拟齐次子水平集的体积：两种具有收敛速率的线性代数确定性算法
Volume of quasi-homogeneous sublevel sets: Two linear algebra deterministic algorithms with convergence rates
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中文总结 AI 辅助
针对正拟齐次多项式的单位子水平集体积计算问题，提出两种仅依赖数值线性代数的确定性算法，分别以多项式和指数速率收敛，改进了多变量矩-SOS体积层级的收敛速率。
中文摘要 AI 辅助
我们考虑正拟齐次多项式的单位子水平集的勒贝格体积计算问题。通过多项式将环境边界框的勒贝格测度向前推进，可将这个高维体积问题简化为一维矩问题。这一处理将环境维度从优化过程中移除，仅将其限制在计算多项式在边界框上的矩这一预处理阶段，对于稀疏或可分多项式而言，该预处理的计算复杂度为环境维度的多项式级。针对由此产生的单变量松弛问题，我们提出两种确定性算法，每种算法均返回体积的经过验证的上下界。两种算法均完全绕过半定优化，仅依赖标准数值线性代数。第一种算法用切比雪夫多项式逼近分段常数函数，使得每次松弛简化为快速余弦变换，且在松弛阶数上以多项式速率收敛；第二种算法从包含矩矩阵和定位矩阵的单个广义特征值问题中提取体积上下界，该矩阵的大小随松弛阶数线性增长，且以指数速率收敛，其收敛速率的比值由多项式在边界框上的先验上界决定。最后，两种算法生成的单变量多项式均适用于多变量矩-SOS体积层级，因此代数和几何速率可传递到该层级本身，从而改进了其已知的最佳收敛速率。
英文摘要
We consider the problem of computing the Lebesgue volume of the unit sublevel set of a positive quasi-homogeneous polynomial. Pushing the Lebesgue measure of an ambient bounding box forward through the polynomial reduces this high-dimensional volume to a one-dimensional moment problem. This removes the ambient dimension from the optimization and confines the dimension to a single preprocessing stage, computing the moments of the polynomial over the box, which is polynomial in the ambient dimension for sparse or separable polynomials. We propose two deterministic algorithms for the resulting univariate relaxations, each returning certified upper and lower bounds on the volume. Both bypass semidefinite optimization entirely and rely only on standard numerical linear algebra. The first approximates a piecewise-constant function by a Chebyshev polynomial, so that each relaxation reduces to a fast cosine transform, and converges at a polynomial rate in the relaxation order. The second extracts the volume bounds from a single generalized eigenvalue problem involving moment and localizing matrices whose size grows linearly with the relaxation order, and converges at an exponential rate; the ratio governing this rate is determined by an a priori upper bound on the polynomial over the bounding box. Finally, the univariate polynomials produced by either algorithm are feasible for the multivariate moment-SOS volume hierarchy. The algebraic and geometric rates therefore transfer to the hierarchy itself, improving on its best known convergence rates.
