Signal model Given the generating low-rank component Γ and dictionary Φ, our signal model further depends on six coefficient parameters, e Γ - the energy of the low-rank coefficients, b Γ - defining the decay factor of the low-rank coefficients, S - the sparsity level, b S - defining the decay factor of the sparse coefficients, ρ - the noise level and s m - the maximal signal scale. Given these parameters, we choose a low-rank decay factor c Γ uniformly at random in the interval [1−b Γ ,1]. We set $${v}(\ell) =\sigma _{\ell } c_{\Gamma }^{\ell }$$ for 1≤ℓ≤L, where σ ℓ are iid uniform ± 1 Bernoulli variables, and renormalise the sequence to have norm ∥v∥2=e Γ . Similarly, we choose a decay factor c S for the sparse coefficients uniformly at random in the interval [1−b S ,1]. We set $$x(k) = \sigma _{k} c_{S}^{k}$$ for 1≤k≤S, where σ ℓ are iid uniform ± 1 Bernoulli variables, and renormalise the sequence to have norm ∥x∥2=1−e Γ . Finally, we choose a support set I={i 1…i S } uniformly at random as well as a scaling factor s uniformly at random from the interval [0,s m ] and according to our signal model in (2) set $$y= {s} \cdot \frac {\Gamma {v} + \Phi _{I} x +{r}}{\sqrt {1+\|{r} \|_{2}^{2}}},$$ where r is a Gaussian noise vector with variance ρ 2 if ρ>0.