Machine learning methods in the computational biology of cancer

Proc. R. Soc. A 470, 20140081 (8 July 2014; Published online 23 April 2014) (doi:10.1098/rspa.2014.0081)

There are a couple of unfortunate errors in equations (4.2) and (4.3) of the article. Specifically, the ${\ell }_{\mathrm{\infty }}$-norm in both equations should be replaced by the ℓ₁-norm. The correct versions of these equations are as follows:

$$\left.\begin{array}{rl}& \underset{w,\theta ,y,z}{min}(1-\lambda )\hspace{0.17em}\left[\sum _{i=1}^{m_1}{y}_i+\sum _{i=1}^{m_2}{z}_i\right]+\lambda \sum _{i=1}^n|w_i|\text{s.t. }\\ & x^{i}w-\theta +y_i\ge 1\hspace{0.17em}\mathrm{\forall }i\in {\mathcal{M}}_1,x^{i}w-\theta -z_i\le -1\hspace{0.17em}\mathrm{\forall }i\in {\mathcal{M}}_2,\\ & y\ge {\mathbf{\text{0}}}_{m_1},z\ge {\mathbf{\text{0}}}_{m_2}.\end{array}\right\}$$ 4.2

This can be converted to

$$\left.\begin{array}{rl}& \underset{w,\theta ,y,z}{min}(1-\lambda )\hspace{0.17em}\left[\sum _{i=1}^{m_1}{y}_i+\sum _{i=1}^{m_2}{z}_i\right]+\lambda \sum _{i=1}^n{v}_i\\ & x^{i}w-\theta +y_i\ge 1\hspace{0.17em}\mathrm{\forall }i\in {\mathcal{M}}_1,x^{i}w-\theta -z_i\le -1\hspace{0.17em}\mathrm{\forall }i\in {\mathcal{M}}_2,\\ & y\ge {\mathbf{\text{0}}}_{m_1},\hspace{0.17em}z\ge {\mathbf{\text{0}}}_{m_2},v_i\ge w_i\hspace{0.17em}\mathrm{\forall }i,\hspace{0.17em}{v}_i\ge -w_i\hspace{0.17em}\mathrm{\forall }i.\end{array}\right\}$$ 4.3