近似点集的度量幅度
Approximating Metric Magnitude of Point Sets
AI总结：
本文研究点集度量幅度的高效近似，提出快速迭代算法与子集选择方法，并验证其在泛化差距分析、神经网络正则化和聚类中的应用。
摘要：
Metric magnitude（度量幅度）是衡量点云“大小”的一种度量，具有许多理想的几何性质。它已被应用于多种数学场景，近期研究表明它可以增强机器学习和优化算法。但当数据集较大或必须反复执行计算（例如在模型训练中）时，其计算成本限制了可用性。本文研究幅度计算问题，并给出近似该量的高效方法。我们证明它可以表述为凸优化问题，但不能表述为子模优化问题。论文描述了两种新算法：一种收敛迅速且准确的迭代近似算法，以及一种使计算更快的子集选择方法。此前已有研究提出，随机梯度下降过程中生成的模型序列的幅度与泛化差距相关。利用我们更具可扩展性的算法扩展该结果后发现，更长的序列实际上具有更高的相关性。我们还描述了幅度在机器学习中的新应用：作为神经网络训练的有效正则化项，以及作为一种新颖的聚类准则。
英文摘要：
Metric magnitude is a measure of the "size" of point clouds with many desirable geometric properties. It has been adapted to various mathematical contexts and recent work suggests that it can enhance machine learning and optimization algorithms. But its usability is limited due to the computational cost when the dataset is large or when the computation must be carried out repeatedly (e.g. in model training). In this paper, we study the magnitude computation problem, and show efficient ways of approximating it. We show that it can be cast as a convex optimization problem, but not as a submodular optimization. The paper describes two new algorithms - an iterative approximation algorithm that converges fast and is accurate, and a subset selection method that makes the computation even faster. It has been previously proposed that magnitude of model sequences generated during stochastic gradient descent is correlated to generalization gap. Extension of this result using our more scalable algorithms shows that longer sequences in fact bear higher correlations. We also describe new applications of magnitude in machine learning - as an effective regularizer for neural network training, and as a novel clustering criterion.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
