稀疏傅里叶和的最优外推界
Optimal Extrapolation Bounds for Sparse Fourier Sums
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中文总结 AI 辅助
研究任意实频率上\(k\)稀疏傅里叶和的最优外推界，给出相关不等式改进前人结果。作为算法结果，提高了聚类频率恢复算法的聚类中心分辨率，还获得稀疏傅里叶特征空间的外部杠杆得分和转移界。
中文摘要 AI 辅助
我们证明了关于任意实频率上的\(k\)稀疏傅里叶和的最优外推定理，无需任何分离假设，界定了在观察其能量的区间之外该和能有多大。对于每个\(g(t)=\sum_{j = 1}^k v_j e^{i\lambda_jt}\)，其中\(\lambda_j\in\mathbb{R}\)且每个\(x\geq1\)，有\(|g(x)|\leq k^{O(1)}\exp(O(k\mathop{\mathrm{arcosh}} x))\|g\|_{L^2[-1,1]}\)。在端点情形下，细化为明确的界\(|g(1 + \delta)|\leq O(k)\exp(O(k\sqrt\delta))\|g\|_{L^2[-1,1]}\)，\(0\leq\delta\leq1\)。这改进了Chen和Price（ICALP 2019）的\(\exp(O(k^2\log k\cdot\delta))\)增长估计，且指数缩放直至\(k\)中的常数和多项式因子是最优的。作为算法结果，我们将Chen - Price的聚类频率恢复算法的聚类中心分辨率提高了\(k\)倍，同时保持其样本复杂度直至对数因子。我们还获得了稀疏傅里叶特征空间的外部杠杆得分和转移界，将域内主动回归保证转换为采样区间之外基本精确的预测保证。
英文摘要
We prove an optimal extrapolation theorem for $k$-sparse Fourier sums over arbitrary real frequencies, without any separation assumption, bounding how large such a sum can be just outside an interval on which its energy is observed. For every $g(t)=\sum_{j=1}^k v_j e^{iλ_jt}$ with $λ_j\in\mathbb R$ and every $x\ge1$, $$ |g(x)|\le k^{O(1)}\exp(O(k\mathop{\mathrm{arcosh}} x))\|g\|_{L^2[-1,1]} . $$ In the endpoint regime, this refines to the explicit bound $$ |g(1+δ)|\le O(k)\exp(O(k\sqrtδ))\|g\|_{L^2[-1,1]}, \qquad 0\leδ\le1 . $$ This improves on the $\exp(O(k^2\log k\cdotδ))$ growth estimate of Chen and Price (ICALP 2019), and the exponential scaling is optimal up to constants and polynomial factors in $k$. As an algorithmic consequence, we improve the cluster-center resolution of Chen--Price's clustered-frequency recovery algorithm by a factor of $k$, while preserving its sample complexity up to logarithmic factors. We also obtain exterior leverage-score and transfer bounds for sparse Fourier feature spaces, converting in-domain active-regression guarantees into essentially sharp prediction guarantees just outside the sampling interval.
