Thanks Igor,

I find sampling theory fascinating. So much so, that I did my PhD research on it years ago. I got a degree in engineering but it could have been as well a PhD in applied math since sampling is what it was all about.

If it weren’t because I have made a few controversial remarks in a blog of an influential Fields Medalist, I wouldn’t mind to provide a link to my own work :).

I stopped working on sampling as soon as I graduated, but I still try to follow as much as I can, that’s why I was eager to learn about what Emmanuel Candès had to say at the ICM.

I know that most mathematicians by training consider applied math a sort of waste, but in this day and age, applied math is more relevant than it has ever been given the complexity of the systems we build and the amount of information these systems need to process.

Besides, the beauty of mathematics from my point of view, and here comes yet another controversial remark, is that mathematical concepts are as real as the physical world, only the physical world is a “subset” of all the mathematical reality, the subset that our sensors perceive. Richard Feynman said that nature speaks the language of mathematics, which is true, but I think it is because nature is a subset of the larger mathematical realm: God himself.

]]>As Terry said some of the other development are not as dramatic as “saving children’s life through faster MRI sampling” because of hardware. MRI does not require a change of hardware to experiment with the sampling reductionbrought forth by compressive sensing Pretty much everything else does. When that is the case, Compressive Sensing just becomes of the ways to get an answer from detectors and sometimes it is difficult to make it shine against well oiled signal processing chains already in place (CT scanners is one example Ihave in mind). I have highlighted some of the other breakthroughs earlier (structured sparsity, asymptotic sparsity and advanced marix factorization. In terms of hardware, it currently either blends in to current signal processing chains or it may open some totally different avenues ( shameful self promotion, http://www.nature.com/srep/2014/140709/srep05552/full/srep05552.html).

Igor.

]]>Thanks Terence (or Terry, whichever you prefer 🙂 ),

I skimmed through the ISIT presentation that you provided. I think that both are interesting applications, but the one that is probably most likely to gain traction, if compressed sensing comes with a practical implementation, is superresolution. Digital imaging is everywhere and I see it could be a very interesting application: getting high resolution images from from low resolution ones (subject to some application dependent constrains that would solve the uncertainty inherent to the problem) .

In a previous life (like 8 years ago), I did some experiments with the existing methods at the time of reconstructing audio signals from the magnitude of the Fourier coefficients. Not an easy problem by any means. My conclusion was that it is something one should try to avoid doing unless there is no choice. There are very few practical situations in which those phase coefficients are not available in any way or that the phase coefficients cannot be reasonably estimated from time delays.

I must add that I am humbled that you thought that addressing my question was worth your time :).

]]>I haven’t followed developments closely in the last few years, but among recent developments have been to use compressed sensing methods to perform phase reconstruction and superresolution (see e.g. Candes’ slides at http://www.itsoc.org/resources/media/isit-2013-istanbul/CandesISIT2013.pdf , which are focused on more recent applications than the ones in his ICM talk). The applications here are not as dramatic as those in MRI, but it may take a few more years for the applied side of these new developments to bear fruit.

]]>This is a good question.

While I had watched Candès’ lecture before it was referred to here -and I enjoyed it very much- I must say that I was left with the feeling that compressed sensing seems to be stuck exactly in the same place where it was circa 2010/2011.

There have been a lot of incremental research but the basic idea, the applications targeted, and the most general results seem to be the same. Nothing really revolutionary seems to have been produced since that would make all other sampling methods obsolete. Compressed sensing seems to have found its niche in MRI -which is onto itself significant-, but that’s about it.

The Netflix Prize was awarded in 2009. While privacy considerations have prevented newer editions, it doesn’t seem to me that there has been any significant progress towards beating the methods used by the team who won the 2009 edition using compressed sensing ideas.

Insight from people current with the state of the art in compressed sensing would be appreciated.

]]>I’d say a quite few by using additional structure in the signal.

]]>To get a bird’s eye view of the development in these two areas, compressive sensing and advanced matrix factorizations, you might want to check these two pages I curate;

Advanced matrix factorizations:

https://sites.google.com/site/igorcarron2/matrixfactorizations

Compressive Sensing, The Big Picture

https://sites.google.com/site/igorcarron2/cs

Since all these problems are NP-hard in general, we have seen a lot of activities on how to relax those and have had a slew of different algorithms to implement each of these relaxations. In my view, what has really changed the view of most people is the discovery that beyond asymptotics, most of these relaxations yields good results up to sharp phase transitions and instead of usual long infightings about asymptotics and attendant constants which used to be an issue for the longest time, these phase transitions are pretty much the acid test of certain problems yielding true insights (see http://nuit-blanche.blogspot.fr/2013/11/sunday-morning-insight-map-makers.html).

And “…yes, we expect the mathematicians to clear the waters there…” ( http://nuit-blanche.blogspot.fr/2013/10/sunday-morning-insight-watching-p-vs-np.html )

Cheers,

Igor.

]]>is not accurate, as the problem can be stated without the mention of the word “equation”. ]]>