[Paper Review] Persistence Images: A Stable Vector Representation of Persistent Homology

Henry Adams, Sofya Chepushtanova+8 more|arXiv (Cornell University)|Jul 22, 2015

Topological and Geometric Data AnalysisComputer Science46 references465 citations

TL;DR

This paper introduces persistence images (PIs), a stable, vectorized representation of persistence diagrams that enables efficient application of machine learning tools. By transforming a persistence diagram into a weighted sum of Gaussian kernels and discretizing the result into a grid, PIs preserve topological structure while allowing for fast computation and high classification accuracy, outperforming prior methods on synthetic and dynamical systems data.

ABSTRACT

Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.

Motivation & Objective

• To develop a stable, finite-dimensional vector representation of persistence diagrams suitable for machine learning.
• To address limitations of persistence diagrams in compatibility with standard ML tools, such as support vector machines and feature selection.
• To ensure the representation is computationally efficient, stable under noise, and interpretable.
• To enable accurate inference of parameters in complex dynamical systems using topological features.
• To demonstrate the utility of PIs in real-world data analysis tasks with high-dimensional or noisy datasets.

Proposed method

• Map a persistence diagram to a persistence surface via a weighted sum of 2D Gaussian kernels centered at each point in the diagram.
• Discretize the persistence surface over a grid to form a matrix of pixel values, creating a finite-dimensional vector representation.
• Use customizable weighting functions to emphasize features of high persistence or other topological significance.
• Apply standard machine learning techniques—such as sparse linear support vector machines—on the resulting vectorized PIs.
• Enable feature selection to identify discriminative topological regions in the persistence diagram.
• Validate the method using both synthetic data and real dynamical systems, including the linked twist map and anisotropic Kuramoto-Sivashinsky equation.

Experimental results

Research questions

• RQ1 Can a stable, vectorized representation of persistence diagrams be constructed that supports efficient machine learning while preserving topological information?
• RQ2 How does the performance of persistence images compare to persistence landscapes and raw persistence diagrams in classification tasks under varying noise levels?
• RQ3 To what extent can persistence images capture subtle topological differences in pattern-forming dynamical systems for parameter inference?
• RQ4 How robust are classification results to hyperparameter choices in PI construction, such as Gaussian variance and image resolution?
• RQ5 Can persistence images enable effective feature selection that maps back to interpretable regions in the original persistence diagram?

Key findings

• Persistence images achieved classification accuracy of 97.3% on the anisotropic Kuramoto-Sivashinsky equation data at time t=10, outperforming variance-based classifiers (77.62%) and low-resolution surface approximations (19.3%).
• The use of both H₀ and H₁ PIs together improved classification accuracy to 97.3%, compared to 94.7% for H₀ alone and 93.3% for H₁ alone.
• Classification accuracy remained stable across PI parameter variations: changing Gaussian variance from 0.0001 to 0.1 altered H₀ accuracy by less than 1 percentage point.
• PIs enabled feature selection via sparse linear SVM, with selected pixels corresponding directly to interpretable regions in the persistence diagram.
• Computing distances between PIs was significantly faster than between persistence diagrams, with comparable speed to persistence landscape methods.
• The method demonstrated high robustness to noise, maintaining strong performance across multiple noise levels in synthetic data experiments using K-medoids clustering.