遍历FFT计算树求解与时间无关的稀疏傅里叶变换
Traversing the FFT Computation Tree for Dimension-Independent Sparse Fourier Transforms
摘要：
我们认为，研究稀疏傅立叶变换问题，其中一个目的是快速恢复近似傅立叶$k$ -稀疏向量$\widehat{x} \in \mathbb{C}^{n^d}$从观察其时域表示$x$。在精确的$k$ -稀疏情况下，最好的已知的维度无关算法在$k$中以接近立方时间运行，并且不清楚是否可能有更快的算法，如在低维度中。除此之外，所有已知的方法要么遭受对维度$d$的指数依赖性，要么只能容忍微不足道的噪声量。这与Cooley和Tukey的经典FFT形成鲜明对比，后者是稳定的，对输入向量的维数完全不敏感：它的运行时间是$O(N\log N)$，在任何维数$d$中，对于$N=n^d$。我们的工作旨在解决上述问题。首先，我们提供了一个翻译/减少正是$k$稀疏FT问题的具体树探索任务，要求恢复$k$叶在一个完整的二叉树在一定的探索规则。随后，我们提供了（a）一个几乎二次的$k$时间算法的这项任务，和（b）证据表明，通过这种方法的稀疏FT的强次二次时间可能是不可能的。我们通过证明稀疏多项式多点评估（经典的非等间距稀疏FT）的条件二次时间下限来实现后者，这是上述翻译中的核心例程。因此，我们的结果相结合，可以看作是一个几乎完整的理解这种方法，这是唯一已知的方法，产生次线性时间维无关的稀疏FT算法。随后，我们提供了我们的算法的鲁棒性，产生一个强大的立方时间算法下有界$\ell_2$噪声。这就需要证明新的结构特性，最近推出的自适应混叠滤波器结合各种新的技术和思想。
英文摘要：
We consider the well-studied Sparse Fourier transform problem, where one aims to quickly recover an approximately Fourier $k$-sparse vector $\widehat{x} \in \mathbb{C}^{n^d}$ from observing its time domain representation $x$. In the exact $k$-sparse case the best known dimension-independent algorithm runs in near cubic time in $k$ and it is unclear whether a faster algorithm like in low dimensions is possible. Beyond that, all known approaches either suffer from an exponential dependence on the dimension $d$ or can only tolerate a trivial amount of noise. This is in sharp contrast with the classical FFT of Cooley and Tukey, which is stable and completely insensitive to the dimension of the input vector: its runtime is $O(N\log N)$ in any dimension $d$ for $N=n^d$. Our work aims to address the above issues. First, we provide a translation/reduction of the exactly $k$-sparse FT problem to a concrete tree exploration task which asks to recover $k$ leaves in a full binary tree under certain exploration rules. Subsequently, we provide (a) an almost quadratic in $k$ time algorithm for this task, and (b) evidence that a strongly subquadratic time for Sparse FT via this approach is likely impossible. We achieve the latter by proving a conditional quadratic time lower bound on sparse polynomial multipoint evaluation (the classical non-equispaced sparse FT) which is a core routine in the aforementioned translation. Thus, our results combined can be viewed as an almost complete understanding of this approach, which is the only known approach that yields sublinear time dimension-independent Sparse FT algorithms. Subsequently, we provide a robustification of our algorithm, yielding a robust cubic time algorithm under bounded $\ell_2$ noise. This requires proving new structural properties of the recently introduced adaptive aliasing filters combined with a variety of new techniques and ideas.
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