Note: My answer will of course be influenced by the realm of my own research

Do note that has a very good list of topics. I'm going to explain some of the topics I think are interesting.

**Compressed Sensing:**

The big elephant in the room has already been excellently introduced by . From a mathematical perspective, compressed sensing gives us conditions under which we can approximately solve a sparse linear problem (think \mathsf{A} \vec{x} = \vec{y}). This has a lot of implications for things like Fourier Transforms and general linear optimization. We've already seen some amazing applications to MRI, but what's next for Compressed Sensing?

- Connection to Harmonic Analysis: Heavy-weights of the field such as David Donoho, Emmanuel Candés and of course Terence Tao [0] have shown that compressed sensing in very closely tied to Harmonic Analysis. Using fairly sophisticated techniques from Real Analysis and Probability Theory, Tao and Candés were able to show that some very compelling bounds on the sampling frequency and spectra of certain sparse linear operators.
- Applications to Non-linear problems: Can we only solve linear problems with Compressed Sensing? Probably not! It might be possible to find non-linear generalizations of the techniques in Compressed Sensing to things like the Wavelet Transform, giving applications a lot more flexibility. Moreover, by extending Compressed Sensing to non-linear problems, a larger class of image processing and recognition problems will become feasible. In terms of applications, algorithms for Feature Detection in Images and Videos might be revolutionized
- Relationship to Concentration of Measure: Oddly enough, Compressed Sensing showed up during Candés and Donoho's investigation into large dimensional statistical problems. Roughly speaking, since the 1960s, people had believed that it is impossible to correctly sample probability distributions on large dimensional spaces. As we've been collection an ever increasing collection of large dimensional data that don't necessarily fit in the the framework of classical frequentist statistics (e.g. DNA microarrays), one might think that our analysis of this data is hopeless. However, there is a phenomena known as concentration of measure [*] which roughly means that as one increases the dimension of their data, the non-zero probability mass is located (concentrated) on a smaller volume of space. When one has concentration phenomena, one can find sparsity and wavelet-like structures in this large-N limit, and an interesting question to ask is, Do Compressed Sensing methods always work when there is a concentration phenomena? [1] These phenomena might give a sufficient condition for Compressed Sensing and might help us understand the limits of this technique

Source: www.quora.com

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