Abstract
Most developments in compressed sensing have revolved around the exploitation of signal structures that can be expressed and understood most easily using a geometrical interpretation. This geometric point of view not only underlies many of the initial theoretical developments on which much of the theory of compressed sensing is built, but has also allowed ideas to be extended to much more general recovery problems and structures. A unifying framework is that of non-convex, low-dimensional constraint sets in which the signal to be recovered is assumed to reside. The sparse signal structure of traditional compressed sensing translates into a union of low dimensional subspaces, each subspace being spanned by a small number of the coordinate axes. The union of subspaces interpretation is readily generalised and many other recovery problems can be seen to fall into this setting. For example, instead of vector data, in many problems, data is more naturally expressed in matrix form (for example a video is often best represented in a pixel by time matrix). A powerful constraint on matrices are constraints on the matrix rank. For example, in low-rank matrix recovery, the goal is to reconstruct a low-rank matrix given only a subset of its entries. Importantly, low-rank matrices also lie in a union of subspaces structure, although now, there are infinitely many subspaces (though each of these is finite dimensional). Many other examples of union of subspaces signal models appear in applications, including sparse wavelet-tree structures (which form a subset of the general sparse model) and finite rate of innovations models, where we can have infinitely many infinite dimensional subspaces. In this chapter, I will provide an introduction to these and related geometrical concepts and will show how they can be used to (a) develop algorithms to recover signals with given structures and (b) allow theoretical results that characterise the performance of these algorithmic approaches.
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Notes
- 1.
In this chapter, we use two, somewhat different, meanings for the term vector. On the one hand, we call any one dimensional array of real or complex numbers a vector, this is the meaning used here. Below, we will introduce a more abstract definition of vectors as elements of some mathematical space. Which of these two definitions is appropriate at any one point in this chapter should be clear from the context.
- 2.
A subset \(\mathcal{S }\subset \mathcal{H }\) is called closed if every sequence with elements in \(\mathcal{S }\) that converges to an element of \(\mathcal{H }\) has a limit in the subset \(\mathcal{S }\) itself.
- 3.
That is, a signal whose Fourier transform \(\mathcal{X }(f)\) is assumed to be zero apart from the set \(S\subset [-B_N,\ B_N]\).
- 4.
We assume here that we use the norm \(\sqrt{\sum x_i^2}\), though other Euclidean norms are treated with equal ease.
- 5.
Note that for \(\delta <\frac{\Vert P_\mathcal{S }(\mathbf{x })\Vert }{\Vert \widetilde{\mathbf{e }}\Vert }\) and for \(\mu \) and \(\alpha \) as in the theorem, both, the numerator and the denominator in the iteration count are negative numbers, so that a decrease in delta leads to an increase in the required number of iterations. If we were to choose \(\delta \) such that the numerator becomes positive, we would get a negative number of iterations. This has to be interpreted as meaning that we actually don’t need to run the algorithm at all, as the associated estimate error is already achieved by the estimate \({\hat{\mathbf{x }}}=\mathbf{x }^0=\mathbf{0 }\).
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Acknowledgments
This work was supported in part by the UKs Engineering and Physical Science Research Council grants EP/J005444/1 and D000246/1 and a Research Fellowship from the School of Mathematics at the University of Southampton.
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Blumensath, T. (2014). The Geometry of Compressed Sensing. In: Carmi, A., Mihaylova, L., Godsill, S. (eds) Compressed Sensing & Sparse Filtering. Signals and Communication Technology. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38398-4_2
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