Editing Open Problems:25
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==Update== | ==Update== | ||
− | It has been shown {{cite|AndoniKR-14}} that for '''normed spaces''' the above implication is true: if a snowflake of a normed space does not embed into $\ell_2$ (in fact, more generally, does not ''uniformly embed'' into a Hilbert space), then, there is a non-trivial communication lower bound for distinguishing small and large distances. | + | It has been shown {{cite|AndoniKR-14}} that for '''normed spaces''' the above implication is true: if a snowflake of a normed space does not embed into $\ell_2$ (in fact, more generally, does not ''uniformly embed'' into a Hilbert space), then, there is a non-trivial communication lower bound for distinguishing small and large distances. Furtermore, {{cite|KhotN-19}} show that a similar statement is not true for metrics in general: there exist metrics which are sketchable with constant approximation and space, but do not $O(1)$-embed into $\ell_{1-\epsilon}$ for any $\epsilon<1/2$. |