A fundamental question in streaming complexity is whether every space-efficient turnstile algorithm is implicitly a linear sketch. The landmark work of Li, Nguyen, and Woodruff [LNW14] established an equivalence between the two, but their reduction requires a stream length that is at least doubly exponential in the dimension $n$. In the opposite direction, results by Kallaugher and Price [KP20] demonstrate a separation for streams of linear length, showing that the equivalence does not hold in general. The most natural and practically relevant regime -- polynomial-length streams -- has therefore remained open. We show that polynomial-length turnstile algorithms admit linear-sketch simulations. More precisely, if a turnstile algorithm uses $S$ bits of space and succeeds on all streams of length $\mathrm{poly}(D, n)$, then on final vectors $x$ with $\|x\|_2 \le D$, its output can be recovered from $O(S)$ linear measurements of $x$, using $O(S \log S)$ bits overall. For smooth problems under appropriate input distributions, a mollified version of the reduction yields a bounded-entry sketch with $O(S / \log D)$ measurements and optimal $O(S)$ total space. Our results extend to strict turnstile streams and non-uniform Read-Once Branching Programs (ROBPs). Our proof departs from prior transition-graph based machinery, relying instead on a Fourier-analytic framework and tools from additive combinatorics to extract discrete linear measurements. Our analysis shows that any $S$-bit algorithm can only be sensitive to a low-dimensional lattice of heavy Fourier frequencies, which we then use to construct the rows of the sketching matrix. Consequently, we obtain new lower bounds for polynomial-length streams via existing real sketching and communication lower bounds.

翻译：流式复杂度中的一个基本问题是：每个空间高效的旋转栅门算法是否本质上都是一个线性草图。Li、Nguyen 和 Woodruff [LNW14] 的开创性工作确立了二者之间的等价性，但他们的归约要求流长度至少为维度 $n$ 的双指数级。相反方向，Kallaugher 和 Price [KP20] 的结果展示了线性长度流下的分离性，表明该等价性一般不成立。最自然且实践相关的区间——多项式长度流——因此仍然悬而未决。我们证明多项式长度的旋转栅门算法可接受线性草图模拟。更精确地说，若一个旋转栅门算法使用 $S$ 位空间且对所有长度为 $\mathrm{poly}(D, n)$ 的流都成功，则对于满足 $\|x\|_2 \le D$ 的最终向量 $x$，其输出可从 $x$ 的 $O(S)$ 个线性测量中恢复，总计使用 $O(S \log S)$ 位。对于适当输入分布下的光滑问题，归约的光滑版本可生成一个具有 $O(S / \log D)$ 个测量和最优 $O(S)$ 总空间的有界条目草图。我们的结果扩展到严格旋转栅门流和非均匀一次读取分支程序（ROBP）。我们的证明脱离了先前基于转移图的机制，转而依赖傅里叶分析框架和加性组合学工具来提取离散线性测量。分析表明，任何 $S$ 位算法仅对重傅里叶频率的低维格敏感，我们利用这一性质构造草图矩阵的行。因此，通过现有的实草图和通信下界，我们获得了多项式长度流的新下界。