AI 中文总结
本文针对受对抗噪声干扰的非网格k-傅里叶稀疏信号学习问题，提出改进的插值算法，大幅降低了样本复杂度，还基于切比雪夫多项式增长假设优化了算法的样本与时间复杂度。
AI 中文摘要
稀疏傅里叶变换中的一个经典问题至少可以追溯到1795年Prony的工作，即学习一个k-傅里叶稀疏信号x(t):=∑_{j=1}^k α_j e^{2πi f_j t}，其中f_1,…,f_k为任意频率。我们研究在固定时间窗口[-T,T]、受有界ℓ₂范数的对抗噪声干扰下学习x(t)的问题，其中频率f_1,…,f_k可能是“非网格”的——任意位于给定带宽[-F,F]内。特别地，我们的目标是输出一个稀疏插值结果x̃，使得x̃(t)在时间窗口[-T,T]内近似于x(t)。1. 我们的第一个结果表明，插值的样本复杂度为k²·O(log(kFT/ε))²，虽然其运行时间为(kFT/ε)^(O(k))，但这大幅改进了之前样本复杂度的上界k⁴·(log FT)^(O(1))，且与下界Ω(k log FT)存在约k的差距。2. 我们的第二个结果提供了插值x(t)的高效算法。第一个算法需要m=k^3.75·(log FT)^(O(1))个样本，运行时间为m^(ω+o(1))（ω为矩阵乘法指数）。假设任何k-傅里叶稀疏信号的增长幅度不会显著超过次数为(k-1)的切比雪夫多项式的增长幅度——具体而言，对于任何t∉[-T,T]，有|x(t)|≤e^(k·O(√(|t|/T -1)))·max_{s∈[-1,1]}|x(s)|——第二个算法进一步将样本复杂度改进为m'=k³·(log FT)^(O(1))，时间复杂度改进为(m')^(ω+o(1))。
英文摘要
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 $x(t):=\sum_{j=1}^k α_j e^{2 π\mathbf{i} f_j t}$ 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 $k^2 \cdot O(\log \frac{k FT}ε)^2$. While its running time is $(\frac{k FT}ε)^{O(k)}$, 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 $Ω(k \log FT)$. 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 $m^{ω+o(1)}$ time ($ω$ 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 $(m')^{ω+o(1)}$.
