径向裁剪的精确联合偏差-能量包络
A Sharp Joint Bias-Energy Envelope for Radial Clipping
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中文总结 AI 辅助
该研究确定了仅存在p阶矩时径向裁剪的精确联合偏差-能量包络，分析了极值构型的转变，得到最优常数及统计应用。
中文摘要 AI 辅助
径向裁剪同时完成两项操作：去除向量在球外的部分，保留球内的有界向量。被去除的部分会产生偏差，而保留部分的平方范数会产生能量。当仅存在p阶矩（1<p≤2）时，我们确定了这些效应的精确联合代价。该问题最终是一维的：所有内容都由射线上一个点的位置决定。在α=pβ处，极值构型发生变化：一侧可在裁剪球上选择一个最坏点；另一侧唯一的正极值半径向外移动。当p=2时，在第一种 regime（区域）中所有内半径都相等，得到的常数在确定性不等式和对应的希尔伯特空间随机问题中都是最优的。在第一种 regime 中，对称律达到该界；在第二种 regime 中，上确界未被达到，但带有一个越来越罕见的异常值的两点族趋近于它。我们还展示了相同的转变如何决定精确的偏差-能量前沿，最后总结了统计结果及在在线学习中的应用。
英文摘要
Radial clipping does two things at once: it removes the part of a vector outside a ball and retains a bounded vector inside the ball. The removed part produces bias, while the squared norm of the retained part produces energy. We determine the exact joint price of these effects when only a $p$-moment, $1<p\le2$, is available. The problem turns out to be one-dimensional: everything is decided by the position of one point along a ray. At $α=pβ$, the extremal configuration changes. On one side a worst point can be chosen on the clipping sphere; on the other the unique positive extremal radius moves outside. When $p=2$, all inner radii tie in the first regime. The resulting constant is optimal both in the deterministic inequality and in the corresponding Hilbert-space stochastic problem. In the first regime, a symmetric law attains the bound. In the second, the supremum is not attained, but a two-point family with an increasingly rare outlier approaches it. We also show how the same transition determines the sharp bias-energy frontier. We conclude with statistical consequences and an application to online learning.
