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.

在高维背景下，本文通过对数据的随机压缩版本进行训练和全秩度量，在距离度量学习的错误方面提供了理论保证，这不依赖于环境维度，同时假定数据来自有界支持并自动在存在良好几何结构时收缩。合成和真实数据集的实验结果支持我们在高维背景下的理论发现。

内在维度对于压缩下度量学习的影响

The Effect of Intrinsic Dimension on Metric Learning under Compression

As an alternative to variable selection or shrinkage in high dimensional
regression, we propose to randomly compress the predictors prior to analysis.
This dramatically reduces storage and computational bottlenecks, performing
well when the predictors can be projected to a low dimensional linear subspace
with minimal loss of information about the response. As opposed to existing
Bayesian dimensionality reduction approaches, the exact posterior distribution
conditional on the compressed data is available analytically, speeding up
computation by many orders of magnitude while also bypassing robustness issues
due to convergence and mixing problems with MCMC. Model averaging is used to
reduce sensitivity to the random projection matrix, while accommodating
uncertainty in the subspace dimension. Strong theoretical support is provided
for the approach by showing near parametric convergence rates for the
predictive density in the large p small n asymptotic paradigm. Practical
performance relative to competitors is illustrated in simulations and real data
applications.

本文提出一种基于随机压缩预测变量的高维回归方法，并使用模型平均来减少方法对于随机投影矩阵的敏感度，可用于实际数据应用。