Theoretical Validation of the Latent Optimally Partitioned-${\ell }_2/{\ell }_1$ Penalty with Application to Angular Power Spectrum Estimation

Abstract

This paper demonstrates that, in both theory and practice, the latent optimally partitioned (LOP)-${\ell }_2/{\ell }_1$ penalty is effective for exploiting block-sparsity without knowledge of the concrete block structure. More precisely, we first present a novel theoretical result showing that the optimized block partition in the LOP-${\ell }_2/{\ell }_1$ penalty satisfies a condition required for accurate recovery of block-sparse signals. Motivated by this result, we present a new application of the LOP-${\ell }_2/{\ell }_1$ penalty to estimation of angular power spectrum, which is block-sparse with unknown block partition, in MIMO communication systems. Numerical simulations show that the proposed use of block-sparsity with the LOP-${\ell }_2/{\ell }_1$ penalty significantly improves the estimation accuracy of the angular power spectrum.