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No good dimension reduction in the trace class norm Schechtman, Gideon
Description
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.
Item Metadata
Title |
No good dimension reduction in the trace class norm
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2020-02-13T16:55
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Description |
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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Extent |
30.0 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: Weizmann Institute
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Series | |
Date Available |
2020-08-12
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0392701
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International