For vector $\boldsymbol{x},\boldsymbol{y} \in \mathcal{R}^{n}$, if \begin{equation} \| \boldsymbol{x} \|_0 = \| \boldsymbol{y} \|_0 \end{equation} What relationship will $\| \boldsymbol{x} \|_1$ and $ \| \boldsymbol{y} \|_1$(or $\| \boldsymbol{x} \|_2$ and $ \| \boldsymbol{y} \|_2$) have?
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$\begingroup$The $L_0$ norm is the number of non-zero elements in a vector. Then it is not strictly a measure of a distance, then you couln't say the equality directly implies a relation between $ \Vert x \Vert _1, \Vert y \Vert _1 $ ... It is more likely to be interpreted as a measure of sparsity, to find the sparsest solution to a set of equations.
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