序列欧几里得连接的最优指数内存：边幂代价与相变
Optimal exponential memory for sequential Euclidean connections: edge-power costs and phase transitions
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中文总结 AI 辅助
该研究针对序列欧几里得连接规则，分析了其边幂代价的最优参数，在不同输入序列和幂次下推导了优化器的尺度、相变及渐近代价等关键结果。
中文摘要 AI 辅助
我们考虑一种在线几何连接规则，该规则存储一个状态点。处理 $p_i$ 后，状态按 $x_i=\gamma x_{i-1}+(1-\gamma)p_i$ 更新，且保留从 $x_{i-1}$ 到 $x_i$ 以及从 $x_i$ 到 $p_i$ 的两段线段。设计参数 $\gamma$ 控制状态的持久性。我们在独立均匀输入和任意输入序列下，最小化所得边长度的 $\alpha$ 次幂之和。对于单位球中的均匀点，该平稳问题在 $\alpha=1$ 处存在相变；其连续扩展在 $0<\alpha\leq1$ 时于边界处取最小值，而在 $\alpha>1$ 时所有全局极小值均位于内部。主要结果确定了联合窗口 $\alpha_N=1+\varepsilon_N$（$\varepsilon_N\log N\to\lambda$）内的有限优化器：低于显式阈值时，优化器处于 $N^{-1/2}$ 尺度；在阈值处，其尺度为 $\sqrt{\log N/(N\log\log N)}$；高于阈值时，优化器趋近于显式平稳根并带有两个可计算修正项。第二个阈值确定哪个修正项支配位置，且通过微分估计证明最终的唯一性。在 $\alpha=3d+8$ 处，平稳端点的线性系数改变符号，一条严格局部极大值分支进入参数区间。在任意输入序列下，对于 $0<\alpha\leq3$，最优参数及每个处理点的渐近最坏情况边幂代价是显式的；在高次幂下，周期块与分离论证表明，该优化代价渐近于 $2\log2/\log\alpha$。二次和四次幂的精确结果、所有偶次幂的有理递归以及高维扩展，为优化器提供了额外描述。
英文摘要
We study the edge-power cost of the labelled tree generated by the $γ$-strategy, a constant-gain rule for sequential Euclidean connections. Starting with $x_0=p_0$, each input point $p_i$ is attached to $x_{i-1}$, and the state is updated by $x_i=γx_{i-1}+(1-γ)p_i$. Retaining $x_i$ subdivides the insertion segment into a spine edge and a leaf edge. The memory parameter $γ$ controls how long earlier points influence subsequent attachment points. We minimize the sum of the $α$-powers of the edge lengths under independent uniform input and arbitrary input sequences. For uniform points in the unit ball, the stationary problem has a transition at $α=1$. Its continuous extension is minimized at the boundary for $0<α\leq1$, while every global minimizer is interior for $α>1$. We determine the finite optimizer in the joint window $α_N=1+\varepsilon_N$, $\varepsilon_N\log N\toλ$. Below an explicit threshold it lies on the $N^{-1/2}$ scale, at the threshold its scale is $\sqrt{\log N/(N\log\log N)}$, and above the threshold it approaches an explicit stationary root with two computable corrections. A second threshold identifies the governing correction, and differentiated estimates prove eventual uniqueness. At $α=3d+8$, the linear coefficient at the stationary endpoint changes sign and a branch of strict local maxima enters the parameter interval. For arbitrary input sequences, the optimal parameter and asymptotic worst-case edge-power cost per point are explicit for $0<α\leq3$. At high powers, periodic antipodal block inputs give explicit lower bounds which, with a separation argument, show that the optimized cost is asymptotic to $2\log2/\logα$. Exact results for powers two and four, a rational recursion for every even power, and a high-dimensional expansion complete the analysis.
