随机置换正交乘积与快速降维
Randomly Permuted Orthogonal Products and Fast Dimension Reduction
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中文总结 AI 辅助
本文研究随机符号置换正交矩阵乘积的性质，将其应用于快速降维，解决了Jain等人的问题，改进了ORA构造的运行时间，还扩展结果至结构化无限模型。
中文摘要 AI 辅助
我们研究随机符号置换对正交矩阵乘积的影响及其在快速降维中的应用。设 $A,B \in \mathbb{R}^{d\times d}$ 为正交矩阵，$\Sigma \in \mathbb{R}^{d\times d}$ 为均匀随机符号置换矩阵。我们分析随机正交矩阵 $U=A \Sigma B$，并证明在 $A$ 和 $B$ 的元素大小满足温和假设时，$\max_{i,j=1,\ldots,d} |U_{ij}| =O\left ( \sqrt{\frac{\log d}{d}}\right )$ 以高概率成立。作为应用，我们表明 ORA（一种每次更新为 $\pi/4$ 旋转的 Kac 游走类似物）在 $O(d\log d)$ 次更新后达到相同的最大元素尺度，这解决了 Jain 等人的一个问题并改进了他们构造的运行时间；我们还证明并行 ORA 在 $O(\log d)$ 轮后达到该尺度。随后我们研究随机嵌入 $\Phi = \sqrt{\frac{d}{m}}\\, P_I U D_{\xi'}$，其中 $P_I$ 限制到 $m$ 个坐标，$\xi'$ 为独立 Rademacher 向量。我们确定了控制范数保持的两个参数，并证明在相应的可容许范围内，$\Phi$ 达到最优嵌入维数 $m\asymp\varepsilon^{-2}\log(N)$。最后，我们将结果扩展到结构化无限模型，包括稀疏向量、低秩矩阵和子空间的有限并集。
英文摘要
We study the effect of random signed permutations on products of orthogonal matrices and their applications to fast dimension reduction. Let $A,B \in \mathbb{R}^{d\times d}$ be orthogonal matrices and let $Σ\in \mathbb{R}^{d\times d}$ be a uniformly random signed permutation matrix. We analyze the random orthogonal matrix \[ U=A ΣB, \] and show that, under mild assumptions on the size of the entries of $A$ and $B$, \[ \max_{i,j=1,\ldots,d} |U_{ij}| =O\left ( \sqrt{\frac{\log d}{d}}\right ) \] with high probability. As an application, we show that ORA, an analogue of the Kac walk in which every update is a $π/4$ rotation, reaches the same maximal entry scale after $O(d\log d)$ updates. This resolves a question of Jain et al. and improves the running time of their construction. We also show that parallel ORA reaches this scale after $O(\log d)$ rounds. We then study the random embedding \[ Φ = \sqrt{\frac{d}{m}}\, P_I U D_{ξ'}, \] where $P_I$ restricts to $m$ coordinates and $ξ'$ is an independent Rademacher vector. We identify two parameters controlling norm preservation and show that, throughout the corresponding admissible range, $Φ$ achieves optimal embedding dimension $m\asymp\varepsilon^{-2}\log(N)$. Finally, we extend the result to structured infinite models, including sparse vectors, low-rank matrices, and finite unions of subspaces.
