单仓傅里叶挑战：用于无维度重建认证
One-Bin Fourier Challenges for Dimension-Free Reconstruction Certification
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中文总结 AI 辅助
针对部分傅里叶重建中输出不可信的问题，提出基于秘密相位掩码和随机残差检查的提交后验收测试，实现误差的统计保证，支持求解器验收与模型选择。
中文摘要 AI 辅助
部分傅里叶测量是计算成像的基础，其重建结果越来越多地来自迭代或学习型求解器，这些求解器的恢复保证依赖于信号模型、采样规律和求解器精度。这些保证均不适用于特定的最终输出：该输出可能满足所有采集到的系数，但在未测量的零空间中却严重错误。原生傅里叶留出法无法修复此问题，因为频谱集中的误差除非其频段被抽取，否则会被遗漏，因此均匀最坏情况风险需要与维度成比例的隐藏频段数量。我们提出一种提交后验收测试：秘密有限相位掩码将一次固定的傅里叶读数转化为密集的随机残差检查，中位数均值阈值处理校准的读数噪声。我们证明，一旦$q\ge C(1+N\sigma^2/E_{\rm rej})^2\log(1/\delta)$，则高于$E_{\rm rej}$的误差被拒绝，低于$E_{\rm acc}$的误差以置信度$1-\delta$被接受，明确揭示了单仓分辨率下限$N\sigma^2/\sqrt q$；在零噪声极限下，QPSK提供$4^{-q}$的健全性，而对于最坏情况普通仓误差，最小最大误接受率为$97.9\\%$。在固定噪声阈值比下，测量的筛选误接受率从16次检查时的约$21\\%$降至64次时的$5\\%$；一个数据一致的、NMSE为$0.200$的学习型零空间故障被$96.9\\%$的挑战库拒绝，而16个普通仓在$98.9\\%$的情况下遗漏了该故障；在16次检查时，误差统计从128维到$65{,}536$维保持稳定。该测试支持求解器验收、模型选择和停止认证，并提供经过认证的新鲜库误差区间。
英文摘要
Partial-Fourier measurement underlies computational imaging, and its reconstructions increasingly come from iterative or learned solvers whose recovery guarantees are conditional on a signal model, a sampling law, and solver accuracy. None of those guarantees transfers to a particular committed output: it can satisfy every acquired coefficient while remaining badly wrong in the unmeasured nullspace. Native Fourier holdout does not repair this, since a spectrally concentrated error is missed unless its bin is drawn, so uniform worst-case risk needs a number of hidden bins proportional to the dimension. We propose a post-commit acceptance test: secret finite-phase masks turn one fixed Fourier readout into dense randomized residual checks, and a median-of-means threshold handles calibrated readout noise. We prove that errors above $E_{\rm rej}$ are rejected and errors below $E_{\rm acc}$ are accepted with confidence $1-δ$ once $q\ge C(1+Nσ^2/E_{\rm rej})^2\log(1/δ)$, explicitly exposing the one-bin resolution floor $Nσ^2/\sqrt q$; in the zero-noise limit, QPSK gives $4^{-q}$ soundness versus $97.9\%$ minimax false acceptance for a worst-case ordinary-bin error. At a fixed noise-to-threshold ratio, measured screening false acceptance fell from about $21\%$ at 16 checks to $5\%$ at 64; a data-consistent learned nullspace failure with NMSE $0.200$ was rejected by $96.9\%$ of challenge banks while 16 ordinary bins missed it $98.9\%$ of the time; and error statistics at 16 checks stayed stable from 128 through $65{,}536$ dimensions. The test supports solver acceptance, model selection, and stopping certification, with a certified fresh-bank error interval.
