We combine standard persistent homology with image persistent homology to
define a novel way of characterizing shapes and interactions between them. In
particular, we introduce: (1) a mixup barcode, which captures
geometric-topological interactions (mixup) between two point sets in arbitrary
dimension; (2) simple summary statistics, total mixup and total percentage
mixup, which quantify the complexity of the interactions as a single number;
(3) a software tool for playing with the above.
As a proof of concept, we apply this tool to a problem arising from machine
learning. In particular, we study the disentanglement in embeddings of
different classes. The results suggest that topological mixup is a useful
method for characterizing interactions for low and high-dimensional data.
Compared to the typical usage of persistent homology, the new tool is sensitive
to the geometric locations of the topological features, which is often
desirable.

我们通过结合标准持久化同调与图像持久化同调，定义了一种描述形状及其之间相互作用的新方法。特别地，我们引入了混合条码（Mixup Barcode），它能够捕捉任意维度中两个点集之间的几何 - 拓扑相互作用（混合效果）；我们还提出了简单的总混合（Total Mixup）和总混合百分比（Total Percentage Mixup）两种统计指标，用于量化相互作用的复杂性；同时，我们还开发了一种软件工具，可用于处理上述问题。作为概念验证，我们将该工具应用于机器学习中的一个问题，即研究不同类别嵌入表示中的解耦（Disentanglement）问题。结果表明，拓扑混合是一种有用的方法，可以用于描述低维和高维数据的相互作用。与持久化同调的典型应用相比，这个新工具更加敏感于拓扑特征的几何位置，这在许多情况下是非常可取的。