Improved Algorithms for Learning Fourier-sparse Signals
A classical problem in sparse Fourier transforms, which dates back to the work by Prony in 1795 at least, is to learn a $k$-Fourier-sparse signal $x(t):=\sum_{j=1}^k \alpha_j e^{2 \pi \mathbf{i} f_j t}$ with arbitrary frequencies $f_1,\ldots,f_k$. We study this problem of learning $x(t)$ in a fixed time window $[-T,T]$ under adversarial noise with bounded $\ell_2$ norm, where the frequencies $f_1,\ldots,f_k$ may be"off-grid"-- arbitrarily located in a given bandlimit $[-F,F]$. In particular, our goal is to output a sparse interpolation $\tilde{x}$ such that $\tilde{x}(t) \approx x(t)$ in the time window $[-T,T]$. 1. Our first result shows that the sample complexity of interpolation is $k^2 \cdot O(\log \frac{k FT}{\epsilon})^2$. While its running time is $(\frac{k FT}{\epsilon})^{O(k)}$, this improves the previous upper bound $k^{4} \cdot (\log FT)^{O(1)}$ on the sample complexity substantially and leaves a gap of about $k$ to the lower bound $\Omega(k \log FT)$. 2. Our second result provides efficient algorithms to interpolate $x(t)$. The first algorithm takes $m=k^{3.75} \cdot (\log FT)^{O(1)}$ samples and $m^{\omega+o(1)}$ time ($\omega$ is the matrix multiplication exponent). Assuming that the growth of any $k$-Fourier-sparse signal cannot be significantly larger than the growth of the degree-$(k-1)$ Chebyshev polynomial -- specifically, $x(t) \le e^{k \cdot O\big( \sqrt{\frac{|t|}{T}-1} \big)} \cdot \underset{s \in [-1,1]}{\max} |x(s)|$ for any $t \notin [-T,T]$, the second algorithm further improves the sample complexity to $m'=k^{3} \cdot (\log FT)^{O(1)}$ and the time complexity to $(m')^{\omega+o(1)}$.