Swing 滤波器的重启退化及一种简单修复
Restart Degeneration of the Swing Filter and a Simple Repair
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中文总结 AI 辅助
针对 Swing 滤波器在粗糙输入上的重启退化问题，提出半预算斜率走廊、真实端点弦和单样本桥的简单修复，恢复二次压缩尺度并保持误差界与内存。
中文摘要 AI 辅助
Swing 滤波器通过其先前记录延伸一条可行线，并在最后一个被接受的样本处从拟合值重启。我们表明，这种继承的端点会导致粗糙输入上的压缩出现奇异损失。对于固定容差下均匀采样的布朗运动，每个固定数量的记录时间收敛到第一个连续可行性边界，并伴有交替的边界误差。最长首段之后的段长在概率上趋于零。每当该边界先于观测视界时，相对于通过真实采样节点进行连续近似的段数比几乎必然发散。对于单位高斯增量、约束最小二乘且无最大滞后，第一平均段长具有容差 $E$ 的二次阶，第二平均段长具有线性阶，而从第三段起的每个固定平均段长具有 $E\log E$ 阶。我们通过条件生存尾部解释对数项，并证明在大容差极限下随后逼近平稳系数的几何界。平稳均值满足 $C_E\sim\gamma_{\mathrm{Swing}}E\log E$。一个显式的一维核指定了单一系数 $\gamma_{\mathrm{Swing}}\approx1.8120703$，并允许 $3/4$ Wasserstein 收缩和收敛的评估括号。该小数不是经过认证的误差区间。半预算斜率走廊、真实端点弦和单样本桥修复了退化，同时保留了原始误差界、连续输出和恒定工作内存。包括桥成本在内，每个输出段的长流平均跨度渐近于 $21\zeta(3)E^2/(8\pi^2)$。其显式系数由已知的布朗锚定寿命得出。因此，对记录规则的更改恢复了二次尺度。
英文摘要
The Swing filter extends a feasible line through its previous recording and restarts from a fitted value at the last accepted sample. We show that this inherited endpoint causes a singular loss of compression on rough input. For uniformly sampled Brownian motion at fixed tolerance, every fixed number of recording times converges to the first continuous feasibility boundary, with alternating boundary errors. The longest post-first segment tends to zero in probability. The segment-count ratio against a continuous approximation through true sampled knots diverges almost surely whenever that boundary precedes the observation horizon. For unit Gaussian increments, constrained least squares and no maximum lag, the first mean segment length has quadratic order in the tolerance $E$, the second has linear order, and every fixed mean from the third onward has order $E\log E$. We explain the logarithm through the conditional survival tail and prove a geometric bound on the subsequent approach to the stationary coefficient in the large-tolerance limit. The stationary mean satisfies $C_E\simγ_{\mathrm{Swing}}E\log E$. An explicit one-dimensional kernel specifies the single coefficient $γ_{\mathrm{Swing}}\approx1.8120703$ and admits a $3/4$ Wasserstein contraction and convergent evaluation brackets. The decimal is not a certified error interval. A half-budget slope corridor, true endpoint chords and one-sample bridges repair the degeneration while preserving the original error bound, continuous output and constant working memory. Including bridge costs, the long-stream mean span per output segment is asymptotic to $21ζ(3)E^2/(8π^2)$. Its explicit coefficient follows from the known Brownian anchored lifetime. Thus a change to the recording rule restores the quadratic scale.
