Given $n$ observations from two balanced classes, consider the task of
labeling an additional $m$ inputs that are known to all belong to \emph{one} of
the two classes. Special cases of this problem are well-known: with complete
knowledge of class distributions ($n=\infty$) the problem is solved optimally
by the likelihood-ratio test; when $m=1$ it corresponds to binary
classification; and when $m\approx n$ it is equivalent to two-sample testing.
The intermediate settings occur in the field of likelihood-free inference,
where labeled samples are obtained by running forward simulations and the
unlabeled sample is collected experimentally. In recent work it was discovered
that there is a fundamental trade-off between $m$ and $n$: increasing the data
sample $m$ reduces the amount $n$ of training/simulation data needed. In this
work we (a) introduce a generalization where unlabeled samples come from a
mixture of the two classes -- a case often encountered in practice; (b) study
the minimax sample complexity for non-parametric classes of densities under
\textit{maximum mean discrepancy} (MMD) separation; and (c) investigate the
empirical performance of kernels parameterized by neural networks on two tasks:
detection of the Higgs boson and detection of planted DDPM generated images
amidst CIFAR-10 images. For both problems we confirm the existence of the
theoretically predicted asymmetric $m$ vs $n$ trade-off.

在这项工作中，我们介绍了一种推广的情况，其中未标记的样本来自两类的混合物，研究了在最大均值差异（MMD）分离条件下非参数密度类的极小样本复杂性，并且通过用神经网络参数化的核函数在两个任务上进行了实证性能研究，即在探测希格斯玻色子和检测植入的 DDPM 生成图像与 CIFAR-10 图像之间。对于这两个问题，我们确认了理论预测的非对称的 m 与 n 之间的交换比例存在。