半空间截断下高斯分布学习的最优样本复杂度快速算法

Computer Science > Data Structures and Algorithms

arXiv:2606.27298v1 (cs) [Submitted on 25 Jun 2026]

Title:Fast algorithms for learning a Gaussian under halfspace truncation with optimal sample complexity

Authors:Haitong Liu, Deepak Narayanan Sridharan, David Steurer, Manuel Wiedmer View a PDF of the paper titled Fast algorithms for learning a Gaussian under halfspace truncation with optimal sample complexity, by Haitong Liu and 3 other authors View PDF HTML (experimental)

Abstract: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.

Comments:	88 pages; accepted at the 39th Annual Conference on Learning Theory (COLT 2026)
Subjects:	Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Statistics Theory (math.ST); Machine Learning (stat.ML)
Cite as:	arXiv:2606.27298 [cs.DS]
	(or arXiv:2606.27298v1 [cs.DS] for this version)
	https://doi.org/10.48550/arXiv.2606.27298 Focus to learn more arXiv-issued DOI via DataCite

Submission history

From: Manuel Wiedmer [view email]
[v1] Thu, 25 Jun 2026 17:14:57 UTC (80 KB)
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