论文标题
估计高维度的措施
Estimating Barycenters of Measures in High Dimensions
论文作者
论文摘要
Barycentric平均是总结措施人群的原则方法。用于估计barycenters的现有算法通常将其作为加权总和的参数，并优化其权重和/或位置。但是，由于维度的诅咒，这些方法并未扩展到高维设置。在本文中，我们提出了一种可扩展和一般算法，用于估计高维度中的措施的重中心。关键思想是将对措施的优化转变为对生成模型的优化，引入诱导性偏差，从而使方法可以扩展，同时仍然准确地估算了barycenters。我们证明，在对差异的轻度假设下，局部收敛，表明该方法已得到良好的态度。我们证明我们的方法很快，在低维问题上取得了良好的性能，并缩放到高维设置。特别是，我们的方法是第一个用于估计数千个维度的barycenters的方法。
Barycentric averaging is a principled way of summarizing populations of measures. Existing algorithms for estimating barycenters typically parametrize them as weighted sums of Diracs and optimize their weights and/or locations. However, these approaches do not scale to high-dimensional settings due to the curse of dimensionality. In this paper, we propose a scalable and general algorithm for estimating barycenters of measures in high dimensions. The key idea is to turn the optimization over measures into an optimization over generative models, introducing inductive biases that allow the method to scale while still accurately estimating barycenters. We prove local convergence under mild assumptions on the discrepancy showing that the approach is well-posed. We demonstrate that our method is fast, achieves good performance on low-dimensional problems, and scales to high-dimensional settings. In particular, our approach is the first to be used to estimate barycenters in thousands of dimensions.
