On the structure of low-rank matrices that approximate the identity matrix
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910743104847872 |
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| author | Malykhin, Yuri |
| author_facet | Malykhin, Yuri |
| contents | Consider a matrix $A$ of rank $n$ that approximates the $N\times N$ identity matrix with elementwise error at most $1/3$. We give a lower bound on the number of elements s.t. $|A_{i,j}|>γ$, for a certain threshold. Two corollaries are obtained.
1. If $n \le K\log N$ with some $K$, then at least $c(K)N^2$ elements satisfy $|A_{i,j}|>c(K)n^{-1/2}$. This answers a question of B.S. Kashin.
2. The number of nonzero elements in $A$ is at least $c\log(N)/(n\log(2+n/\log N))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09302 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the structure of low-rank matrices that approximate the identity matrix Malykhin, Yuri Functional Analysis Metric Geometry Consider a matrix $A$ of rank $n$ that approximates the $N\times N$ identity matrix with elementwise error at most $1/3$. We give a lower bound on the number of elements s.t. $|A_{i,j}|>γ$, for a certain threshold. Two corollaries are obtained. 1. If $n \le K\log N$ with some $K$, then at least $c(K)N^2$ elements satisfy $|A_{i,j}|>c(K)n^{-1/2}$. This answers a question of B.S. Kashin. 2. The number of nonzero elements in $A$ is at least $c\log(N)/(n\log(2+n/\log N))$. |
| title | On the structure of low-rank matrices that approximate the identity matrix |
| topic | Functional Analysis Metric Geometry |
| url | https://arxiv.org/abs/2412.09302 |