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- W4309802323 abstract "We present a sublinear query algorithm for outputting a near-optimal low-rank approximation to any positive semidefinite Toeplitz matrix $T in mathbb{R}^{d times d}$. In particular, for any integer rank $k leq d$ and $epsilon,delta > 0$, our algorithm makes $tilde{O} left (k^2 cdot log(1/delta) cdot text{poly}(1/epsilon) right )$ queries to the entries of $T$ and outputs a rank $tilde{O} left (k cdot log(1/delta)/epsilonright )$ matrix $tilde{T} in mathbb{R}^{d times d}$ such that $| T - tilde{T}|_F leq (1+epsilon) cdot |T-T_k|_F + delta |T|_F$. Here, $|cdot|_F$ is the Frobenius norm and $T_k$ is the optimal rank-$k$ approximation to $T$, given by projection onto its top $k$ eigenvectors. $tilde{O}(cdot)$ hides $text{polylog}(d) $ factors. Our algorithm is emph{structure-preserving}, in that the approximation $tilde{T}$ is also Toeplitz. A key technical contribution is a proof that any positive semidefinite Toeplitz matrix in fact has a near-optimal low-rank approximation which is itself Toeplitz. Surprisingly, this basic existence result was not previously known. Building on this result, along with the well-established off-grid Fourier structure of Toeplitz matrices [Cybenko'82], we show that Toeplitz $tilde{T}$ with near optimal error can be recovered with a small number of random queries via a leverage-score-based off-grid sparse Fourier sampling scheme." @default.
- W4309802323 created "2022-11-29" @default.
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- W4309802323 date "2022-11-21" @default.
- W4309802323 modified "2023-09-26" @default.
- W4309802323 title "Toeplitz Low-Rank Approximation with Sublinear Query Complexity" @default.
- W4309802323 doi "https://doi.org/10.48550/arxiv.2211.11328" @default.
- W4309802323 hasPublicationYear "2022" @default.
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