发表机构
University of Minnesota, Twin Cities（明尼苏达大学双城分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出Lee的直接刘易斯不动点迭代，可对所有p>2高精度计算刘易斯权重，其复杂度优于现有优化方法，合成与真实实验验证了该方法的有效性。
AI 中文摘要
矩阵的ℓₚ-刘易斯权重由一个不动点方程定义。对于p<4，Cohen与Peng[CP15]证明，迭代该方程的等价重排形式可高精度计算刘易斯权重；对于p≥4，此前的高精度方法采用基于优化的 approaches。本文证明，Lee[Lee16]论文中提出的直接刘易斯不动点迭代，可对所有p>2高精度计算刘易斯权重。对于划分为行块A₍₁₎,…,A₍ₖ₎的矩阵A∈ℝ^(m×n)，当p>2时，我们在O(p log(p√(∑ᵢ₌₁ᵏ rank(A₍ᵢ₎))/ε)轮精确杠杆得分向量计算中，逐坐标计算ε-近似的ℓₚ块刘易斯权重；对于普通刘易斯权重，该复杂度变为O(p log(p√m)/ε)，相较于Gribling、Sidford与Zhang[GSZ26]针对p≥4给出的O(p² log(m/ε)) bound有所改进。我们的核心发现是，每次直接刘易斯更新会将与真实权重的KL散度以1−2/p的比例压缩。我们还通过体积采样与熵独立性对该压缩给出了另一种解释。合成实验与我们预测的局部压缩率高度吻合，迭代次数随p近似线性增长；真实数据实验则展示了有限p块刘易斯设计的信息浓度权衡关系。
英文摘要
The $\ell_p$-Lewis weights of a matrix are defined by a fixed-point equation. For $p<4$, Cohen and Peng [CP15] showed that iterating an equivalent rearrangement of this equation computes Lewis weights to high precision; for $p\geq4$, prior high-precision methods instead use optimization-based approaches. We show that the direct Lewis fixed-point iteration, appearing in the thesis of Lee [Lee16], computes Lewis weights to high precision for every $p>2$. For a matrix $\mathbf{A}\in\mathbb{R}^{m\times n}$ partitioned into row blocks $\mathbf{A}_{[1]},\ldots,\mathbf{A}_{[k]}$, we compute, for $p>2$, coordinatewise $\varepsilon$-approximate $\ell_p$ block Lewis weights in $O\left(p\log\frac{p\sqrt{\sum_{i=1}^k\operatorname{rank}(\mathbf{A}_{[i]})}}{\varepsilon}\right)$ rounds of exact leverage-score-vector computations. For ordinary Lewis weights, this becomes $O\left(p\log\frac{p\sqrt{m}}{\varepsilon}\right)$, improving the $O\left(p^2\log(m/\varepsilon)\right)$ bound of Gribling, Sidford, and Zhang [GSZ26] for $p\geq4$. Our main observation is that each direct Lewis update contracts the KL divergence to the true weights by a factor of $1-\frac{2}{p}$. We also give an alternate explanation of this contraction through volume sampling and entropic independence. Synthetic experiments closely match our predicted local contraction rates and iteration counts grow approximately linearly with $p$; real-data experiments illustrate the information-concentration tradeoff of finite-$p$ block Lewis designs.
Comments25 pages, 8 figures
