Here in this paper: https://ideas.repec.org/a/eee/ejores/v292y2021i3p1004-1018.html ,
the author has written that $\phi (\cdot )$ represents the feature map from input space $\mathcal{X}$ to the high dimensional feature space $\mathcal{H}$, and the associated inner product can be calculated by evaluating its associated kernel function $K(u, v) = \langle \phi (u),\phi (v)\rangle $ and $u\in \mathcal{D}$ which is the dataset. After that he introduced this kernel function: $K_m (u,v)=1-|\frac{q_m^T (u-v)}{c_m \cdot \kappa}|$. My question is that what is the feature map $\phi_m (\cdot )$? He has used $\phi_m$ in his model but how can I obtain $\phi_m$ from $K_m$ (feature mp from kernel function here in this special case)?
Determining the actual feature map is, in general, impossible. The theory surrounding reproducing kernel hilbert spaces (RKHS) guarantees that such a feature map exists; however, there's not necessarily an easy way to determine it. For additional info see Kernel Method.