Persistence Images: A Stable Vector Representation of Persistent Homology

By Henry Adams, T. Emerson, M. Kirby, Rachel A. Neville, C. Peterson, Patrick D. Shipman, Sofya Chepushtanova, Eric M. Hanson, Francis C. Motta, Lori Ziegelmeier. Journal of machine learning research, 2015

873 citations recorded by the source corpus.

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.

Subfields: Biophysics, Computational Theory and Mathematics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/223841a71f5bce4cb03040e229d13e9a71b78ec3

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Citations in the corpus

Topology-Preserving Deep Image Segmentation (2019, neutral): “For completeness, we also refer to other existing works on topological features and their kernels [1, 33, 23, 5].”
Deep Learning with Topological Signatures (2017, background): “Specifically, this layer is designed such that its output is stable with respect to the 1-Wasserstein distance (similar to [27] or [1]).”
An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists (2017, methodology): “(2016) and combined with quantization and pooling methods to address non rigid shape analysis problems; Betti curves extracted from persistence diagrams are used with 1-dimensional Convolutional Neural Networks (CNN) to analyze time dependent data and recognize human activities from inertial sensors”
Wasserstein Stability for Persistence Diagrams (2020, neutral): “(1) sliced Wasserstein kernel, CT = 1, see [8] (2) persistent images, CT = 1, see [1] (3) persistent scale space, CT = 1, see [34, 30] (4) weighted Betti curves see [29, 39, 15], (5) learned/optimized representations [25, 26], (6) persistent homology rank function, CT = 1, [35] Related results for g”
What are higher-order networks? (2021, background): “6, the 1-chain [3, 4]+[4, 5]− [3, 5] is clearly a cycle, but it is not the boundary”
Persistent Convolution: A Topological Framework for AI Alignment Testing and Semantic Space Characterization (2026, unknown): “This is a helpful tuning parameter, and aligns with more recent representations of persistence diagrams that rely on Kernel Density Estimation (KDE), like persistence images (Adams et al., 2017).”