Table 4 Joint constrained sparse haplotype frequency estimation algorithm in the presence of noisy measurements

1

Set k = 0

2

Set ρ > 0

3

Set X⁰ = 0, Z⁰ = 0, ${\mathbf{U}}_1^0=0$, ${\mathbf{U}}_2^0=0$

4

U₄ = (I−H^T(I+H H^T)⁻¹H)

5

Repeat

6

  k = k + 1

7

  ${\mathbf{u}}_3^T=\frac{1^T{\mathbf{U}}_4\left({\mathbf{H}}^H\left({\mathbf{U}}_2^k-{\mathbf{Z}}_1^k\right)+{\mathbf{U}}_1-{\mathbf{Z}}_2^k\right)-1^T}{1^T{\mathbf{U}}_{4}1}$

8

  ${\mathbf{X}}^{k+1}={\mathbf{U}}_4\left({\mathbf{U}}_1^k-{\mathbf{Z}}_1^k+{\mathbf{H}}^T({\mathbf{U}}_2^k-{\mathbf{Z}}_2^k)-1{\mathbf{u}}_3^T\right)$

9

  ${\mathbf{Z}}_1^{k+1}=max\left(\text{Shrink}\left({\mathbf{X}}^{k+1}+\frac{1}{\rho }{\mathbf{U}}_1^k,\frac{1}{\rho }\right),0\right)$

10

  ${\mathbf{Z}}_2^{k+1}=\text{proj}(\mathbf{H},{\mathbf{X}}^{k+1},{\mathbf{U}}_2^k,\mathbf{A},\delta )$

11

  ${\mathbf{U}}_1^{k+1}={\mathbf{U}}_1^k+{\mathbf{X}}^{k+1}-{\mathbf{Z}}_1^{k+1}$

12

  ${\mathbf{U}}_2^{k+1}={\mathbf{U}}_2^k+\mathbf{H}{\mathbf{X}}^{k+1}-{\mathbf{Z}}_2^{k+1}$

13

until convergence