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BIRS Workshop Lecture Videos

No good dimension reduction in the trace class norm Schechtman, Gideon


I'll present a result of Assaf Naor, Gilles Pisier and myself: Let $S_1$ denote the Schatten--von Neumann trace class, i.e., the Banach space of all compact operators $T:\ell_2\to \ell_2$ whose trace class norm $\|T\|_{S_1}=\sum_{j=1}^\infty\sigma_j(T)$ is finite, where $\{\sigma_j(T)\}_{j=1}^\infty$ are the singular values of $T$. We prove that for arbitrarily large $n\in \mathbb{N}$ there exists a subset $S\subset S_1$ with $|S|=n$ that cannot be embedded with bi-Lipschitz distortion $O(1)$ into any $n^{o(1)}$-dimensional linear subspace of $S_1$. This extends a well known result of Brikmann and Charikar (2003) who proved a similar result with $\ell_1$ replacing $S_1$. It stand in sharp dichotomy with the Johnson--Lindenstrauss lemma (1984) which says that the situation in $\ell_2$ is very different.

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