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Table 1 JAMMIT optimization algorithm

From: Joint analysis of multiple high-dimensional data types using sparse matrix approximations of rank-1 with applications to ovarian and liver cancer

1. Let \( \mathfrak{D}=\left\{{D}_1,{D}_2,\cdots, {D}_K\right\} \) be a MMDS
2. Form pre-processed super-matrix \( D= stack\left(\mathfrak{D}\right) \).
3. Compute best rank-1 approximation, \( \left({u}_0,{v}_0\right) \) of D such that \( D\approx {u}_0{v}_0^T \).
4. Compute 1 -regularization \( {u}_1 \) of \( {u}_0 \): \( {u}_1=\underset{u}{ \arg \min}\left({\left\Vert D-u{v}_0^T\right\Vert}_2^2+\lambda {\left\Vert u\right\Vert}_1\right) \).
5. Compute \( {v}_1={D}^T{u}_1 \) to obtain solution \( \left({u}_1,{v}_1\right) \).
6. Assign \( {u}_0\leftarrow {u}_1 \) and \( {v}_0\leftarrow {v}_1 \).
7. Repeat steps 4–6 until convergence to final solution \( \left(u,v\right) \) where \( v={D}^Tu \).
8. Form MMSIG \( \zeta \) composed of variables selected by the non-zero entries of \( u \).
9. Parse \( \zeta \) according to stacking order of the \( {D}_k \) in \( D \) to obtain \( {\zeta}_k \) for each \( {D}_k \).
10. Parse \( u \) according to stacking order of \( {D}_k \) in \( D \) to obtain \( {u}_k \) for each \( {D}_k \).
11. Compute sequence of sparse rank-1 approximations \( \widehat{D}=\left\{{\widehat{D}}_1,{\widehat{D}}_2,\cdots, {\widehat{D}}_K\right\} \) where \( {\widehat{D}}_k\approx {u}_k{v}^T \) for \( k=1,2,\cdots, K \).
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