Abstract
Recent theory of compressed sensing (CS) tells us that sparse signals can be reconstructed from a small number of random samples. In reconstruction of sparse signals, greedy algorithms, such as the orthogonal matching pursuit (OMP), have been shown to be computationally efficient. In this paper, the performance of OMP is shown to be dependent on how well information of the underlying signals is preserved in the residual vector. Further, to improve the information preservation, we present a modification of OMP, called oblique projection matching pursuit (ObMP), which updates the residual in a oblique projection manor. Using the restricted isometric property (RIP), we build a solid yet very intuitive grasp of the more accurate phenomenon of ObMP. We also show from empirical experiments that the ObMP achieves improved reconstruction performance over the conventional OMP algorithm in terms of support detection ratio and mean squared error (MSE).
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Acknowledgments
This work was supported in part by the institute for Information and Communications Technology Promotion Grant Funded by the Korean Government (MSIP) under Grant B0126-16-1017 and in part by the research professor program of Korea University, Korea.
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Appendices
Appendix A: Proof of Proposition 1
Proof
Let Ω = T k∪Λ α . Then, the projection of y onto span (Φ Ω) can be given by
Following the argument in [17], we can show that
Since P Ω y lies in the column span of \(\boldsymbol {\Phi }_{\Omega } = [\boldsymbol {\Phi }_{T^{k}}, \boldsymbol {\Phi }_{{\Lambda }_{\alpha }}]\) where \(\boldsymbol {\Phi }_{T^{k}}\) and \(\boldsymbol {\Phi }_{{\Lambda }_{\alpha }}\) are disjoint, it has a unique linear representation with \(\boldsymbol {\Phi }_{T^{k}}\) and \(\boldsymbol {\Phi }_{{\Lambda }_{\alpha }}\). That is,
Furthermore, from Table 1, the residual vector of ObMP becomes
which completes the proof. □
Appendix B: Proof of Proposition 2
Proof
First, for the OMP algorithm, \(\| \mathbf {r}^{k} - \mathbf {r}_{0}^{k} \|_{2}\) can be lower bounded as
where (a) is from Lemma 5.
We next consider the ObMP algorithm. \(\| \mathbf {r}_{1}^{k} - \mathbf {r}_{0}^{k} \|_{2}\) can be upper bounded as follows:
where (a) is from Lemma 3, (b) is due to Lemma 4, and (c) is because α∈(0,1].
The proof is now complete. □
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Wang, J., Wang, F., Dong, Y. et al. Oblique Projection Matching Pursuit. Mobile Netw Appl 22, 377–382 (2017). https://doi.org/10.1007/s11036-016-0773-x
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DOI: https://doi.org/10.1007/s11036-016-0773-x