Optimal exponential memory for sequential Euclidean connections: edge-power costs and phase transitions
Abstract
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 x0=p0, each input point pi is attached to xi−1, and the state is updated by xi=γxi−1+(1−γ)pi. Retaining xi 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≤1, while every global minimizer is interior for α>1. We determine the finite optimizer in the joint window αN=1+εN, εNlog N→λ. Below an explicit threshold it lies on the N−1/2 scale, at the threshold its scale is √(log N/(Nloglog 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≤3. At high powers, periodic antipodal block inputs give explicit lower bounds which, with a separation argument, show that the optimized cost is asymptotic to 2log2/logα. Exact results for powers two and four, a rational recursion for every even power, and a high-dimensional expansion complete the analysis.
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BibTeX
@misc{castro2026optimal,
title = {{Optimal exponential memory for sequential Euclidean connections: edge-power costs and phase transitions}},
author = {Pedro M. M. de Castro},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.27777},
eprint = {2608.27777},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27777},
}
RIS
TY - UNPB
TI - Optimal exponential memory for sequential Euclidean connections: edge-power costs and phase transitions
AU - Pedro M. M. de Castro
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2608.27777
UR - https://arxiv.org/abs/2608.27777
AB - 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≤1, while every global minimizer is interior for α>1. We determine the finite optimizer in the joint window α_N=1+ε_N, ε_Nlog N→λ. Below an explicit threshold it lies on the N^−1/2 scale, at the threshold its scale is √(log N/(Nloglog 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≤3. At high powers, periodic antipodal block inputs give explicit lower bounds which, with a separation argument, show that the optimized cost is asymptotic to 2log2/logα. Exact results for powers two and four, a rational recursion for every even power, and a high-dimensional expansion complete the analysis.
ER -