Given a matrix $\mathbf{A} \in \mathbb{R}^{k \times n}$, a partitioning of $[k]$ into groups $S_1,\dots,S_m$, an outer norm $p$, and a collection of inner norms such that either $p \ge 1$ and $p_1,\dots,p_m \ge 2$ or $p_1=\dots=p_m=p \ge 1/\log n$, we prove that there is a sparse weight vector $\mathbf{\beta} \in \mathbb{R}^{m}$ such that $\sum_{i=1}^m \beta_i \cdot \|\mathbf{A}_{S_i}\mathbf{x}\|_{p_i}^p \approx_{1\pm\varepsilon} \sum_{i=1}^m \|\mathbf{A}_{S_i}\mathbf{x}\|_{p_i}^p$, where the number of nonzero entries of $\mathbf{\beta}$ is at most $O_{p,p_i}(\varepsilon^{-2}n^{\max(1,p/2)}(\log n)^2(\log(n/\varepsilon)))$. When $p_1\dots,p_m \ge 2$, this weight vector arises from an importance sampling procedure based on the block Lewis weights, a recently proposed generalization of Lewis weights. Additionally, we prove that there exist efficient algorithms to find the sparse weight vector $\mathbf{\beta}$ in several important regimes of $p$ and $p_1,\dots,p_m$. Our main technical contribution is a substantial generalization of the change-of-measure method that Bourgain, Lindenstrauss, and Milman used to obtain the analogous result when every group has size $1$. Our generalization allows one to analyze change of measures beyond those implied by D. Lewis's original construction, including the measure implied by the block Lewis weights and natural approximations of this measure.

翻译：给定矩阵 $\mathbf{A} \in \mathbb{R}^{k \times n}$，将 $[k]$ 划分为 $S_1,\dots,S_m$ 组，外范数 $p$ 以及一组内范数满足要么 $p \ge 1$ 且 $p_1,\dots,p_m \ge 2$，要么 $p_1=\dots=p_m=p \ge 1/\log n$，我们证明存在一个稀疏权重向量 $\mathbf{\beta} \in \mathbb{R}^{m}$ 使得 $\sum_{i=1}^m \beta_i \cdot \|\mathbf{A}_{S_i}\mathbf{x}\|_{p_i}^p \approx_{1\pm\varepsilon} \sum_{i=1}^m \|\mathbf{A}_{S_i}\mathbf{x}\|_{p_i}^p$，其中 $\mathbf{\beta}$ 的非零项数目至多为 $O_{p,p_i}(\varepsilon^{-2}n^{\max(1,p/2)}(\log n)^2(\log(n/\varepsilon)))$。当 $p_1,\dots,p_m \ge 2$ 时，该权重向量来源于基于分块刘易斯权重（一种近期提出的刘易斯权重的推广）的重要性采样过程。此外，我们证明在 $p$ 和 $p_1,\dots,p_m$ 的若干重要情形下存在高效算法来求解稀疏权重向量 $\mathbf{\beta}$。我们的主要技术贡献在于实质性地推广了 Bourgain、Lindenstrauss 和 Milman 用于处理每组大小为 $1$ 时类似结果的测度变换方法。这一推广使我们能够分析超出 D. Lewis 原始构造所隐含测度之外的测度变换，包括由分块刘易斯权重及其自然近似所隐含的测度。