AI 中文总结
该研究针对分数布朗运动的平衡偏差问题，在H=1/2处发现相变，给出不同H值下的偏差界，刻画了解空间几何，还提出了对应偏差界的随机多项式时间算法。
AI 中文摘要
我们研究在区间[0,1]上具有Hurst指数H∈(0,1)的n条独立分数布朗运动样本路径的平衡偏差，这是平衡高斯向量的无限维类似问题。我们在H=1/2处建立了一个相变：以高概率，偏差为Ω(n^{1/2-H})，且为O(n^{1/2-H}(log n)^{c(H)})，其中当H≥1/2时c(H)=H+1/2，否则c(H)=1/2。在临界指数H=1/2处，我们证明当n→∞时，偏差以恒定概率为Θ(1)。在该区域中，我们进一步通过计算局部极小值的期望数量、在存在阈值附近建立重叠间隙性质，并在每个发散最优性阈值处证明其不存在，来刻画解空间的几何结构。我们还给出了随机多项式时间算法，该算法以高概率计算出具有如下偏差的符号：对于H<1/2，偏差为O(n^{1/2-H}√log n)；对于H=1/2，偏差为O((log n)^{3/2})；对于H>1/2，偏差为O(√log n)。我们的分析结合了基于分数布朗运动小波表示的截断平衡论证与概率方法。
英文摘要
We study the discrepancy of balancing $n$ independent sample paths of fractional Brownian motion with Hurst exponent $H\in(0,1)$ on $[0,1]$, an infinite-dimensional analogue of balancing Gaussian vectors. We establish a phase transition at $H=1/2$: with high probability, the discrepancy is $Ω(n^{1/2-H})$ and $\mathcal O(n^{1/2-H}(\log n)^{c(H)})$, where $c(H)=H+1/2$ if $H\geq 1/2$ and $c(H)=1/2$ otherwise. At the critical exponent $H=1/2$, we show that the discrepancy is $Θ(1)$ with constant probability as $n\to\infty$. In this regime, we further characterize the geometry of the solution space by computing the expected number of local minima, establishing an overlap gap property near the existence threshold, and proving its absence at every diverging optimality threshold. We also give randomized polynomial-time algorithms that compute signings with discrepancy $\mathcal O(n^{1/2-H}\sqrt{\log n})$ for $H<1/2$, $\mathcal O((\log n)^{3/2})$ for $H=1/2$, and $\mathcal O(\sqrt{\log n})$ for $H>1/2$, with high probability. Our analysis combines a truncated balancing argument based on a wavelet representation of fractional Brownian motion with probabilistic methods.
Comments39 pages
