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Persistence Images: A Stable Vector Representation of Persistent Homology

Henry Adams, Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, Lori Ziegelmeier

arXiv:1507.06217v3cs.CGmath.ATstat.ML

TL;DR

The paper addresses the need for machine-learning-compatible representations of persistence diagrams. It constructs persistence images by smoothing and discretizing diagrams into weighted vectors, then establishes stability and evaluates their use across classification, feature selection, and dynamical-parameter inference. The reported results show improved classification over persistence landscapes and diagrams, interpretable discriminatory pixels, and utility for dynamical-system applications.

  • Problem

    Many machine-learning methods require more structure than persistence-diagram metrics provide, motivating a vector representation that is stable, efficient, interpretable, and flexible.

  • Method

    The paper maps each persistence diagram to a weighted Gaussian persistence surface, integrates that surface over a grid to form a persistence image, and applies vector-based machine-learning methods.

  • Results

    Persistence images improve classification over persistence landscapes and diagrams, support interpretable feature selection, and encode topological information for dynamical-parameter inference.

  • Takeaways & Limitations

    Persistence images provide a stable vector representation that connects topological data analysis with a broad range of machine-learning tools and supports multiple homological dimensions.

  • Takeaways & Limitations

    For general distributions, the persistence-image kernel is not stable with respect to W_p when 1 < p ≤∞, and one stability bound worsens as image resolution increases.

Abstract

from arXiv · show

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.

1 Introduction

Persistence images convert persistence diagrams into finite-dimensional vectors designed for machine learning while retaining interpretability, stability, computational efficiency, and adjustable feature importance.

  • Motivation: Persistence diagrams encode topological features across scales, but their metric structure alone does not support many standard machine-learning tools.These tools include support vector machines, decision trees, neural networks, feature selection, and dimension reduction.
  • Problem Statement: The paper seeks a persistence-diagram representation that is vector-valued, stable to noise, efficient, interpretable, and adjustable across regions.These requirements are listed explicitly in the problem statement.
  • Proposed Representation: Persistence images are finite-dimensional vector representations obtained by converting persistence diagrams into integrable weighted Gaussian surfaces and integrating over grid boxes.The resulting matrix of pixel values vectorizes the diagram.
  • Machine-Learning Use: Vector-valued persistence images enable established machine-learning and statistical tools, including norms, angle-based similarities, and sparse methods.The representation is intended to provide access to tools already developed for data in R^n.
  • Design Flexibility: Weighting functions can emphasize different persistence regions, including high-persistence features or, in some applications, medium-persistence features.The paper allows broad weighting choices rather than requiring a single importance rule.

2 Related Work

Related work provides stable and interpretable representations of persistence diagrams, while positioning persistence images as Euclidean vectors with broad machine-learning compatibility and flexible weighting.

  • Persistence Landscapes: Persistence landscapes offer stable functional representations and unique means, whereas persistence images have the advantage of living in Euclidean space.Euclidean structure makes persistence images amenable to a broader range of machine-learning techniques.
  • Comparative Performance: Persistence images behave comparably or better than persistence diagrams in shared classification settings, are more computationally efficient, and outperform persistence landscapes in most tasks.These comparisons are reported for synthetic data sampled from common topological spaces.
  • Stability: Grid-count feature vectors are interpretable but unstable because small point movements can change bins and points emerging from the diagonal cause discontinuities.Smoothing addresses bin-boundary sensitivity, while weighting is introduced to address diagonal-related instability.
  • Positioning Persistence Images: Persistence images use non-negatively weighted Gaussians, support stable vector-based learning, and permit weighting schemes that differ from distance-from-diagonal emphasis.The paper contrasts these properties with a reflected-Gaussian surface and its associated kernel limitations.
  • Applications: In rock-surface classification, persistence images outperform an aggregate descriptor both directly and when combined with CLBP preprocessing.Adding persistence images to eleven non-topological descriptors improved classification more than adding the aggregate descriptor.

3 Background on Persistent Homology

Persistent homology tracks homological features through nested spaces and records their births and deaths in persistence diagrams, which can be computed from point clouds or real-valued functions.

  • Homology: Homology encodes holes through homology groups, whose ranks are Betti numbers counting independent holes of each dimension.Examples include connected components, loops, and trapped volumes.
  • Persistent Homology: Persistent homology tracks homology classes as a scale or time parameter increases through a nested sequence of topological spaces.Inclusions induce linear maps between corresponding homology groups.
  • Persistence Diagrams: A persistence diagram is a multiset of birth-death points, with each point recording when a homological feature appears and disappears.Points lie on or above the diagonal, and points near the diagonal are often treated as noise.
  • Input Data: Point-cloud data uses Vietoris–Rips filtrations, while real-valued functions use sublevel-set filtrations to generate persistence diagrams.The paper applies the function-based construction to surfaces produced by the Kuramoto-Sivashinsky equation.
  • Metrics: Wasserstein and bottleneck metrics quantify similarity between persistence diagrams and support stability analysis under perturbations.These metrics measure dissimilarity between the homological characteristics of datasets.

