证书治理的CRT稀疏FFT：显式解码模型下验证的全局标签候选构建

中文摘要 AI 辅助

我们提出了一种确定性的基于CRT的稀疏傅里叶架构，用于在工程化变换长度下精确恢复无噪声、网格上、至多k稀疏的频谱。变换长度N被选为小的两两互质阶段素数之精确乘积，使得每个阶段能够使用规模随稀疏度而非环境维度扩展的变换。三次互素增量时间移位提供了一种单边单例一致性筛选：每个真正的单例通过，而通过的碰撞箱仍为临时候选。每个候选箱通过总标签发射规则映射至至多一个全局频率标签，并且各阶段的标签集相交，为每个输入产生确定性的O(k)候选界。候选构建使用O(k log N)个样本。在具有精确算术和精确相位评估但不具有单位成本根索引解码的比较实数RAM模型中，它需要O(k log^2 N / log k)次算术运算；在更强的根索引预言机模型中，算术成本为O(k log N)。在所有阶段真正的单例存活下，候选集包含真实支撑。然而，精确性并不依赖于该存活条件：连续样本残差验证器仅在精确时才接受稀疏重建，所有失败均路由至密集FFT。因此，所得到的混合算法在所述无噪声、网格上、至多k稀疏模型中为每个输入返回精确频谱，最坏情况运行时间为O(N log N)。验证的稀疏路径在根索引预言机模型中额外产生O(k^2)的验证器成本。两种计算模型均非比特复杂度模型。

英文摘要

We present a deterministic CRT-based sparse Fourier architecture for exact recovery of noiseless, on-grid, at-most-\(k\)-sparse spectra with an engineered transform length. The transform length \(N\) is selected as an exact product of small pairwise-coprime stage primes, allowing each stage to use transforms whose size scales with the sparsity rather than the ambient dimension. Three co-prime-increment time shifts provide a one-sided singleton-consistency screen: every genuine singleton passes, while passing collision bins remain provisional candidates. Each candidate bin is mapped by a total label-emission rule to at most one global frequency label, and per-stage label sets are intersected, yielding a deterministic \(O(k)\) candidate bound for every input. Candidate construction uses \(O(k\log N)\) samples. In a comparison real-RAM model with exact arithmetic and exact phase evaluation but without unit-cost root-index decoding, it requires \(O(k\log^2 N/\log k)\) arithmetic operations; in a stronger root-index-oracle model the arithmetic cost is \(O(k\log N)\). Under all-stage genuine-singleton survival, the candidate set contains the true support. Exactness, however, does not depend on that survival condition: a consecutive-sample residual verifier accepts a sparse reconstruction only when it is exact, and all failures route to a dense FFT. The resulting hybrid algorithm therefore returns the exact spectrum for every input in the stated noiseless, on-grid, at-most-\(k\)-sparse model, with \(O(N\log N)\) worst-case runtime. The verified sparse path additionally incurs an \(O(k^2)\) verifier cost in the root-index-oracle model. Neither computational model is a bit-complexity model.

发表机构

• SparseTech

机构由 AI 辅助整理，请以论文原文为准。

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42 pages, 1 figure, 6 tables