A totally unimodular view of structured sparsity

In this section, we compare the performance of minimizing the TU relaxation g∗∗ G,G of the proposed Sparse G-group cover (c.f., Section 5.4) in problem (2), which we will call Sparse latent group lasso (SLGL), with Basis pursuit (BP) and Sparse group Lasso (SGL). Recall the SGL criteria is (1 − α) ∑ G∈G √ |G|‖xG‖q + α‖xG‖1, with q = 2 in [3]. We compare also against SGL∞ where we set q = ∞, which is better suited for signals with equal valued non-zero coefficients. We generate a sparse signal x in dimensions p = 200, covered by G = 5 groups, randomly chosen from the M = 29 groups. The groups generated are interval groups, of equal size of 10 coefficients, and with an overlap of 3 coefficients between each two consecutive groups. The true signal x has 3 non-zero coefficients (all set to one) in each of its 5 active groups (cf., Figure 2). Note that these groups lead a TU group structure G, so the TU relaxation in this case is tight. We recover x from its compressive measurements y = Ax +w, where the noise w is a random Gaussian vector of variance σ = 0.01 and A is a random column normalized Gaussian matrix. We encode the data via ‖y − Ax‖2 ≤ ‖w‖2 using the true `2-norm of the noise. We produce the data randomly 10 times and report the averaged results.