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. |