A Topological Regularizer for Classifiers via Persistent Homology
A Topological Regularizer for Classifiers via Persistent Homology
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发表时间：
2018-06
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通讯作者：
Chao Chen;Xiuyan Ni;Qinxun Bai;Yusu Wang
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作者：
Chao Chen;Xiuyan Ni;Qinxun Bai;Yusu Wang
正则化在监督学习中起着至关重要的作用。大多数现有的方法都以结构不可知的方式执行全局正则化。在本文中，我们提出了一个新的方向，并建议加强结构简单的分类边界的拓扑复杂性的正则化。特别是，我们对拓扑复杂性的测量包含了拓扑特征的重要性（例如，连接的组件、句柄等），并提供对伪拓扑结构的直接控制。我们将新的测量作为训练分类器的拓扑惩罚。我们还提出了一个有效的算法来计算这种惩罚的梯度。我们的方法提供了一种新的方法来拓扑简化模型的全局结构，而不必牺牲太多的模型的灵活性。我们证明了我们的新拓扑正则化的有效性在一系列的合成和真实世界的数据集。
Regularization plays a crucial role in supervised learning. Most existing methods enforce a global regularization in a structure agnostic manner. In this paper, we initiate a new direction and propose to enforce the structural simplicity of the classification boundary by regularizing over its topological complexity. In particular, our measurement of topological complexity incorporates the importance of topological features (e.g., connected components, handles, and so on) in a meaningful manner, and provides a direct control over spurious topological structures. We incorporate the new measurement as a topological penalty in training classifiers. We also propose an efficient algorithm to compute the gradient of such penalty. Our method provides a novel way to topologically simplify the global structure of the model, without having to sacrifice too much of the flexibility of the model. We demonstrate the effectiveness of our new topological regularizer on a range of synthetic and real-world datasets.