集深δ公式的拟多项式击中集
Quasi-polynomial Hitting-set for Set-depth-Delta Formulas
摘要：
如果存在（未知）划分（X_1，.，X_d），即C（x）= \sum_{i=1}^k \prod_{j=1}^{d} f_{i,j}（x_{X_j}），其中f_{i,j}是F[ x_{X_j} ]中的稀疏多项式。将此定义扩展到任何深度-我们称深度-Delta公式C（由Sigma和Pi门交替层组成，顶部有一个Sigma门）为集合深度-Delta公式，如果C中的每个Pi层都遵守变量的（未知）分区;如果Delta是偶数，则最底部Pi层的乘积门可以计算任意单项式。在这项工作中，我们给出了一个击中集生成器的集深度-Delta公式（在任何字段）与运行时间多项式在exp（（{Delta}^2 log s）^{Delta - 1}），其中s是大小上的输入集深度-Delta公式的限制。换句话说，我们给出了这类常深度公式的拟多项式时间黑盒多项式恒等式检验。以前，Delta=3的非常特殊的情况（也称为集合多线性深度3电路）没有已知的次指数时间命中集合生成器。Shpilka & Yehudayoff（FnT-TCS 2010）将其宣布为开放问题; Nisan & Wigderson（FOCS 1995）首先研究了该模型。我们的工作解决了这个问题，不仅深度为3，但深度为ε. log s / loglog s，对于一个固定的常数ε < 1。该技术是通过深度-（Delta-1）公式在Hadamard代数上，在变量上应用“移位”后，研究深度-Delta公式。我们提出了一个新的代数猜想的低支持度的秩浓度在后者的公式，并设法证明它的情况下，集深度Delta公式。
英文摘要：
We call a depth-4 formula C set-depth-4 if there exists a (unknown) partition (X_1,...,X_d) of the variable indices [n] that the top product layer respects, i.e. C(x) = \sum_{i=1}^k \prod_{j=1}^{d} f_{i,j}(x_{X_j}), where f_{i,j} is a sparse polynomial in F[x_{X_j}]. Extending this definition to any depth - we call a depth-Delta formula C (consisting of alternating layers of Sigma and Pi gates, with a Sigma-gate on top) a set-depth-Delta formula if every Pi-layer in C respects a (unknown) partition on the variables; if Delta is even then the product gates of the bottom-most Pi-layer are allowed to compute arbitrary monomials. In this work, we give a hitting-set generator for set-depth-Delta formulas (over any field) with running time polynomial in exp(({Delta}^2 log s)^{Delta - 1}), where s is the size bound on the input set-depth-Delta formula. In other words, we give a quasi-polynomial time blackbox polynomial identity test for such constant-depth formulas. Previously, the very special case of Delta=3 (also known as set-multilinear depth-3 circuits) had no known sub-exponential time hitting-set generator. This was declared as an open problem by Shpilka & Yehudayoff (FnT-TCS 2010); the model being first studied by Nisan & Wigderson (FOCS 1995). Our work settles this question, not only for depth-3 but, up to depth epsilon.log s / loglog s, for a fixed constant epsilon < 1. The technique is to investigate depth-Delta formulas via depth-(Delta-1) formulas over a Hadamard algebra, after applying a `shift' on the variables. We propose a new algebraic conjecture about the low-support rank-concentration in the latter formulas, and manage to prove it in the case of set-depth-Delta formulas.
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