发表机构
Nanyang Technological University; University of Technology Sydney（南洋理工大学; 悉尼科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了有限图结构如何控制图矩阵的尖锐谱增长，通过两个有限割优化给出多项式和对数指数，并应用于SoS可行性与高斯随机张量网络。
AI 中文摘要
图矩阵编码了由共享随机变量构建的随机矩阵中的依赖关系，并出现在谱算法、平方和（SoS）及高维统计中。我们确定了有限图结构如何控制其尖锐谱增长。对于稠密Rademacher模型中的每个固定简单图形状，包括重叠或空矩阵边界，我们证明了$\mathbb E\\|M_\alpha\\|=\Theta_\alpha(n^{(v+h-s)/2}(\log n)^{a_*/2})$，其中$v$计顶点数，$h$计孤立求和顶点，$s$为最小边界分隔符大小，$a_*$在最小分隔符上最大化活跃分量计数。因此，两个有限割优化决定了多项式和对数指数。该公式填补了分隔符界限中的多对数间隙，且一个具有相同粗参数但不同范数的无限族表明对数指数记录了真正的新结构。证明通过将标签损失转换为分隔符盈余来控制增长迹矩中的所有缺陷层；条件展平和同步波动产生匹配的下界。我们将分析扩展到指定的独立因子混沌、局部权重、不等维度、有界非对称噪声、高斯输入和固定次数Hermite输入。应用包括$9\le k\le c\sqrt n$的四次团SoS可行性而无需渐近对数损失，以及高斯随机张量网络：偏差阈值、尖锐期望尺度、熵估计，以及对于连通无环等维网络，重缩放的最大输出特征值收敛到极限律的精确右边缘。这些结果将有限结构与尖锐增长尺度联系起来，并将额外的代数和谱结构与完全可行性和精确极限常数联系起来。
英文摘要
Graph matrices encode dependencies in random matrices built from shared random variables and arise in spectral algorithms, sum-of-squares (SoS), and high-dimensional statistics. We determine how finite graph structure controls their sharp spectral growth. For every fixed simple graph shape in the dense Rademacher model, including overlapping or empty matrix boundaries, we prove $\mathbb E\|M_α\|=Θ_α(n^{(v+h-s)/2}(\log n)^{a_*/2})$, where $v$ counts vertices, $h$ isolated summation vertices, $s$ the minimum boundary-separator size, and $a_*$ maximizes an active-component count over minimum separators. Thus two finite cut optimizations determine both the polynomial and logarithmic exponents. The formula closes the polylogarithmic gap in separator bounds, and an infinite family with identical coarse parameters but different norms shows that the logarithmic exponent records genuinely new structure. The proof controls all defect layers in growing trace moments by converting label loss into separator excess; conditional flattening and synchronized fluctuations yield matching lower bounds. We extend the analysis to specified independent-factor chaoses, local weights, unequal dimensions, bounded asymmetric noise, Gaussian inputs, and fixed-degree Hermite inputs. Applications include degree-four clique SoS feasibility for $9\le k\le c\sqrt n$ without an asymptotic logarithmic loss, and Gaussian random tensor networks: deviation thresholds, sharp expected scales, entropy estimates, and, for connected loopless equal-dimensional networks, convergence of the rescaled largest output eigenvalue to the exact right edge of the limiting law. These results connect finite structure to sharp growth scales, and additional algebraic and spectral structure to full feasibility and exact limiting constants.
Comments79 pages
