ROABP的击中集与集多线性回路的和
Hitting-sets for ROABP and Sum of Set-Multilinear circuits
摘要：
针对阶数未知的只读一次不经意代数分支程序（ROABP）给出了一个$n^{O(\log n)}$ -时间（$n$为输入长度）黑盒多项式恒等式检验算法.该类已知的最佳结果是由Forbes-Saptharishi-Shpilka（STOC 2014）得出的$n^{O(\log^2 n)}$，并且仅适用于多线性ROABP。我们摆脱了它们对个体度的指数依赖。这样，我们将未知阶ROABP的时间复杂度与已知阶ROABP（由于Forbes-Shpilka（FOCS 2013））以及深度- $3$集-多线性电路（由于Agrawal-Saha-Saxena（STOC 2013））进行匹配。我们的证明更简单，涉及一种新的技术，称为基础隔离。深度- $3$模型最近变得非常重要，因为它已成为理解通用算术电路的垫脚石。它对多重线性的限制有已知的指数下界，但没有非平凡的黑盒恒等式检验。在本文中，我们采取了一个步骤，设计这样的命中集。我们给出了第一个次指数白盒PIT的总和不断许多集多线性深度$3$电路。为了实现这一点，我们定义了距离和基集的概念。对于多线性深度- $3$电路，距离度量分区距离单纯的细化有多远。针对$d$ -距离，设计了一个按时间的命中集$n^{O(d \log n)}$.此外，我们给我们的结果的模型的距离是大的，但它是小的，当限制到某些基集（变量）的扩展。我们还探讨了一种新的模型ROABP的因子矩阵是可逆的（称为可逆因子ROABP）。针对宽度- $w$可逆因子ROABP，设计了一个击中集时间多项式（$n^{w^2}$）.此外，当$w=2$。之前，宽度- $2$ ROABP的最佳结果是准多项式时间（Forbes-Saptharishi-Shpilka，STOC 2014）。
英文摘要：
We give a $n^{O(\log n)}$-time ($n$ is the input size) blackbox polynomial identity testing algorithm for unknown-order read-once oblivious algebraic branching programs (ROABP). The best result known for this class was $n^{O(\log^2 n)}$ due to Forbes-Saptharishi-Shpilka (STOC 2014), and that too only for multilinear ROABP. We get rid of their exponential dependence on the individual degree. With this, we match the time-complexity for the unknown order ROABP with the known order ROABP (due to Forbes-Shpilka (FOCS 2013)) and also with the depth-$3$ set-multilinear circuits (due to Agrawal-Saha-Saxena (STOC 2013)). Our proof is simpler and involves a new technique called basis isolation. The depth-$3$ model has recently gained much importance, as it has become a stepping-stone to understanding general arithmetic circuits. Its restriction to multilinearity has known exponential lower bounds but no nontrivial blackbox identity tests. In this paper, we take a step towards designing such hitting-sets. We give the first subexponential whitebox PIT for the sum of constantly many set-multilinear depth-$3$ circuits. To achieve this, we define notions of distance and base sets. Distance, for a multilinear depth-$3$ circuit, measures how far are the partitions from a mere refinement. We design a hitting-set in time $n^{O(d \log n)}$ for $d$-distance. Further, we give an extension of our result to models where the distance is large but it is small when restricted to certain base sets (of variables). We also explore a new model of ROABP where the factor-matrices are invertible (called invertible-factor ROABP). We design a hitting-set in time poly($n^{w^2}$) for width-$w$ invertible-factor ROABP. Further, we could do without the invertibility restriction when $w=2$. Previously, the best result for width-$2$ ROABP was quasi-polynomial time (Forbes-Saptharishi-Shpilka, STOC 2014).
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