AI 中文总结
本研究通过重随机化SRHT（两个Walsh变换与符号对角）构造子空间嵌入，证明样本量$k=\min\{n,\lceil Cr/\varepsilon^2\rceil\}$以高概率保持范数，解答开放问题TR-01。
AI 中文摘要
本研究探讨了由两个归一化实Walsh变换、两个独立符号对角矩阵以及无放回均匀坐标采样所获得的子空间嵌入。主要结果表明，对于通用常数$C$，规定的样本量$k=\min\{n,\lceil Cr/\varepsilon^2\rceil\}$足以在概率至少为0.99的情况下，将每个固定的$r$维子空间上的所有平方范数保持在$1\pm\varepsilon$范围内。该结果适用于所有环境Walsh维度和所有秩，并解答了数值线性代数开放问题库中的问题TR-01。证明通过连通图收缩和Walsh特征恒等式控制变换后投影的联合条目累积量。这些估计随后界定了中心化投影乘积的偶数次幂的迹的期望。一个双投影分解将该估计转化为对两个谱边缘的控制。任意密度的Bernoulli采样以及接近全采样时的确定性上界产生了规定的坐标数量。
英文摘要
This work studies subspace embeddings obtained by two normalized real Walsh transforms, two independent sign diagonals, and uniform coordinate sampling without replacement. The main result shows that the prescribed sample size $k=\min\{n,\lceil Cr/\varepsilon^2\rceil\}$, for a universal constant $C$, suffices to preserve all squared norms on each fixed $r$-dimensional subspace within $1\pm\varepsilon$ with probability at least $0.99$. The result holds for every ambient Walsh dimension and all ranks, and answers Problem TR-01 in the Open Problems in Numerical Linear Algebra repository. The proof controls joint entry cumulants of the transformed projection through connected graph contractions and Walsh character identities. These estimates then bound the expected trace of even powers of a product of centered projections. A two-projection decomposition converts this estimate into control of both spectral edges. Bernoulli sampling at arbitrary densities and a deterministic upper bound near full sampling yield the prescribed number of coordinates.
Comments42 pages. Lean 4 formalization and verification records: https://github.com/yuningyang19/OpenProblemsInNLA_TR-01
