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For real matrix $X$ where $d_{i,j}^2(X)$ indicates the euclidean distance squared between the rows $i,j$ of $X$, if $d_{i,j}^2(X)=||f(X_i.)-f(X_j.)||_H$ then what would the function $f(.)$ be? Is there only one such function? Note that $X_i.$ indicates the row $i$ of $X$ and the norm is the hilbert norm.

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You can use the versors. Given that $X \in \mathbb{R}^{N \times M}$, then you can use the versors in $\mathbb{R}^N$ which are $$ e_i = \left [0 ~ 0 \cdot \cdot \cdot 0 ~ 1 ~0 \cdot \cdot \cdot 0 ~ 0 \right]^T$$ where the only "1" is at position $i$. Then, your $f$ is a $f_i$ defined as follows: $$ f_i(X) = e_i^T X $$ In this way you obtain your $i$-th row vector.

Finally, you have that $d_{ij}^2(X) = \|f_i(X) - f_j(X)\|_H^2$

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