Testing Convex Truncation.
A. De and S. Nadimpalli and R. Servedio.
ACM-SIAM Symposium on Discrete Algoithms (SODA), pp. 4050-4082, 2023.

Abstract:

We study the basic statistical problem of testing whether normally distributed n-dimensional data has been truncated, i.e. altered by only retaining points that lie in some unknown truncation set $S \subseteq \R^n$. As our main algorithmic results,

  • We give a computationally efficient $O(n)$-sample algorithm that can distinguish the standard normal distribution $N(0,I_n)$ from $N(0,I_n)$ conditioned on an unknown and arbitrary convex set $S$.
  • We give a different computationally efficient $O(n)$-sample algorithm that can distinguish $N(0,I_n)$ from $N(0,I_n)$ conditioned on an unknown and arbitrary mixture of symmetric convex sets.

    • These results stand in sharp contrast with known results for learning or testing convex bodies with respect to the normal distribution or learning convex-truncated normal distributions, where state-of-the-art algorithms require essentially $n^{\sqrt{n}}$ samples. An easy argument shows that no finite number of samples suffices to distinguish $N(0,I_n)$ from an unknown and arbitrary mixture of general (not necessarily symmetric) convex sets, so no common generalization of the above two results is possible.

    We also prove lower bounds on the sample complexity of distinguishing algorithms (computationally efficient or otherwise) for various classes of convex truncations; in some cases these lower bounds match our algorithms up to logarithmic or even constant factors.

    pdf of full version, including strengthened lower bound from preliminary conference version

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