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 $\ell_2$ 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 $\ell_2$.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2016
- DOI:
- 10.48550/arXiv.1609.01361
- arXiv:
- arXiv:1609.01361
- Bibcode:
- 2016arXiv160901361C
- Keywords:
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- Computer Science - Data Structures and Algorithms
- E-Print:
- FOCS 2016