交叉多胞体的低次多项式逼近
Low-Degree Polynomial Approximation of the Cross-Polytope
- University of New South Wales（新南威尔士大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文确定了交叉多胞体低次多项式逼近的度-失真权衡，证明非负形式与平方和最优值均为Θ(√(d/t))，并给出匹配的构造及对一般对称多胞体的推广。
AI中文摘要：
我们确定了 $d$ 维交叉多胞体 $B_1^d$ 的多项式逼近的度-失真权衡。对于每个 $1\le t\le d$，每个在原点之外为正的全局非负 $2t$ 次形式具有至少 $(2e)^{-1/2}\sqrt{d/t}$ 的乘法夹逼失真，而一个显式的平方和（SoS）形式实现至多 $(2e)^{1/2}\sqrt{d/t}$ 的失真。因此，非负形式和 SoS 的最优值均为 $\Theta(\sqrt{d/t})$，并且度 $\Theta(d)$ 对于恒定失真是必要且充分的。下界与表示无关。对符号置换取平均并在 $\ell_1$ 球的平坦点上评估，将每个候选简化为支撑大小分布 $V(k)=k^{-2t}Q(k)$，其中 $\deg Q\le t$ 且 $Q(0)=0$。在替换 $u=1/k$ 后，拉格朗日插值表明，该度预算无法在 $d$ 个支撑尺度上保持分布近似恒定。一个匹配的 SoS 构造对符号向量面法线的偶次幂取平均，并将上界简化为拉德马赫矩。我们还分离出一个加权倒数网格引理，推导出对 $\ell_p$ 球的推论，并对比极立方体。对于一般的对称多胞体，加权面幂次产生一个单边证书，其边界-底目标函数是凹的，其最坏方向预言机归结为凸对偶范数问题；在二次度时，其优化器恢复了经典的最优设计和 John 椭球。这是一个预言机模型证书优化，而非端到端复杂性结果或对完整 SoS 最优值的刻画。
英文摘要：
We determine the degree-distortion tradeoff for polynomial approximation of the $d$-dimensional cross-polytope $B_1^d$. For every $1\le t\le d$, every globally nonnegative degree-$2t$ form that is positive away from the origin has multiplicative sandwich distortion at least $(2e)^{-1/2}\sqrt{d/t}$, while an explicit sum-of-squares (SoS) form achieves distortion at most $(2e)^{1/2}\sqrt{d/t}$. Hence both the nonnegative-form and SoS optima are $Θ(\sqrt{d/t})$, and degree $Θ(d)$ is necessary and sufficient for constant distortion. The lower bound is representation-free. Averaging over signed permutations and evaluating on flat points of the $\ell_1$ sphere reduces every candidate to a support-size profile $V(k)=k^{-2t}Q(k)$ with $°Q\le t$ and $Q(0)=0$. After the substitution $u=1/k$, Lagrange interpolation shows that this degree budget cannot keep the profile nearly constant across $d$ support scales. A matching SoS construction averages even powers of sign-vector facet normals and reduces the upper bound to a Rademacher moment. We also isolate a weighted reciprocal-grid lemma, derive consequences for $\ell_p$ balls, and contrast the polar cube. For a general symmetric polytope, weighted facet powers yield a one-sided certificate whose boundary-floor objective is concave and whose worst-direction oracle reduces to convex dual-norm problems; at degree two, its optimizers recover classical optimal design and the John ellipsoid. This is an oracle-model certificate optimization, not an end-to-end complexity result or a characterization of the full SoS optimum.
