How many samples are needed to reconstruct a sparse signal?

Well, there are many, many results around some of which you probably know (at least if you are following this blog or this one). Today I write about a neat result which I found quite some time ago on reconstruction of nonnegative sparse signals from a semi-continuous perspective.

**1. From discrete sparse reconstruction/compressed sensing to semi-continuous **

The basic sparse reconstruction problem asks the following: Say we have a vector which only has non-zero entries and a fat matrix (i.e. ) and consider that we are given measurements . Of course, the system is underdetermined. However, we may add a little more prior knowledge on the solution and ask: Is is possible to reconstruct from if we know that the vector is sparse? If yes: How? Under what conditions on , , and ? This question created the expanding universe of compressed sensing recently (and this universe is expanding so fast that for sure there has to be some dark energy in it). As a matter of fact, a powerful method to obtain sparse solutions to underdetermined systems is -minimization a.k.a. Basis Pursuit on which I blogged recently: Solve

and the important ingredient here is the -norm of the vector in the objective function.

In this post I’ll formulate semi-continuous sparse reconstruction. We move from an -vector to a finite *signed measure* on a closed interval (which we assume to be for simplicty). We may embed the -vectors into the space of finite signed measures by choosing points , from the interval and build with the point-masses (or Dirac measures) . To a be a bit more precise, we speak about the space of Radon measures on , which are defined on the Borel -algebra of and are finite. Radon measures are not very scary objects and an intuitive way to think of them is to use Riesz representation: Every Radon measure arises as a continuous linear functional on a space of continuous functions, namely the space which is the closure of the continuous functions with compact support in with respect to the supremum norm. Hence, Radon measures work on these functions as . It is also natural to speak of the support of a Radon measure and it holds for any continuous function that

An important tool for Radon measures is the Hahn-Jordan decomposition which decomposes into a positive part and a negative part , i.e. and are non-negative and . Finally the variation of a measure, which is

provides a norm on the space of Radon measures.

Example 1For the measure one readily calculates thatand hence

In this sense, the space of Radon measures provides a generalization of .

We may sample a Radon measure with linear functionals and these can be encoded by continuous functions as

This sampling gives a bounded linear operator . The generalization of Basis Pursuit is then given by

This was introduced and called “Support Pursuit” in the preprint Exact Reconstruction using Support Pursuit by Yohann de Castro and Frabrice Gamboa.

More on the motivation and the use of Radon measures for sparsity can be found in Inverse problems in spaces of measures by Kristian Bredies and Hanna Pikkarainen.

**2. Exact reconstruction of sparse nonnegative Radon measures **

Before I talk about the results we may count the degrees of freedom a sparse Radon measure has: If with some than is defined by the weights and the positions . Hence, we expect that at least linear measurements should be necessary to reconstruct . Surprisingly, this is almost enough if we know that the measure is nonnegative! We only need one more measurement, that is and moreover, we can take fairly simple measurements, namely the monomials: (with the convention that ). This is shown in the following theorem by de Castro and Gamboa.

Theorem 1Let with , and let , be the monomials as above. Define . Then is the unique solution of the support pursuit problem, that is of

*Proof:* The following polynomial will be of importance: For a constant define

The following properties of will be used:

- for
- has degree and hence, is a linear combination of the , , i.e. .
- For small enough it holds for that .

Now let be a solution of (SP). We have to show that . Due to property 2 we know that

Due to property 1 and non-negativity of we conclude that

Moreover, by Lebesgue’s decomposition we can decompose with respect to such that

and is singular with respect to . We get

and we conclude that and especially . This shows that is a solution to . It remains to show uniqueness. We show the following: If there is a with support in such that , then . To see this, we build, for any , the sets

and assume that there exists such that ( denoting the restriction of to ). However, it holds by property 3 of that

and consequently

which is a contradiction. Hence, for all and this implies . Since has its support in we conclude that . Hence the support of is exactly . and since and hence . This can be written as a Vandermonde system

which only has the zero solution, giving .

**3. Generalization to other measurements **

The measurement by monomials may sound a bit unusual. However, de Castro and Gamboa show more. What really matters here is that the monomials for a so-called Chebyshev-System (or Tchebyscheff-system or T-system – by the way, have you ever tried to google for a T-system?). This is explained, for example in the book “Tchebycheff Systems: With Applications in Analysis and Statistics” by Karlin and Studden. A T-system on is simply a set of functions such that any linear combination of these functions has at most zeros. These systems are called after Tchebyscheff since they obey many of the helpful properties of the Tchebyscheff-polynomials.

What is helpful in our context is the following theorem of Krein:

Theorem 2 (Krein)If is a T-system for , and are in the interior of , then there exists a linear combination which is non-negative and vanishes exactly the the point .

Now consider that we replace the monomials in Theorem~1 by a T-system. You recognize that Krein’s Theorem allows to construct a “generalized polynomial” which fulfills the same requirements than the polynomial is the proof of Theorem~1 as soon as the constant function 1 lies in the span of the T-system and indeed the result of Theorem~1 is also valid in that case.

**4. Exact reconstruction of -sparse nonnegative vectors from measurements **

From the above one can deduce a reconstruction result for -sparse vectors and I quote Theorem 2.4 from Exact Reconstruction using Support Pursuit:

Theorem 3Let , , be integers such that and let be a complete T-system on (that is, is a T-system on for all ). Then it holds: For any distinct reals and defined as

Basis Pursuit recovers all nonnegative -sparse vectors .

**5. Concluding remarks **

Note that Theorem~3 gives a deterministic construction of a measurement matrix.

Also note, that nonnegativity is crucial in what we did here. This allowed (in the monomial case) to work with squares and obtain the polynomial in the proof of Theorem~1 (which is also called “dual certificate” in this context). This raises the question how this method can be adapted to all sparse signals. One needs (in the monomial case) a polynomial which is bounded by 1 but matches the signs of the measure on its support. While this can be done (I think) for polynomials it seems difficult to obtain a generalization of Krein’s Theorem to this case…

November 3, 2011 at 5:35 pm

The pursuit appears like a fake. Actually every (!) nonnegative measure minimizes the norm subject to appropriate data. The analysis is only taking into account nonnegativity. If you would minimize any other functional subject to nonnegativity you get the same results.

November 3, 2011 at 8:04 pm

Probably there is a misunderstanding: Nonnegativity is not a constraint in the minimization. “Support pursuit” minimizes over all signed Radon measures. However, the “exact reconstruction” from 2s+1 only holds for measures which are nonnegative.

November 3, 2011 at 8:54 pm

No. If nonnegativity is a constraint it would be completely trivial, since the scalar product with the function 1 is fixed by the data, i.e. the 1-norm for positive measures.

However, since the constant function 1 is always part of the system, one trivially has the constant function 1 in the range of the adjoint operator. Thus, the source condition is satisfied for any nonnegative measure, thus it is a norm-minimizing solution. It suffices to check that for s-sparse solutions there is no other. The same theory can be built for nonpositive solutions, the main thing is that this kind of systems does not promote sign-changes.

November 4, 2011 at 8:27 am

Right, one needs to know that the “strict source condition” (or existence of a dual certificate) implies uniqueness of the solution – and in the case of nonnegativity (or nonpositivity) the construction of the dual certificate is trivial. I would say: Simple proof for a cool fact…

November 9, 2011 at 9:28 pm

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