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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.06154 |
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| _version_ | 1866916241525964800 |
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| author | Bandeira, Afonso S. Mixon, Dustin G. Steinerberger, Stefan |
| author_facet | Bandeira, Afonso S. Mixon, Dustin G. Steinerberger, Stefan |
| contents | We prove the existence of a positive semidefinite matrix $A \in \mathbb{R}^{n \times n}$ such that any decomposition into rank-1 matrices has to have factors with a large $\ell^1-$norm, more precisely $$ \sum_{k} x_k x_k^*=A \quad \implies \quad \sum_k \|x_k\|^2_{1} \geq c \sqrt{n} \|A\|_{1},$$ where $c$ is independent of $n$. This provides a lower bound for the Balan--Jiang matrix problem. The construction is probabilistic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_06154 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A lower bound for the Balan--Jiang matrix problem Bandeira, Afonso S. Mixon, Dustin G. Steinerberger, Stefan Functional Analysis We prove the existence of a positive semidefinite matrix $A \in \mathbb{R}^{n \times n}$ such that any decomposition into rank-1 matrices has to have factors with a large $\ell^1-$norm, more precisely $$ \sum_{k} x_k x_k^*=A \quad \implies \quad \sum_k \|x_k\|^2_{1} \geq c \sqrt{n} \|A\|_{1},$$ where $c$ is independent of $n$. This provides a lower bound for the Balan--Jiang matrix problem. The construction is probabilistic. |
| title | A lower bound for the Balan--Jiang matrix problem |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2405.06154 |