Fourier-Sparse Interpolation without a Frequency Gap

Abstract

We consider the problem of estimating a Fourier-sparse signal from noisy samples, where the sampling is done over some interval [0, T ] and the frequencies can be "off-grid". Previous methods for this problem required the gap between frequencies to be above 1/T , the threshold required to robustly identify individual frequencies. We show the frequency gap is not necessary to estimate the signal as a whole: for arbitrary k-Fourier-sparse signals under 2 bounded noise, we show how to estimate the signal with a constant factor growth of the noise and sample complexity polynomial in k and logarithmic in the bandwidth and signal-to-noise ratio.As a special case, we get an algorithm to interpolate degree d polynomials from noisy measurements, using O(d) samples and increasing the noise by a constant factor in 2 .

Sample Efficient Toeplitz Covariance Estimation

“…For continuous Fourier operators, a bound similar to (37) was proven in [CP18] and improved in [AKM + 19] using a different technique, which we adapt. We prove (38) using techniques from [CKPS16] which establishes a similar bound for continuous operators. These results were very recently improved by an s factor in [CP19], so it might also be possible to improve our bound as well.…”

“…Beginning with the results of [CKPS16], bounds like Lemma C.1, on the other hand, can be used to establish similar results for off-grid Fourier matrices. In contrast to earlier work on "offgrid" sparse recovery problems (e.g., [TBSR13, CFG14, TBR15, BCG + 15]), results based on such bounds require no assumptions on the frequencies in S, including no "separation" assumption that |f i − f j | is not too small for all i, j.…”

“…Most theoretical bounds for "off-grid" sparse recovery require strong separation assumptions that no frequencies are too close together [TBSR13, CFG14, TBR15, BCG + 15]. A recent exception is [CKPS16], which we build on in this work since a lack of separation assumptions is essential to obtaining bounds for general low-rank Toeplitz matrices.…”

“…All of our estimators run in polynomial time except for Algorithm 3, which is used to obtain the bound of Theorem 2.10. An interesting open question is if one can use techniques from e.g., [CKPS16] to obtain polynomial time estimators for this setting.…”

“…In contrast to earlier work on "offgrid" sparse recovery problems (e.g., [TBSR13, CFG14, TBR15, BCG + 15]), results based on such bounds require no assumptions on the frequencies in S, including no "separation" assumption that |f i − f j | is not too small for all i, j. We refer the reader to [CKPS16] and [PS15] for a more in-depth discussion of this issue. In the context of our work, Lemma C.1 allows us to avoid a separation assumption on the frequencies in T 's Vandermonde decomposition, which in necessary in some related work [CCG15].…”