学习多频带信号与傅里叶稀疏信号
Learning Multiband Signals and Fourier-sparse Signals
- University of Science and Technology of China（中国科学技术大学）
- Hefei National Laboratory（合肥国家实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究提出了无需先验频带位置的多频带信号高效恢复与插值算法，并证明k傅里叶稀疏信号存在多频带近似，给出了采样复杂度达统计上界量级的插值算法。
AI中文摘要：
我们研究用于学习多频带信号和傅里叶稀疏信号的高效算法。多频带信号的傅里叶变换的支撑集为有限个区间，即$I_1 \u0026cup; I_2 \u0026cdots; \u0026cup; I_n$。关于多频带信号已有大量研究，其中Avron等人提出了一种采样复杂度近乎最优的高效重构算法。但以往所有多频带信号算法都针对重构问题，即$I_1,\u2026,I_n$的位置是先验已知的。另一方面，尽管学习具有$k$个任意频率的傅里叶稀疏信号的问题最早可追溯至1795年的Prony，但设计高效且鲁棒的学习算法仍是一个开放问题。目前最先进的成果来自Cai等人近期的工作，其高效算法需要$\tilde{O}(k^3)$个样本和$\tilde{O}(k^{3 \u03c9})$的时间，而统计上界为$\tilde{O}(k^2)$个样本。设$[-1,1]$为时间窗，其中噪声满足$\u2113_2$有界。1. 我们提出了一种高效算法，可在$\tilde{O}(n+\u2211_i |I_i|)$个样本和$\tilde{O}(n+\u2211_i |I_i|)$时间内恢复$\u02c6{x}$中各频带$I_1,\u2026,I_n$的位置。此外，将该算法与Avron等人的重构算法结合，可得到一种采样复杂度为$\tilde{O}(n+\u2211_i |I_i|)$的高效插值算法。2. 我们证明，任意$k$傅里叶稀疏信号$x$都存在一个多频带近似$z$，其傅里叶变换的支撑集大小为$|\u02c6{z}|=\tilde{O}(k^2)$。此外，我们提出了一种针对$k$傅里叶稀疏信号的插值算法，需要$\tilde{O}(k^2)$个样本和$\tilde{O}(k^5)$的时间。
英文摘要：
We consider efficient algorithms to learn multiband signals and Fourier-sparse signals. A mutliband signal has a Fourier transform supported by a bounded number of intervals, say $I_1 \cup I_2 \cdots \cup I_n$. There is a long line of research on multiband signals. In particular, Avron et al. showed an efficient reconstructing algorithm whose sample complexity is almost optimal. However, all previous algorithms for multiband signals consider the reconstructing problem in which the locations of $I_1,\ldots,I_n$ are given as a priori knowledge. On the other hand, although the problem of learning Fourier-sparse signals with $k$ arbitrary frequencies dates at least to Prony in 1795, designing efficient and robust learning algorithms is still an open problem. The state-of-the-art is an efficient algorithm of $\tilde{O}(k^3)$ samples and $\tilde{O}(k^{3 ω})$ time from the very recent work by Cai et al., while the statistical upper bound is $\tilde{O}(k^2)$ samples. Let $[-1,1]$ be the time window in which the noise is $\ell_2$ bounded. 1. We show an efficient algorithm to recover the locations of the bands $I_1,\ldots,I_n$ in $\hat{x}$ within $\tilde{O}(n+\sum_i |I_i|)$ samples and $\tilde{O}(n+\sum_i |I_i|)$ time. Furthermore, combining this with the reconstructing algorithm by Avron et al. provides an efficient interpolation algorithm within $\tilde{O}(n+\sum_i |I_i|)$ samples. 2. We show that every $k$-Fourier-sparse signal $x$ admits a multiband approximation $z$ whose Fourier transform is of support size $|\mathrm{supp}(\hat{z})|=\tilde{O}(k^2)$. Furthermore, we show an interpolation algorithm for $k$-Fourier-sparse signals with $\tilde{O}(k^2)$ samples and $\tilde{O}(k^5)$ time.
