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

2026-06-25

AI Summary

This work addresses the problem of efficiently learning a high-dimensional Gaussian distribution under truncation by an unknown halfspace. By introducing a novel interpretation of low-order moments through relative truncation parameters, the authors propose a direct parameter recovery method that bypasses projected stochastic gradient descent. The algorithm recovers the original Gaussian to within total variation distance ε using only Õ(d²/ε²) samples. Its runtime is dominated by covariance matrix computation, and it achieves information-theoretically optimal sample complexity as well as optimal time complexity—establishing, for the first time, a doubly optimal guarantee for this problem setting.

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.

Problem

• Gaussian learning
• halfspace truncation
• truncated distributions
• high-dimensional statistics
• sample complexity

Innovation

• truncated Gaussian
• halfspace truncation
• optimal sample complexity
• relative truncation parameter
• efficient parameter recovery