[论文解读] Deep Learning with Topological Signatures

Christoph D. Hofer, Roland Kwitt+2 more|arXiv (Cornell University)|Jul 13, 2017

Topological and Geometric Data Analysis参考文献 26被引用 102

一句话总结

本论文提出一种可训练的输入层，将持久性图（拓扑特征）映射到任务优化的表征，使深度网络能够进行端到端学习并利用拓扑特征。它在图分类和二维形状识别方面显示出改进，尤其在社交网络图上显著超越了现有方法。

ABSTRACT

Inferring topological and geometrical information from data can offer an alternative perspective on machine learning problems. Methods from topological data analysis, e.g., persistent homology, enable us to obtain such information, typically in the form of summary representations of topological features. However, such topological signatures often come with an unusual structure (e.g., multisets of intervals) that is highly impractical for most machine learning techniques. While many strategies have been proposed to map these topological signatures into machine learning compatible representations, they suffer from being agnostic to the target learning task. In contrast, we propose a technique that enables us to input topological signatures to deep neural networks and learn a task-optimal representation during training. Our approach is realized as a novel input layer with favorable theoretical properties. Classification experiments on 2D object shapes and social network graphs demonstrate the versatility of the approach and, in case of the latter, we even outperform the state-of-the-art by a large margin.

研究动机与目标

• Motivate the use of topological data analysis (TDA) to capture robust topological and geometric information for varied data modalities.
• Develop a differentiable input layer that projects persistence diagrams into learnable representations compatible with deep networks.
• Ensure stability of the layer with respect to the 1-Wasserstein distance to persistent diagrams.
• Demonstrate the approach on diverse tasks (2D shape classification and social network graph classification) to show versatility.
• Show that learning task-tailored representations from topological signatures can outperform existing methods on challenging benchmarks.

提出的方法

• Introduce a parametrized projection layer that takes a persistence diagram and projects it using a learnable set of structure elements.
• Use a rotated coordinate system to map diagram points to a 1D persistence axis and a 1D birth-persistence axis.
• Define a family of differentiable structure functions s_mu,sigma,nu that produce a scalar projection per element and concatenate across N elements.
• Prove stability: the layer is Lipschitz with respect to the 1-Wasserstein distance, ensuring robustness to diagram perturbations.
• Train the layer end-to-end with standard backpropagation, leveraging differentiability of s_mu,sigma,nu with respect to mu and sigma.
• Demonstrate that the approach handles diagrams with varying cardinality and can be integrated as a general-purpose multiset input layer.

实验结果

研究问题

• RQ1 Can a trainable input layer map persistence diagrams into representations that are both stable (Wasserstein) and task-optimized for deep learning?
• RQ2 Does incorporating topological signatures via end-to-end learning improve performance on heterogeneous tasks such as 2D shape classification and graph classification?
• RQ3 To what extent do essential (never-dying) topological features contribute to discriminative power in practice?
• RQ4 How does the proposed layer compare to traditional vectorizations or kernel-based approaches in terms of scalability and accuracy?

主要发现

• The authors propose S_theta,nu as a trainable, differentiable layer that projects persistence diagrams into a fixed-size vector determined by learnable structure elements.
• The layer is proven Lipschitz continuous with respect to the 1-Wasserstein distance, ensuring stability of topological signatures under perturbations.
• In experiments, the approach demonstrates versatility by applying to 2D shape classification and social network graph classification, achieving strong results and outperforming the state-of-the-art on the graph task.
• The method handles diagrams of varying cardinality and can be seen as a general way to incorporate multisets of real-valued vectors into deep networks.
• Using essential topological features (as opposed to ignoring them) provides notable performance gains in graph classification.