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Table 1 Signal model

From: Fast dictionary learning from incomplete data

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, where σ are iid uniform ± 1 Bernoulli variables, and renormalise the sequence to have norm v2=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≤kS, where σ are iid uniform ± 1 Bernoulli variables, and renormalise the sequence to have norm x2=1−e Γ . Finally, we choose a support set I={i 1i 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.