[论文解读] A Polynomial Degree Bound on Equations for Non-Rigid Matrices and Small Linear Circuits

Mrinal Kumar, Ben Lee Volk|arXiv (Cornell University)|Jan 1, 2021

Complexity and Algorithms in Graphs参考文献 2被引用 2

一句话总结

该论文为定义非刚性矩阵和小型线性电路的方程建立了多项式次数上界，通过证明此类矩阵满足次数至多为 $n^3$ 的非零多项式，解决了Gesmundo等人提出的一个猜想。证明方法结合了Shpilka和Volkovich的低次多项式映射与维数计数，从而从多项式恒等式测试的去随机化中导出新的电路下界。

ABSTRACT

We show that there is an equation of degree at most poly(n) for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field 𝔽, there is a non-zero n²-variate polynomial P ∈ 𝔽[x_{1, 1}, …, x_{n, n}] of degree at most poly(n) such that every matrix M which can be written as a sum of a matrix of rank at most n/100 and a matrix of sparsity at most n²/100 satisfies P(M) = 0. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer and Landsberg [Fulvio Gesmundo et al., 2016] and improves the best upper bound known for this problem down from exp(n²) [Abhinav Kumar et al., 2014; Fulvio Gesmundo et al., 2016] to poly(n). We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices M such that the linear transformation represented by M can be computed by an algebraic circuit with at most n²/200 edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded. Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [Amir Shpilka and Ilya Volkovich, 2015] to construct low degree "universal" maps for non-rigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a low degree annihilating polynomial completes the proof. As a corollary, we show that any derandomization of the polynomial identity testing problem will imply new circuit lower bounds. A similar (but incomparable) theorem was proved by Kabanets and Impagliazzo [Valentine Kabanets and Russell Impagliazzo, 2004].

研究动机与目标

• 解决Gesmundo、Hauenstein、Ikenmeyer和Landsberg关于非刚性矩阵存在低次方程的猜想。
• 为可由小型线性电路计算的矩阵建立方程次数的多项式上界，即使在无深度限制的情况下也成立。
• 证明去随机化多项式恒等式测试可导致新的电路下界，推进代数自然证明的前沿。

提出的方法

• 利用Shpilka和Volkovich提出的低次多项式映射，为非刚性矩阵和小型线性电路构造通用参数化。
• 应用维数计数论证，表明任何此类多项式映射都必须具有低次消去多项式。
• 利用低次参数化的存在性，为非刚性矩阵的代数簇导出一个次数至多为 $n^3$ 的非平凡多项式方程。
• 将该方法扩展至具有小线性电路的矩阵的代数簇，证明在无深度约束下也存在类似的次数上界。
• 将方程寻找问题转化为维度为 $\exp(\text{poly}(n))$ 的空间上的线性系统求解，从而实现PSPACE构造。
• 采用变换将电路转为近乎最小深度（几乎-MD）形式，以保持多项式规模并支持高效求值。

实验结果

研究问题

• RQ1 非刚性矩阵的定义方程的次数是否可被 $n$ 的多项式有界，如Gesmundo等人所猜想？
• RQ2 即使电路深度无界，是否存在一个多项式次数的方程来描述可由小型线性电路计算的矩阵的代数簇？
• RQ3 去随机化多项式恒等式测试是否意味着线性电路或刚性矩阵的新电路下界？

主要发现

• 该论文证明了非刚性矩阵（非 $(\varepsilon n, \varepsilon n^2)$-刚性）的代数簇由一个次数至多为 $n^3$ 的非零多项式定义，证实了Gesmundo等人的猜想，且相比此前的指数界有显著改进。
• 对于大小至多为 $n^2/200$ 的线性电路可计算的矩阵，即使在无深度限制下，也建立了类似的多项式次数上界，此前该结果尚不为人所知。
• 该构造产生一个PSPACE算法，其输入为 $1^n$ 时，可输出非刚性矩阵的次数为 $n^3$ 的方程的系数。
• 若多项式恒等式测试（PIT）属于P，则要么存在PSPACE构造的难多项式，要么存在NP预言机构造的矩阵，其线性电路大小为 $\Omega(n^2)$。
• 该结果表明，任何PIT的去随机化都会导致新的电路下界，从而强化了去随机化与复杂性理论之间的联系。
• 该方法亦适用于张量秩，表明在PIT ∈ P的假设下，要么PSPACE中存在难张量，要么可借助NP预言机高效构造张量。