  Kernel dimensionality reduction (KDR) algorithms find a low dimensional
representation of the original data by optimizing kernel dependency measures
that are capable of capturing nonlinear relationships. The standard strategy is
to first map the data into a high dimensional feature space using kernels prior
to a projection onto a low dimensional space. While KDR methods can be easily
solved by keeping the most dominant eigenvectors of the kernel matrix, its
features are no longer easy to interpret. Alternatively, Interpretable KDR
(IKDR) is different in that it projects onto a subspace \textit{before} the
kernel feature mapping, therefore, the projection matrix can indicate how the
original features linearly combine to form the new features. Unfortunately, the
IKDR objective requires a non-convex manifold optimization that is difficult to
solve and can no longer be solved by eigendecomposition. Recently, an efficient
iterative spectral (eigendecomposition) method (ISM) has been proposed for this
objective in the context of alternative clustering. However, ISM only provides
theoretical guarantees for the Gaussian kernel. This greatly constrains ISM's
usage since any kernel method using ISM is now limited to a single kernel. This
work extends the theoretical guarantees of ISM to an entire family of kernels,
thereby empowering ISM to solve any kernel method of the same objective. In
identifying this family, we prove that each kernel within the family has a
surrogate $\Phi$ matrix and the optimal projection is formed by its most
dominant eigenvectors. With this extension, we establish how a wide range of
IKDR applications across different learning paradigms can be solved by ISM. To
support reproducible results, the source code is made publicly available on
\url{https://github.com/chieh-neu/ISM_supervised_DR}.
