Lower Bounds on Adaptive Sensing for Matrix Recovery
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| Format: | Preprint |
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2023
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| author | Kacham, Praneeth Woodruff, David P |
| author_facet | Kacham, Praneeth Woodruff, David P |
| contents | We study lower bounds on adaptive sensing algorithms for recovering low rank matrices using linear measurements. Given an $n \times n$ matrix $A$, a general linear measurement $S(A)$, for an $n \times n$ matrix $S$, is just the inner product of $S$ and $A$, each treated as $n^2$-dimensional vectors. By performing as few linear measurements as possible on a rank-$r$ matrix $A$, we hope to construct a matrix $\hat{A}$ that satisfies $\|A - \hat{A}\|_F^2 \le c\|A\|_F^2$, for a small constant $c$. It is commonly assumed that when measuring $A$ with $S$, the response is corrupted with an independent Gaussian random variable of mean $0$ and variance $σ^2$. Candés and Plan study non-adaptive algorithms for low rank matrix recovery using random linear measurements.
At a certain noise level, it is known that their non-adaptive algorithms need to perform $Ω(n^2)$ measurements, which amounts to reading the entire matrix. An important question is whether adaptivity helps in decreasing the overall number of measurements. We show that any adaptive algorithm that uses $k$ linear measurements in each round and outputs an approximation to the underlying matrix with probability $\ge 9/10$ must run for $t = Ω(\log(n^2/k)/\log\log n)$ rounds showing that any adaptive algorithm which uses $n^{2-β}$ linear measurements in each round must run for $Ω(\log n/\log\log n)$ rounds to compute a reconstruction with probability $\ge 9/10$. Hence any adaptive algorithm that has $o(\log n/\log\log n)$ rounds must use an overall $Ω(n^2)$ linear measurements. Our techniques also readily extend to obtain lower bounds on adaptive algorithms for tensor recovery and obtain measurement-vs-rounds trade-off for many sensing problems in numerical linear algebra, such as spectral norm low rank approximation, Frobenius norm low rank approximation, singular vector approximation, and more. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_17281 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lower Bounds on Adaptive Sensing for Matrix Recovery Kacham, Praneeth Woodruff, David P Data Structures and Algorithms Information Theory We study lower bounds on adaptive sensing algorithms for recovering low rank matrices using linear measurements. Given an $n \times n$ matrix $A$, a general linear measurement $S(A)$, for an $n \times n$ matrix $S$, is just the inner product of $S$ and $A$, each treated as $n^2$-dimensional vectors. By performing as few linear measurements as possible on a rank-$r$ matrix $A$, we hope to construct a matrix $\hat{A}$ that satisfies $\|A - \hat{A}\|_F^2 \le c\|A\|_F^2$, for a small constant $c$. It is commonly assumed that when measuring $A$ with $S$, the response is corrupted with an independent Gaussian random variable of mean $0$ and variance $σ^2$. Candés and Plan study non-adaptive algorithms for low rank matrix recovery using random linear measurements. At a certain noise level, it is known that their non-adaptive algorithms need to perform $Ω(n^2)$ measurements, which amounts to reading the entire matrix. An important question is whether adaptivity helps in decreasing the overall number of measurements. We show that any adaptive algorithm that uses $k$ linear measurements in each round and outputs an approximation to the underlying matrix with probability $\ge 9/10$ must run for $t = Ω(\log(n^2/k)/\log\log n)$ rounds showing that any adaptive algorithm which uses $n^{2-β}$ linear measurements in each round must run for $Ω(\log n/\log\log n)$ rounds to compute a reconstruction with probability $\ge 9/10$. Hence any adaptive algorithm that has $o(\log n/\log\log n)$ rounds must use an overall $Ω(n^2)$ linear measurements. Our techniques also readily extend to obtain lower bounds on adaptive algorithms for tensor recovery and obtain measurement-vs-rounds trade-off for many sensing problems in numerical linear algebra, such as spectral norm low rank approximation, Frobenius norm low rank approximation, singular vector approximation, and more. |
| title | Lower Bounds on Adaptive Sensing for Matrix Recovery |
| topic | Data Structures and Algorithms Information Theory |
| url | https://arxiv.org/abs/2311.17281 |