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A Topological Regularizer for Classifiers via Persistent Homology
Chao Chen, Xiuyan Ni, Qinxun Bai, Yusu Wang
TL;DR
Existing regularization can enforce model simplicity without accounting for the topology of a classification boundary. This paper introduces a robustness-weighted topological penalty and a gradient method, reporting effectiveness on synthetic and real-world datasets. Its treatment focuses on connected components.
Problem
Regularization seeks simplicity to balance training fit and generalization, while this paper identifies boundary topology as an alternative to geometric simplicity.
Source: §1 · p. 1Method
The method penalizes the squared robustness of topological structures in the classifier boundary and computes gradients using a piecewise-linear surrogate.
Source: §3 · pp. 6–7Results
Takeaways & Limitations
The regularizer aims to simplify global model structure without sacrificing too much flexibility and provides direct control over spurious topological structures.
Source: §1 · p. 1For computational efficiency, this paper focuses on connected components; handles and voids are described as extensions.
Source: §1 · p. 2
Abstract
from arXiv · showhide
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 pro- pose an efficient algorithm to compute the gradient of such penalty. Our method pro- vides 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.
1 Introduction
The paper argues that structure-agnostic regularization can force a choice between overfitting and excessive smoothing. It proposes regularizing classification-boundary topology, weighting structures by robustness, and computing a tractable penalty gradient.
1 Introduction
- Structure-agnostic geometric regularizers can leave classifiers overfit or smooth their boundaries enough to reduce accuracy.The authors identify this flexibility trade-off as a limitation of regularizing total classifier or boundary smoothness.Source: §1 · p. 1
- The proposed regularizer penalizes classification-boundary topological complexity, aiming to control spurious structures without over-smoothing the boundary.Its measure incorporates the importance of topological structures and can be combined with geometric regularizers or functional norms.Source: §1 · p. 1
- The penalty sums squared robustness values, distinguishing similarly shaped connected components by how readily perturbations remove them.The paper illustrates that one component can be less robust to noise because it depends on only a few training points.Source: §1 · p. 2
- To compute the penalty gradient, the authors use a piecewise-linear surrogate and show differentiability almost everywhere.The surrogate addresses the lack of closed-form critical points for the classifier function.Source: §1 · p. 2
- The authors report experiments on synthetic and real-world datasets, while focusing this paper's computational treatment on connected components.They state that the framework can be extended to handles and voids.Source: §1 · p. 2
- Where prior methods use topological information as data features, this method uses it as a prior for training the classifier.The paper identifies its approach as the first, to its knowledge, to use topology in this way.Source: §1 · p. 3
2 Level Set, Topology, and Robustness
The section studies binary classifiers through their zero-level-set boundaries and ranks connected components by how much the classifier must be perturbed to remove them. It relates this robustness to critical-point pairings and describes an algorithm for computing the pairings in low- and high-dimensional settings.
Level Set and Topology
- The exposition focuses on binary classification over a compact, simply connected hypercube and treats connected components as its primary topological structures.The authors state that the method can be extended to higher-dimensional structures such as handles and voids.Source: §2 · p. 3
Robustness
- A component’s robustness is the minimum perturbation needed to remove it from the zero-level-set boundary, and is linked to a minimum–saddle persistence pairing.The paired critical points determine how much the classifier function must change to eliminate the component.Source: §2 · p. 4
- A loop’s robustness reflects its function values, not just its geometry: the left loop has robustness 0.21, while the right loop has robustness 0.83.The left loop is less stable because its basin is shallower and a nearby valley lies closer to zero.Source: §2 · p. 4
Algorithm
- For low-dimensional domains, the algorithm computes 0-dimensional persistent homology for f and −f on a discretized grid, using sorting and union-find to build the merge tree.Its stated complexity is O(n log n + mα(n)).Source: §2 · pp. 4–5
- Cutting the merging tree at zero identifies boundary components: a pairing belongs to Πf when its minimum is below zero and its merge saddle is above zero.The global-minimum/global-maximum pair accounts for the component without a corresponding merge pairing.Source: §2 · p. 5
- For high-dimensional feature spaces, the method uses a k-nearest-neighbor graph on training points, so extracted critical points are training data points.The authors justify this approximation with experimental evidence from clustering applications.Source: §2 · p. 5
3 Topological Penalty and Gradient
The section defines a robustness-weighted penalty for classifier-boundary components and derives a piecewise-linear surrogate whose penalty is differentiable almost everywhere.
