Table 1 Signal model

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.