Overview
Research explores a method for sequential Euclidean tree construction that employs an exponentially weighted memory, focusing on the insertion cost during the processing of points in a prescribed order. The system maintains a single $d$-dimensional point of working state, $x_i$, which compresses the previously observed points. This point is updated via the rule $x_0=p_0$ and $x_i=\gamma x_{i-1}+(1-\gamma)p_i$, where $0\leq\gamma\leq1$. The study analyzes the distributional performance and worst-case guarantees associated with this approach.
Research Context
The problem concerns the insertion cost, defined as $\sum_{i=1}^N\lVert p_i-x_{i-1}\rVert^\alpha$, for a sequence of points $p_0,p_1,\ldots,p_N$ within the unit ball of $\mathbb R^d$. The value $x_{i-1}$ is computed from the previously observed points. Existing methods, such as the input-order path, are sensitive to the input distribution and can incur a cost equivalent to the diameter under adversarial conditions. Conversely, a center star construction offers controlled worst-case scale but disregards the sequence of observations. The introduced exponential weighting scheme aims to address these limitations by incorporating sequential memory into the computation of $x_{i-1}$.
Approach
The methodology involves defining a specific memory compression scheme where each $x_i$ is an exponentially weighted average of the input sequence. This scheme is parameterized by $\gamma$, controlling the weighting of past observations relative to the current point $p_i$. The study investigates the behavior of this system under two primary conditions: independent uniform points and arbitrary input sequences.
- For independent uniform points, the analysis focuses on the stationary insertion length and the determination of optimal constant parameters for $N$ insertions.
- For arbitrary input sequences, the investigation centers on the largest asymptotic mean cost for fixed $\gamma$.
Findings
Distributional Performance with Independent Uniform Points
- The stationary insertion length is nonincreasing in the usual stochastic order as $\gamma$ increases.
- For $d\geq2$ and $\alpha>0$, every optimal constant parameter for $N$ insertions, denoted $\gamma_N^*$, satisfies $1-\gamma_N^*=\Theta(N^{-1/2})$.
- The study determines the asymptotic constant for this parameter and the resulting $\sqrt N$ correction, providing explicit bounds dependent on $d$ and $\alpha$.
- When $\alpha=1$, the leading expected tree length is equivalent to that of the center star construction and is strictly smaller than the lengths associated with endpoint constructions.
- For $\alpha=2$, the minimizer for $N\geq2$ is unique, with $1-\gamma_N^*=N^{-1/2}-\tfrac12N^{-1}+O(N^{-3/2})$.
Worst-Case Guarantees for Arbitrary Input Sequences
- For arbitrary input sequences and a fixed $0\leq\gamma<1$, the largest asymptotic mean cost is $(2/(1+\gamma))^\alpha$ for $0<\alpha\leq3$. This value is strictly below the path value when $\gamma>0$.
- Among fixed nonnegative weighting rules where contributing points have the same average distance in the input order from the most recent point, exponential weighting performs within a factor smaller than $1.161^\alpha$ of the best adversarial value in dimension at least two. This ratio approaches one as the average distance grows.