这篇论文给出了学习半空间截断高斯分布的最优算法，样本和时间复杂度都达到理论下界，而且避开了繁琐的随机梯度下降，值得了解。
Lee、Mehrotra和Zampetakis（FOCS'24）首次提出多项式时间算法学习高维截断高斯，但样本与时间非最优。本研究针对非平凡截断，给出高效算法，使用n = Õ(d²/ε²)个样本在总变差距离上达到ε误差。算法时间复杂度主要由计算经验协方差矩阵主导。该样本与时间复杂度在d和ε上均为最优，即使无截断时亦如此。关键创新在于用相对截断参数重新解释截断高斯低阶矩，从而直接恢复参数，避开耗时投影随机梯度下降。
Fast algorithms for learning a Gaussian under halfspace truncation with optimal sample complexity
We study the fundamental problem of learning a high-dimensional Gaussian truncated to an unknown halfspace. Lee, Mehrotra and Zampetakis (FOCS'24) recently obtained the first polynomial time algorithm for this problem, but their resulting sample and time complexity bounds are not optimal. Under non-trivial truncation, for any target accuracy $\varepsilon > 0$ and dimension $d$ we give an efficient algorithm that uses $n = \tilde{O}(d^2/\varepsilon^2)$ samples and learns the underlying Gaussian to error $\varepsilon$ in total variation distance. Our algorithm is also fast: its runtime is dominated by the cost of computing the empirical covariance matrix. Both our sample and time complexity are optimal in terms of $d$ and $\varepsilon$ even without truncation: in this regard, we can learn a Gaussian under halfspace truncation for free. The key ingredient behind our result is a novel reinterpretation of the low-degree moments of the truncated Gaussian in terms of a relative truncation parameter. This relative truncation parameter uniquely determines the parameters of the untruncated Gaussian and enables direct parameter recovery. This reinterpretation allows us to circumvent the time intensive projected stochastic gradient descent procedure that is widely used in learning under truncation.