Sublinear Time Low-Rank Approximation of Toeplitz Matrices
ACM-SIAM Symposium on Discrete Algorithms(2024)
摘要
We present a sublinear time algorithm for computing a near optimal low-rank
approximation to any positive semidefinite (PSD) Toeplitz matrix T∈ℝ^d× d, given noisy access to its entries. In particular,
given entrywise query access to T+E for an arbitrary noise matrix E∈ℝ^d× d, integer rank k≤ d, and error parameter
δ>0, our algorithm runs in time poly(k,log(d/δ)) and
outputs (in factored form) a Toeplitz matrix T∈ℝ^d
× d with rank poly(k,log(d/δ)) satisfying, for some fixed
constant C,
T-T_F ≤ C ·max{E_F,T-T_k_F} + δ·T_F.
Here ·_F is the Frobenius norm and T_k is
the best (not necessarily Toeplitz) rank-k approximation to T in the
Frobenius norm, given by projecting T onto its top k eigenvectors.
Our result has the following applications. When E = 0, we obtain the first
sublinear time near-relative-error low-rank approximation algorithm for PSD
Toeplitz matrices, resolving the main open problem of Kapralov et al. SODA `23,
whose algorithm had sublinear query complexity but exponential runtime. Our
algorithm can also be applied to approximate the unknown Toeplitz covariance
matrix of a multivariate Gaussian distribution, given sample access to this
distribution, resolving an open question of Eldar et al. SODA `20.
Our algorithm applies sparse Fourier transform techniques to recover a
low-rank Toeplitz matrix using its Fourier structure. Our key technical
contribution is the first polynomial time algorithm for discrete time
off-grid sparse Fourier recovery, which may be of independent interest.
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