\[y(\mathbf{x})=w_{0}+\sum_{i=1}^{n} w_{i} x_{i}+\sum_{i=1}^{n} \sum_{j=i+1}^{n}\left\langle\mathbf{v}_{i, f_{j}}, \mathbf{v}_{j, f_{i}}\right\rangle x_{i} x_{j} \]
\[\phi(\mathbf{w}, \mathbf{x})=\sum_{j_{1}, j_{2} \in \mathcal{C}_{2}}\left\langle\mathbf{w}_{j_{1}, f_{2}}, \mathbf{w}_{j_{2}, f_{1}}\right\rangle x_{j_{1}} x_{j_{2}} \]
\[\min _{\mathbf{w}} \sum_{i=1}^{L} \log \left(1+\exp \left\{-y_{i} \phi\left(\mathbf{w}, \mathbf{x}_{i}\right)\right\}\right)+\frac{\lambda}{2}\|\mathbf{w}\|^{2} \]