稀疏化的统一理论
A Unified Theory of Sparsification
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中文总结 AI 辅助
研究实值码的稀疏化能力，提出连续值非冗余性（\(\mathrm{CVNRD}\)）这一核心参数，给出单一结构定理，该定理给出与\(\mathrm{CVNRD}\)近乎线性的稀疏化器，并表明\(\mathrm{CVNRD}\)是坐标方向无偏随机稀疏化方案的下界障碍。
中文摘要 AI 辅助
我们研究实值码的稀疏化能力，这是一种统一的抽象概念，涵盖了组合和连续意义上的稀疏化，包括谱稀疏化。在我们的设定中，码\(C \subseteq \mathbb{R}_{\geq 0}^m\)是一组非负实值向量，对于参数\(\epsilon > 0\)，\(C\)的\((1 \pm \epsilon)\)-稀疏化器是子集\(T \subseteq [m]\)以及权重\(w \in \mathbb{R}_{\geq 0}^T\)，使得对于每个\(c \in C\)，\(\sum_{i \in T} w_i c_i \in (1 \pm \epsilon)\sum_{i=1}^m c_i\)。当\(C \subseteq \{0,1\}^m\)时，这专门化为码稀疏化，进而涵盖了Khanna--Putterman--Sudan（SODA 2024，STOC 2025）和Brakensiek--Guruswami（STOC 2025）研究的CSP稀疏化。同样，对于图\(G=(V,E)\)，若通过\(c^{(x)}_{(u,v)}=(x_u - x_v)^2\)定义\(C=\{c^{(x)}:x\in\mathbb R^V\}\subseteq\mathbb R_{\geq 0}^E\)，则稀疏化\(C\)正是Spielman--Teng（SICOMP 2011）研究的谱图稀疏化。尽管传统上驱动组合和连续稀疏化的技术在很大程度上是不相关的，但我们的主要结果是一个单一的结构定理，用于支配任意实值码\(C\subseteq\mathbb{R}_{\geq 0}^m\)的稀疏化能力。核心参数是连续值非冗余性（\(\mathrm{CVNRD}\)），它是冗余性的实值类似物，捕获了\(C\)中包含的最大近似块对角障碍。我们的定理给出了大小与\(\mathrm{CVNRD}\)近乎线性的稀疏化器，并表明\(\mathrm{CVNRD}\)也是广泛的坐标方向无偏随机稀疏化方案的下界障碍。
英文摘要
We study the sparsifiability of \emph{real-valued codes}, a unifying abstraction that generalizes both combinatorial and continuous notions of sparsification, including spectral sparsification. In our setting, a code $C \subseteq \mathbb{R}_{\geq 0}^m$ is simply a collection of nonnegative real-valued vectors, and for a parameter $ε> 0$, a \emph{$(1 \pm ε)$-sparsifier} of $C$ is a subset $T \subseteq [m]$, together with weights $w \in \mathbb{R}_{\geq 0}^T$, such that, for every $c \in C$, $\sum_{i \in T} w_i c_i \in (1 \pm ε)\sum_{i=1}^m c_i$. When $C \subseteq \{0,1\}^m$, this specializes to code sparsification, and hence captures CSP sparsification, as studied by Khanna--Putterman--Sudan (SODA 2024, STOC 2025) and Brakensiek--Guruswami (STOC 2025). Similarly, for a graph $G=(V,E)$, if one defines $C=\{c^{(x)}:x\in\mathbb R^V\}\subseteq\mathbb R_{\geq 0}^E$ by $c^{(x)}_{(u,v)}=(x_u-x_v)^2$, then sparsifying $C$ is exactly spectral graph sparsification, as studied by Spielman--Teng (SICOMP 2011). Although the techniques driving combinatorial and continuous sparsification have traditionally been largely disjoint, our main result is a single structural theorem governing the sparsifiability of arbitrary real-valued codes $C\subseteq\mathbb{R}_{\geq 0}^m$. The central parameter is \emph{continuous-valued non-redundancy} ($\mathrm{CVNRD}$), a real-valued analogue of non-redundancy that captures the largest approximately block-diagonal obstruction contained in $C$. Our theorem gives sparsifiers of size nearly-linear in $\mathrm{CVNRD}$, and shows that $\mathrm{CVNRD}$ is also a lower-bound obstruction for the broad class of coordinate-wise unbiased randomized sparsification schemes.
