We adapt previous research on topological unsupervised learning to develop a unified functorial perspective on manifold learning and clustering. We first introduce overlapping hierachical clustering algorithms as functors and demonstrate that the maximal and single linkage clustering algorithms factor through an adaptation of the singular set functor. Next, we characterize manifold learning algorithms as functors that map uber-metric spaces to optimization objectives and factor through hierachical clustering functors. We use this characterization to prove refinement bounds on manifold learning loss functions and construct a hierarchy of manifold learning algorithms based on their invariants. We express several state of the art manifold learning algorithms as functors at different levels of this hierarchy, including Laplacian Eigenmaps, Metric Multidimensional Scaling, and UMAP. Finally, we experimentally demonstrate that this perspective enables us to derive and analyze novel manifold learning algorithms.
翻译:我们调整了先前关于地形学的未经监督的学习研究,以形成关于多重学习和集群的统一的传真观点。 我们首先引入了重叠的高频群集算法作为真菌, 并展示了通过调整奇数组真菌来调整最大和单一联系群算法的因素。 其次, 我们将多元学习算法定性为真菌, 绘制微菌空间, 以通过高频群集真菌优化目标和因素。 我们用这种定性来证明多重学习损失功能的精细界限, 并基于其异性构建了多种学习算法的等级。 我们以不同层次( 包括Laplacian Eigenmaps、Metric Moltalistrations和UMAP)的真菌化算法来表达多种艺术的多元学习算法。 最后, 我们实验性地证明这个角度可以让我们生成和分析新的多元学习算法。