Sample Complexity of Low-rank Tensor Recovery from Uniformly Random Entries
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913460441317376 |
|---|---|
| author | Hamaguchi, Hiroki Tanigawa, Shin-ichi |
| author_facet | Hamaguchi, Hiroki Tanigawa, Shin-ichi |
| contents | We show that a generic tensor $T\in \mathbb{F}^{n\times n\times \dots\times n}$ of order $k$ and CP rank $d$ can be uniquely recovered from $n\log n+dn\log \log n +o(n\log \log n) $ uniformly random entries with high probability if $d$ and $k$ are constant and $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$. The bound is tight up to the coefficient of the second leading term and improves on the existing $O(n^{\frac{k}{2}}{\rm polylog}(n))$ upper bound for order $k$ tensors. The bound is obtained by showing that the projection of the Segre variety to a random axis-parallel linear subspace preserves $d$-identifiability with high probability if the dimension of the subspace is $n\log n+dn\log \log n +o(n\log \log n) $ and $n$ is sufficiently large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03504 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sample Complexity of Low-rank Tensor Recovery from Uniformly Random Entries Hamaguchi, Hiroki Tanigawa, Shin-ichi Combinatorics Information Theory Algebraic Geometry Probability We show that a generic tensor $T\in \mathbb{F}^{n\times n\times \dots\times n}$ of order $k$ and CP rank $d$ can be uniquely recovered from $n\log n+dn\log \log n +o(n\log \log n) $ uniformly random entries with high probability if $d$ and $k$ are constant and $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$. The bound is tight up to the coefficient of the second leading term and improves on the existing $O(n^{\frac{k}{2}}{\rm polylog}(n))$ upper bound for order $k$ tensors. The bound is obtained by showing that the projection of the Segre variety to a random axis-parallel linear subspace preserves $d$-identifiability with high probability if the dimension of the subspace is $n\log n+dn\log \log n +o(n\log \log n) $ and $n$ is sufficiently large. |
| title | Sample Complexity of Low-rank Tensor Recovery from Uniformly Random Entries |
| topic | Combinatorics Information Theory Algebraic Geometry Probability |
| url | https://arxiv.org/abs/2408.03504 |