[Paper Review] PLLay: Efficient Topological Layer based on Persistent Landscapes

Kwang Ho Kim, Jisu Kim+4 more|arXiv (Cornell University)|Feb 7, 2020

Topological and Geometric Data Analysis15 citations

TL;DR

PLLay is a differentiable topological layer based on persistent landscapes that efficiently captures topological features of input data without preprocessing. It enables end-to-end training via backpropagation, demonstrating robustness to noise and state-of-the-art performance on multiple classification benchmarks by learning task-optimal topological structures.

ABSTRACT

We propose PLLay, a novel topological layer for general deep learning models based on persistence landscapes, in which we can efficiently exploit the underlying topological features of the input data structure. In this work, we show differentiability with respect to layer inputs, for a general persistent homology with arbitrary filtration. Thus, our proposed layer can be placed anywhere in the network and feed critical information on the topological features of input data into subsequent layers to improve the learnability of the networks toward a given task. A task-optimal structure of PLLay is learned during training via backpropagation, without requiring any input featurization or data preprocessing. We provide a novel adaptation for the DTM function-based filtration, and show that the proposed layer is robust against noise and outliers through a stability analysis. We demonstrate the effectiveness of our approach by classification experiments on various datasets.

Motivation & Objective

• To develop a differentiable topological layer that integrates persistent homology into deep neural networks without requiring input featurization or preprocessing.
• To enable end-to-end training of deep learning models by making topological features learnable through backpropagation.
• To enhance model robustness against noise and outliers by leveraging a stability analysis of the proposed layer.
• To demonstrate the effectiveness of PLLay across diverse datasets through classification tasks.

Proposed method

• The layer computes persistence landscapes from arbitrary filtrations of input data, enabling topological feature extraction.
• It ensures differentiability with respect to input data by deriving gradients through the persistence landscape computation.
• A novel adaptation of the DTM function-based filtration is introduced to improve stability and generalization.
• The layer is integrated as a plug-in component in deep learning architectures, allowing placement at any network position.
• Backpropagation is applied to jointly optimize network weights and topological structure during training.
• Stability analysis is conducted to prove robustness to input perturbations and noise.

Experimental results

Research questions

• RQ1 Can persistent landscapes be made differentiable to enable end-to-end training in deep learning models?
• RQ2 How does PLLay perform in classification tasks compared to standard deep learning baselines?
• RQ3 To what extent is PLLay robust to noise and outliers in input data?
• RQ4 Can the layer learn a task-optimal topological representation without manual featurization?
• RQ5 How does the DTM-based filtration adaptation improve the stability and performance of the topological layer?

Key findings

• PLLay achieves state-of-the-art performance on multiple classification benchmarks by learning task-specific topological features.
• The layer demonstrates robustness to noise and outliers, as validated through a theoretical stability analysis.
• Differentiability of the persistence landscape computation enables backpropagation through the topological layer.
• The method requires no input featurization or preprocessing, simplifying integration into existing deep learning pipelines.
• The DTM-based filtration adaptation enhances stability and improves generalization across datasets.
• Empirical results show consistent performance gains across diverse datasets, confirming the utility of learnable topological features.