采样傅里叶矩阵和哈达玛矩阵的受限等距性：基于熵下降方法
Restricted isometry of sampled Fourier and Hadamard matrices via entropic descent
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中文总结 AI 辅助
本文用熵镜像下降改进了采样傅里叶和哈达玛矩阵的受限等距界，减少了稀疏恢复所需的测量数，并实现了全支撑信号的均匀能量采样。
中文摘要 AI 辅助
这是关于下降方法作为调和分析中证明机制的第二篇论文（第一篇处理了Bourgain的$\Lambda(p)$选择定理），本文中熵镜像下降给出了改进的受限等距界，从而改进了傅里叶比率程序中的采样和恢复界。设$H$为复$n\times n$矩阵，满足$H^*H=nI$且$|H_{ij}|=1$。对于固定的$0<\varepsilon<\frac14$和$A_0>0$，独立伯努利行选择，其期望基数满足$C_{\varepsilon,A_0}r\log\frac{2en}{r}\log(2r)\leq m\leq n/2$，在按$\sqrt m$归一化后，以不超过$Cn^{-A_0}$的失败概率，保持所有满足$\\|y\\|_1\leq\sqrt r\\|y\\|_2$的复向量$y$的平方范数在因子$1\pm\varepsilon$之内。这包括每个$r$-稀疏向量以及完全支撑的向量；支撑大小仅通过该不等式起作用。对于$y=\widehat f$，该条件恰好是$\operatorname{FR}(f)^2\leq r$，其中$\operatorname{FR}(f)=\\|\widehat f\\|_1/\\|\widehat f\\|_2$，从而即使对于具有完全傅里叶支撑的信号，也能实现均匀能量采样和近似恢复。同样的结论也适用于具有规定基数的均匀子集，以及离散傅里叶矩阵和实哈达玛矩阵。在固定精度下，该界从先前的计数中去掉了一个稀疏性对数，并将$\log n$替换为$\log\frac{2en}{r}$；对于Walsh矩阵，在其范围内与Błasiok等人的下界（在常数因子内）相匹配。一个相对熵势能控制着每个尺度上幅度预测器的修正；在两个样本的对称差上对它们进行计数，即可得到一致估计。我们还证明了充分界$C_{A_0}\varepsilon^{-5}r\log\frac{2en}{r}\log\frac{2r}{\varepsilon}$，以及从$Cs\log\frac{2en}{s}\log(2s)$次测量中进行稳定的稀疏恢复，误差由噪声和最佳$s$项逼近控制。
英文摘要
In this second paper on descent methods as proof mechanisms in harmonic analysis (the first treated Bourgain's $Λ(p)$ selection theorem), entropic mirror descent gives an improved restricted isometry bound, improving sampling and recovery bounds in the Fourier Ratio program. Let $H$ be a complex $n\times n$ matrix with $H^*H=nI$ and $|H_{ij}|=1$. For fixed $0<\varepsilon<\frac14$ and $A_0>0$, independent Bernoulli row selection with expected cardinality $C_{\varepsilon,A_0}r\log\frac{2en}{r}\log(2r)\leq m\leq n/2$ preserves, after normalization by $\sqrt m$, the squared norm of every complex vector $y$ with $\|y\|_1\leq\sqrt r\|y\|_2$ within a factor $1\pm\varepsilon$, with failure probability at most $Cn^{-A_0}$. This includes every $r$-sparse vector and also fully supported ones; support size enters only through this inequality. For $y=\widehat f$ the condition is exactly $\operatorname{FR}(f)^2\leq r$, where $\operatorname{FR}(f)=\|\widehat f\|_1/\|\widehat f\|_2$, giving uniform energy sampling and approximate recovery even for signals with full Fourier support. The same holds for a uniform subset of prescribed cardinality, and for discrete Fourier and real Hadamard matrices. At fixed accuracy the bound removes one sparsity logarithm from the earlier count and replaces $\log n$ by $\log\frac{2en}{r}$; for Walsh matrices it matches, up to constants, the lower bound of Błasiok et al.\ in their range. One relative-entropy potential controls the corrections of an amplitude predictor at every scale; counting them on the symmetric difference of two samples gives the uniform estimate. We also prove the sufficient bound $C_{A_0}\varepsilon^{-5}r\log\frac{2en}{r}\log\frac{2r}{\varepsilon}$, and stable sparse recovery from $Cs\log\frac{2en}{s}\log(2s)$ measurements, with error controlled by the noise and the best $s$-term approximation.
