In this paper, we study the orthogonal least squares (OLS) algorithm for sparse recovery. On the one hand, we show that if the sampling matrix $\mathbf{A}$ satisfies the restricted isometry property (RIP) of order $K + 1$ with isometry constant $$ \delta_{K + 1} < \frac{1}{\sqrt{K+1}}, $$ then OLS exactly recovers the support of any $K$-sparse vector $\mathbf{x}$ from its samples $\mathbf{y} = \mathbf{A} \mathbf{x}$ in $K$ iterations. On the other hand, we show that OLS may not be able to recover the support of a $K$-sparse vector $\mathbf{x}$ in $K$ iterations for some $K$ if $$ \delta_{K + 1} \geq \frac{1}{\sqrt{K+\frac{1}{4}}}. $$Comment: To appear in IEEE Transactions on Signal Processin
The Korkine-Zolotareff (KZ) reduction is one of the often used reduction strategies for lattice decoding. In this paper, we first investigate some important properties of KZ reduced matrices. Specifically, we present a linear upper bound on the Hermit constant which is around 7 8 times of the existing sharpest linear upper bound, and an upper bound on the KZ constant which is polynomially smaller than the existing sharpest one. We also propose upper bounds on the lengths of the columns of KZ reduced matrices, and an upper bound on the orthogonality defect of KZ reduced matrices which are even polynomially and exponentially smaller than those of boosted KZ reduced matrices, respectively. Then, we derive upper bounds on the magnitudes of the entries of any solution of a shortest vector problem (SVP) when its basis matrix is LLL reduced. These upper bounds are useful for analyzing the complexity and understanding numerical stability of the basis expansion in a KZ reduction algorithm. Finally, we propose a new KZ reduction algorithm by modifying the commonly used Schnorr-Euchner search strategy for solving SVPs and the basis expansion method proposed by Zhang et al.Simulation results show that the new KZ reduction algorithm is much faster and more numerically reliable than the KZ reduction algorithm proposed by Zhang et al., especially when the basis matrix is ill conditioned.
In many applications including communications, one may encounter a linear model where the parameter vectorx is an integer vector in a box. To estimatex, a typical method is to solve a box-constrained integer least squares (BILS) problem. However, due to its high complexity, the box-constrained Babai integer point x BB is commonly used as a suboptimal solution. In this paper, we first derive formulas for the success probability P BB of x BB and the success probability P OB of the ordinary Babai integer point x OB whenx is uniformly distributed over the constraint box. Some properties of P BB and P OB and the relationship between them are studied. Then, we investigate the effects of some column permutation strategies on P BB . In addition to V-BLAST and SQRD, we also consider the permutation strategy involved in the LLL lattice reduction, to be referred to as LLL-P. On the one hand, we show that when the noise is relatively small, LLL-P always increases P BB and argue why both V-BLAST and SQRD often increase P BB ; and on the other hand, we show that when the noise is relatively large, LLL-P always decreases P BB and argue why both V-BLAST and SQRD often decrease P BB . We also derive a column permutation invariant bound on P BB , which is an upper bound and a lower bound under these two opposite conditions, respectively. Numerical results demonstrate our findings. Finally, we consider a conjecture concerning x OB proposed by Ma et al. We first construct an example to show that the conjecture does not hold in general, and then show that it does hold under some conditions. Index Terms-Box-constrained integer least squares estimation, Babai integer point, success probability, column permutations, LLL-P, SQRD, V-BLAST.
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