3 Topological Penalty and Gradient
- Training minimizes a weighted sum of per-example loss and the topological penalty.The per-example term may be cross-entropy, quadratic, or hinge loss.Source: §3 · p. 6
- The penalty sums squared robustness scores for boundary components, excluding the most robust component.Each component’s robustness is the smaller absolute classifier value at its paired critical points.Source: §3 · p. 6
- A piecewise-linear approximation on a discretized grid makes the penalty gradient computable without directly finding continuous-domain critical points.The surrogate matches classifier values at grid vertices and interpolates linearly along edges.Source: §3 · p. 6
- The surrogate penalty is differentiable almost everywhere; non-unique critical-point selections occur only on a measure-zero subset.Equal function values or equal absolute values can make the selected critical point non-unique and the gradient undefined.Source: §3 · pp. 6–7
- Gradient descent pushes each component’s selected critical-point value toward zero, shrinking the corresponding boundary structure.This can move a saddle toward zero or raise a minimum, depending on which critical point determines robustness.Source: §3 · p. 7
- The method is instantiated for kernel logistic regression and extended to multilabel classification using one scalar boundary function per class.For multilabel classification, the penalty sums robustness across the class-specific functions.Source: §3 · p. 7
4 Experiments
The experiments evaluate TopoReg on synthetic and real-world datasets against standard and geometry-based classifiers. It generally outperforms existing methods and is consistently best or close to best, with a larger advantage on noisy synthetic data.
4 Experiments
- Evaluation covers synthetic noise conditions, UCI and two biomedical datasets, with six-fold testing and five-fold cross-validation for hyperparameter selection.Baselines include KNN, logistic regression, SVM, kernel logistic regression with L1 or L2 norms, EE, and DGR.Source: §4 · pp. 7–8
- The average running time over all datasets for the method is 2.08 seconds; 2D data are normalized to the unit square and discretized on a 300×300 grid, while high-dimensional data use a KNN graph with k=3.Source: §4 · pp. 7–8
- TopoReg generally outperforms existing methods and is consistently best or close to best across the datasets in Table 1.The table reports mean and standard deviation error rates.Source: §4 · p. 8, Table 1 · p. 8
- On noisy synthetic data, TopoReg has a bigger advantage and remains relatively stable under large label noise, unlike the more sensitive geometric regularizers.The authors suggest those geometric regularizers may be more sensitive to initialization and prone to poor local optima.Source: §4 · p. 8
A Background: Persistent Homology
Persistent homology tracks topological features as function values are swept, pairing their creation and disappearance. The paper uses persistence pairings to identify connected components of a classifier's zero-level boundary and measure their robustness.
A Background: Persistent Homology
- Persistent homology tracks feature creation and disappearance during a sublevel-set sweep, pairing critical points whose function values give birth and death times.The difference between those times measures the feature's persistence, or lifetime.Source: A Background: Persistent Homology · pp. 10–11
- In Figure 6, the component created at minimum x3 merges at maximum x4, yielding the critical-point pairing (x3, x4) and diagram point (f3, f4).Source: Figure 6 · p. 11
- Level-set zigzag pairings that span function value zero correspond to connected components of the classifier's zero-level boundary; robustness is the smaller absolute endpoint value.Source: A Background: Persistent Homology · p. 11
- Although general level-set zigzag persistence takes O(n^3), the required pairings can be computed using 0D ordinary persistence for f and -f on the simply connected domain.The paper's argument uses the domain's trivial first homology and connectedness to simplify the computation.Source: A Background: Persistent Homology · pp. 11–12
- The persistence framework can be extended from connected components to higher-dimensional features such as handles and voids.Source: A Background: Persistent Homology · p. 12