encoded through the Bernouilli like matrices was good enough in the first place. That structured signals can now fit more obviously within compressive sensing. We don't need RIP, NSP, or exhaustive dictionary learning with all sorts of guarantees as much anymore. As the figure above shows (from ), we used to not have that comfort. We were unsure that the compressed measurement really was catching the largest component of the signal and its as important dependencies. With this peace of mind, we can now instantiate our favorite machine learning algorithm on the compressed data. Strangely enough, at that point, with some caveat, we may not care about reconstruction anymore.
 Low Energy Wireless Body-Area Networks for Fetal ECG Telemonitoring via the Framework of Block Sparse Bayesian Learning by Zhilin Zhang, Tzyy-Ping Jung. , Scott Makeig , Bhaskar D. Rao .
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