Toeplitz Low-Rank Approximation with Sublinear Query Complexity
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We present a sublinear query algorithm for outputting a near-optimal low-rank approximation to any positive semidefinite Toeplitz matrix T ∈ℝ^d × d. In particular, for any integer rank k ≤ d and ϵ,δ > 0, our algorithm makes Õ (k^2 ·log(1/δ) ·poly(1/ϵ) ) queries to the entries of T and outputs a rank Õ (k ·log(1/δ)/ϵ ) matrix T̃∈ℝ^d × d such that T - T̃_F ≤ (1+ϵ) ·T-T_k_F + δT_F. Here, ·_F is the Frobenius norm and T_k is the optimal rank-k approximation to T, given by projection onto its top k eigenvectors. Õ(·) hides polylog(d) factors. Our algorithm is structure-preserving, in that the approximation T̃ 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 T̃ 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.
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