In compressed sensing, the nullspace property gives necessary and sufficient conditions on the reconstruction of sparse signals using the techniques of -relaxation. The term "nullspace property" originates from Cohen, Dahmen, and DeVore.[1] The nullspace property is often difficult to check in practice, and the restricted isometry property is a more modern condition in the field of compressed sensing.

The technique of -relaxation

The non-convex -minimization problem,

subject to ,

is a standard problem in compressed sensing. However, -minimization is known to be NP-hard in general.[2] As such, the technique of -relaxation is sometimes employed to circumvent the difficulties of signal reconstruction using the -norm. In -relaxation, the problem,

subject to ,

is solved in place of the problem. Note that this relaxation is convex and hence amenable to the standard techniques of linear programming - a computationally desirable feature. Naturally we wish to know when -relaxation will give the same answer as the problem. The nullspace property is one way to guarantee agreement.

Definition

An complex matrix has the nullspace property of order , if for all index sets with we have that: for all .

Recovery Condition

The following theorem gives necessary and sufficient condition on the recoverability of a given -sparse vector in . The proof of the theorem is a standard one, and the proof supplied here is summarized from Holger Rauhut.[3]

Let be a complex matrix. Then every -sparse signal is the unique solution to the -relaxation problem with if and only if satisfies the nullspace property with order .

For the forwards direction notice that and are distinct vectors with by the linearity of , and hence by uniqueness we must have as desired. For the backwards direction, let be -sparse and another (not necessary -sparse) vector such that and . Define the (non-zero) vector and notice that it lies in the nullspace of . Call the support of , and then the result follows from an elementary application of the triangle inequality: , establishing the minimality of .

References

  1. Cohen, Albert; Dahmen, Wolfgang; DeVore, Ronald (2009-01-01). "Compressed sensing and best 𝑘-term approximation". Journal of the American Mathematical Society. 22 (1): 211–231. doi:10.1090/S0894-0347-08-00610-3. ISSN 0894-0347.
  2. Natarajan, B. K. (1995-04-01). "Sparse Approximate Solutions to Linear Systems". SIAM J. Comput. 24 (2): 227–234. doi:10.1137/S0097539792240406. ISSN 0097-5397. S2CID 2072045.
  3. Rauhut, Holger (2011). Compressive Sensing and Structured Random Matrices. CiteSeerX 10.1.1.185.3754.
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