Optimal exponential memory for sequential Euclidean connection: edge-power costs and phase transitions

Abstract

We consider an online geometric connection rule that stores one point of state. After processing $p_i$, the state is updated by $x_i=γx_{i-1}+(1-γ)p_i$, and the two segments from $x_{i-1}$ to $x_i$ and from $x_i$ to $p_i$ are retained. The design parameter $γ$ controls the persistence of the state. We minimize the sum of the $α$-powers of the resulting edge lengths under independent uniform input and under 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 $0<α\leq1$, and every global minimizer is interior for $α>1$. The main result determines 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 which correction governs the location, 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. Under arbitrary input sequences, the optimal parameter and asymptotic worst-case edge-power cost per processed point are explicit for $0<α\leq3$. At high powers, periodic blocks and a separation argument show that this optimized cost is asymptotic to $2\log2/\logα$. Exact results for powers two and four, a rational recursion for all even powers, and a high-dimensional expansion provide additional descriptions of the optimizer.