4 Persistence Images

Persistence images are built by smoothing weighted persistence-diagram points, discretizing the resulting surface, and integrating over grid regions to obtain vectors that can combine homology dimensions.

  • Pipeline: The persistence-image pipeline transforms a persistence diagram into a vector while preserving an interpretable connection to the diagram.The paper illustrates the pipeline from spectral and spatial image information to a persistence image.
  • Persistence Surface: A diagram is transformed into birth-persistence coordinates, where a differentiable probability distribution is centered at each point.The applications use normalized symmetric Gaussians with variance σ^2.
  • Weighting Function: A persistence surface is formed from weighted distributions using a nonnegative weighting function that is zero on the horizontal axis.Continuity and piecewise differentiability are required for the stated stability results.
  • Vectorization: The surface becomes a finite-dimensional vector by integrating it over each of n grid boxes, whose values form the persistence-image pixels.This discretization produces the matrix representation used by machine-learning methods.
  • Multiple Dimensions: Persistence-image vectors from multiple homological dimensions can be concatenated into one machine-learning input vector.The paper describes combining H0, H1, through Hk representations.
  • User Choices: The user selects resolution, distribution parameters, and weighting function; reported classification accuracy is fairly robust to resolution and variance choices.Variance remains an open choice despite low sensitivity in the cited experiments.

5 Stability of Persistence Surfaces and Images

The paper establishes stability of persistence surfaces and their finite-dimensional persistence images under the 1-Wasserstein distance between persistence diagrams. General bounds are supplemented by sharper Gaussian-specific results, while image stability depends on image resolution and assumptions on the weighting function.

  • 5.2 Stability for Gaussian distributions: The stability claims assume a nonnegative weighting function that is zero on the horizontal axis, continuous, and piecewise differentiable.The weighting function is restricted only as needed for the stability results.
  • 5.1 Stability for general distributions: Persistence surfaces are stable with respect to the 1-Wasserstein distance between persistence diagrams.The proof uses a Wasserstein-optimal matching and bounds changes in weighted Gaussian components through gradient norms and point displacements.
  • 5.1 Stability for general distributions: Persistence images inherit stability with respect to the 1-Wasserstein distance between diagrams.The bound depends on the maximum pixel area, total image area, and number of pixels.
  • 5.1 Stability for general distributions: The general L2 stability constant contains √n and therefore grows without bound as image resolution increases.The paper addresses this issue by deriving better constants for Gaussian distributions.
  • 5.2 Stability for Gaussian distributions: The persistence-image inner-product kernel is not stable with respect to Wp for any 1 < p ≤∞.The paper identifies this as a limitation of the kernel formulation, distinct from its 1-Wasserstein stability results.
  • 5.2 Stability for Gaussian distributions: For Gaussian distributions, the paper proves improved stability results for persistence surfaces and persistence images under the 1-Wasserstein distance.The Gaussian analysis controls both L∞ and L1 distances between persistence surfaces, supporting corresponding vector-norm bounds.

6 Experiments

The experiments compare persistence images with other persistence-diagram representations, test parameter robustness and sparse feature selection, and apply PIs to dynamical-system classification. Across these settings, PIs provide accurate, computationally efficient, interpretable, and dimension-reducing representations.

  • Comparison of PDs, PLs, and PIs using K-medoids Classification: PIs achieve consistently high classification accuracy and significantly lower computation times than PD distance matrices in the six-shape K-medoids comparison.The comparison covers PDs, persistence landscapes, and PIs across multiple metrics, homological dimensions, and noise levels.
  • Effect of PI Parameter Choice: PI classification accuracy is insensitive to the tested choices of resolution and variance.This robustness is reported for the six-shape dataset and is also observed in the linked dynamical-system experiment.
  • Differentiating Homological Features by Sparse Support Vector Machine: 100% accuracy is obtained with 5-fold cross-validated SSVM using only a cumulative set of 10 distinct PI pixels.The PIs use resolution 20 × 20, so the original vectors contain 400 pixels; selected H1 pixels correspond to distinct persistence regions for circles and tori.
  • A discrete dynamical model: 82.5% accuracy is achieved by concatenating H0 and H1 PIs, compared with 49.8% using H0 alone and 65.7% using H1 alone.The result comes from a discriminant subspace ensemble averaged over 10 trials.
  • Dynamical-system applications: PIs classify linked-twist-map and anisotropic Kuramoto–Sivashinsky data by capturing subtle structural differences that are difficult to distinguish from raw surfaces or sparse point clouds.For the aKS equation, PI-based inference outperforms low-resolution surface approximations and surface-height variance by at least 20%, including at early times.

7 Conclusion

The paper concludes that persistence images provide stable, flexible vector representations that support improved classification, interpretable feature selection, and high-accuracy inference of dynamical-system parameters.

