[Paper Review] TopoReg: A Topological Regularizer for Classifiers.

Chao Chen, Xiuyan Ni+2 more|arXiv (Cornell University)|Jun 27, 2018

Topological and Geometric Data Analysis4 citations

TL;DR

TopoReg introduces a topological regularizer that simplifies the global structure of classification boundaries by measuring and penalizing topological complexity—such as spurious connected components and handles—using a principled, differentiable loss. The method enables smoother, more generalizable models without sacrificing model flexibility, demonstrating state-of-the-art performance on synthetic and real-world datasets.

ABSTRACT

Regularization plays a crucial role in supervised learning. A successfully regularized model strikes a balance between a perfect description of the training data and the ability to generalize to unseen data. 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 loss in training classifiers. We also propose an efficient algorithm to compute the gradient. 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.

Motivation & Objective

• To address the limitation of existing regularization methods that lack structural awareness by introducing topological simplicity as a new inductive bias.
• To define a meaningful, differentiable measure of topological complexity in classification boundaries that captures features like connected components and handles.
• To develop an efficient gradient computation algorithm for the topological loss to enable end-to-end training of deep classifiers.
• To demonstrate that controlling topological complexity improves generalization without reducing model capacity.

Proposed method

• Proposes a topological complexity measurement based on persistent homology to quantify features such as connected components and loops in the decision boundary.
• Designs a differentiable topological loss function that penalizes the presence of spurious topological features, promoting simpler, more generalizable decision boundaries.
• Introduces an efficient gradient computation method using the subgradient of the topological loss, enabling backpropagation through the topological regularizer.
• Integrates the topological loss as a regularizer in the training objective, combining it with standard cross-entropy loss for supervised learning.
• Employs a differentiable approximation of the topological persistence diagram to enable gradient-based optimization.
• Applies the regularizer during training of neural networks, allowing end-to-end optimization while maintaining model flexibility.

Experimental results

Research questions

• RQ1 Can topological complexity of a classifier's decision boundary be effectively measured and used as a regularizer?
• RQ2 How does enforcing topological simplicity improve generalization in supervised learning?
• RQ3 Can a differentiable topological loss be efficiently computed and integrated into deep learning training pipelines?
• RQ4 Does the proposed regularizer reduce spurious topological structures like isolated components or loops in the decision boundary?
• RQ5 To what extent does TopoReg improve generalization on both synthetic and real-world datasets compared to standard regularization methods?

Key findings

• The topological regularizer significantly reduces spurious topological features such as disconnected components and handles in the decision boundary.
• TopoReg improves generalization performance on both synthetic and real-world datasets, outperforming standard regularization baselines.
• The method maintains model flexibility while simplifying the global structure of the decision boundary, avoiding overfitting.
• The differentiable topological loss enables effective backpropagation, allowing seamless integration into deep learning frameworks.
• Empirical results show that models trained with TopoReg achieve higher test accuracy and better robustness to distribution shift.
• The topological complexity measurement effectively captures structural simplicity, providing a meaningful inductive bias beyond traditional regularization.