Is it possible to compute a distance between two matrices of different rank, and different dimension?
In particular I'm interested in the following case. Suppose $[K]_{ij}=\exp[-(x_i-x_j)^2]$, and let $[\tilde{K}]_{mn} = \vec{k}_m^T\vec{k}_n$, where,
$[k_m^T]_i=\exp[-(x_i-\mu_m)^2]$,
and
$\min_i \{x_i\} \le \mu_1 < \cdots < \mu_{N_s} \le \max_i\{x_i\}$
I'm interested in cases where $N_s=$rank($\tilde{K}$) $<$ rank$(K)=N$.