  • Persistence images improve classification accuracy over persistence landscapes and persistence diagrams across sampled topological spaces and multiple noise levels.The comparison uses K-medoids clustering.
  • Persistence images require less computation time than persistence-diagram distances while offering computation times comparable to persistence landscapes.
  • Sparse support vector machines can select interpretable discriminatory pixels because each pixel corresponds to a region of a persistence diagram.
  • Persistence images support high-accuracy inference of dynamical-system parameters by encoding topological information in a form usable by machine-learning tools.
  • Classification accuracy is robust to persistence-image resolution and variance, reducing the need for large-scale parameter searches.The flexibility of persistence images also permits customization for varied real-world datasets.

A Homology and Data

Homology characterizes topological holes through algebraic structures built from simplicial complexes, while persistent homology tracks these features across scales.

  • Homology characterizes connected components, loops, trapped volumes, and other topological properties using homology groups and Betti numbers.
  • A simplicial complex is built from vertices, edges, triangles, tetrahedra, and higher-dimensional simplices subject to closure and intersection conditions.
  • A chain complex assigns vector spaces to simplices and boundary operators that map each simplex to a sum of its lower-dimensional faces.
  • The relation ∂k ◦∂k+1 = 0 implies that k-boundaries are contained among k-cycles, enabling homology to be defined as Zk/Bk.
  • The k-th Betti number βk is the dimension of Hk(S) and counts independent k-dimensional holes.

A.2 Persistence Diagrams from Point Cloud Data

Persistence diagrams from point clouds arise by computing homology over a Vietoris–Rips filtration, recording when topological features are born and die across scales.

  • Point-cloud persistence diagrams use a simplicial-complex approximation to the topology of data sampled in a metric space.
  • The Vietoris–Rips complex includes a k-simplex when every pair among its k + 1 vertices is within scale ϵ.Choosing ϵ too small produces many connected components, while choosing it too large can obscure structure.
  • Persistent homology computes homology at an increasing sequence of scales and tracks features that persist across the resulting filtered complexes.
  • A persistence diagram is a multiset of points (ϵ, ϵ′), where each point records a feature’s birth scale and death scale.The points lie above the diagonal because features die after they are born.

A.3 Persistence Diagrams from Functions

Persistence diagrams from functions are constructed from sublevel-set filtrations, with simplicial or cubical complexes providing computational realizations for function-valued data.

  • For a real-valued function f, persistent homology studies the topology of sublevel sets f^-1((−∞, ϵ]) across increasing thresholds.
  • Simplicial complexes use lower star filtrations, while cubical complexes provide an analogous construction for grid-based function data.
  • For Kuramoto–Sivashinsky surfaces, the domain is discretized into a 512×512 grid forming a two-dimensional cubical complex.
  • An edge enters the cubical filtration when both of its vertices are present, and a square enters when all four vertices are present.
  • The persistence diagrams are obtained by applying persistent homology to the cubical-complex sublevel-set filtration.
  • Persistence diagrams from point clouds can be viewed as function-derived diagrams using the distance function to the data set and its metric-ball sublevel sets.The Vietoris–Rips filtration approximates the corresponding Čech-complex persistence.

B Examples of Persistence Images

The examples vary persistence images of noisy H1 shapes by resolution and Gaussian variance, illustrating how these parameters affect the representation.

  • Noisy-circle PIs are shown at resolutions 5 × 5 and 50 × 50, with variances σ = 0.01 and σ = 0.2.
  • Noisy-torus PIs use the same resolution and variance combinations to show representations across parameter choices.

C Proofs of Equation (3) and Lemma 3

The proofs characterize the one-dimensional Gaussian crossing used in Equation (3) and reduce the two-dimensional Gaussian case to one dimension through rotation and translation.

  • Proof of Equation (3): Equation (3) expresses the L1 distance between weighted one-dimensional Gaussians centered at u and v as a function F(v − u).
  • Proof of Equation (3): When u ≠ v, the weighted Gaussian curves have a unique intersection z* determined by their parameters.
  • Proof of Equation (3): The proof computes four integrals separately after partitioning the difference between the Gaussian functions at their intersection.
  • Proof of Lemma 3: Lemma 3 applies this reduction to normalized two-dimensional Gaussians defined on R2.
  • Proof of Lemma 3: The two-dimensional Gaussian argument reduces to Gaussians centered at 0 and r = ∥u − v∥2 using polar coordinates and a rotation of the plane.

D SSVM-based Feature Selection

OAA SSVM feature selection identifies sparse subsets of persistence-image pixels that retain discrimination among six synthetic data classes.

  • SSVM feature selection: 100% overall accuracy was obtained with 5-fold cross-validation on six synthetic classes at noise level η = 0.05.
  • SSVM feature selection: The experiment used H1 persistence images with resolution 20 × 20 and variance σ = 0.0001, producing vectors in R400.
  • SSVM feature selection: Features with non-zero SSVM weights were retained, yielding six sparse models containing class-discriminatory pixel subsets.
  • SSVM feature selection: The selected features can be used alone for classification without loss in accuracy and correspond to discriminatory persistence-image pixels.
  • SSVM feature selection: Figure 11 marks the selected pixel locations for the six classes in vectorized persistence images.
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