Sample Complexity of Composite Quantum Hypothesis Testing

This paper investigates symmetric composite binary quantum hypothesis testing (QHT), where the goal is to determine which of two uncertainty sets contains an unknown quantum state. While asymptotic error exponents for this problem are well-studied, the finite-sample regime remains poorly understood. We bridge this gap by characterizing the sample complexity -- the minimum number of state copies required to achieve a target error level. Specifically, we derive lower bounds that generalize the sample complexity of simple QHT and introduce new upper bounds for various uncertainty sets, including of both finite and infinite cardinalities. Notably, our upper and lower bounds match up to universal constants, providing a tight characterization of the sample complexity. Finally, we extend our analysis to the differentially private setting, establishing the sample complexity for privacy-preserving composite QHT.

The Effect of Intrinsic Dimension on Metric Learning under Compression

Metric learning aims at finding a suitable distance metric over the input space, to improve the performance of distance-based learning algorithms. In high-dimensional settings, metric learning can also play the role of dimensionality reduction, by imposing a low-rank restriction to the learnt metric. In this paper, instead of training a low-rank metric on high-dimensional data, we consider a randomly compressed version of the data, and train a full-rank metric there. We give theoretical guarantees on the error of distance-based metric learning, with respect to the random compression, which do not depend on the ambient dimension. Our bounds do not make any explicit assumptions, aside from i.i.d. data from a bounded support, and automatically tighten when benign geometrical structures are present. Experimental results on both synthetic and real data sets support our theoretical findings in high-dimensional settings.