Source-linked AI summary
Explainable topological data analysis using persistence heatmaps
Peter Bubenik, Alexander Wagner, Himanshu Yadav
TL;DR
Representative cycles can aid interpretation of TDA, but may change discontinuously under perturbations, while persistence diagrams and feature maps can be challenging for non-experts to interpret. The paper averages cycle-based contributions over random perturbations to create stable maps whose values sum to the expected learned-function value; the maps are Lipschitz in the data.
Problem
Persistence diagrams and feature maps can be challenging for non-experts to interpret, and representative cycles that could aid interpretation may change discontinuously under data perturbations.
Source: §1 · pp. 1–2Method
The paper distributes a learned function’s value across representative-cycle-based simplex or pixel/voxel contributions and averages them over random perturbations.
Source: §1 · pp. 2–3Results
The expected simplex and pixel/voxel contributions are Lipschitz in the data, satisfy the strong law of large numbers and central limit theorem, and sum to the expected learned-function value.
Source: §1 · p. 3Takeaways & Limitations
Stable visualizations can explain TDA outputs using representative cycles while preserving the expected learned-function value across their displayed contributions.
Source: §1 · pp. 2–3An analytic expression for the convolved mean feature map is not feasible, so the paper approximates it with an empirical procedure.
Source: §6.5 · p. 18
Abstract
from arXiv · showhide
Topological data analysis (TDA) leverages tools from algebraic topology to aid in various machine learning tasks. Numerous TDA constructions are provably stable in the sense that changes in the input produce linearly bounded changes in the output, with a specified bound. We use representative cycles, which are unstable TDA constructions, to produce stable visualizations to aid in explaining TDA. For example, we produce stable heatmaps on images containing the data such that summing the values of the pixels gives the value of a learned regression function.
1. Introduction
TDA supports machine-learning tasks, but its persistence diagrams and feature maps can be difficult for non-experts to interpret, while representative cycles useful for explanation may change discontinuously under data perturbations. The paper distributes a learned function’s value across simplices or pixels/voxels, then averages over random perturbations to obtain stable explanations.
1. Introduction
- Representative cycles offer interpretive information for persistent homology classes, but can change discontinuously when the data are perturbed.The paper uses them to construct stable visualizations, addressing the difficulty that persistence diagrams and feature maps can pose for non-experts.Source: §1 · pp. 1–2
- The goal is to distribute the expected learned-function value across simplices or pixels/voxels in a stable way.The construction preserves the value: summing the resulting vector entries gives the function value for each input.Source: §1 · p. 3
- The expected simplex- and pixel/voxel-level contributions are Lipschitz in the data, and their sums preserve the expected learned-function value.Averaging under Gaussian perturbations also gives componentwise maps satisfying the strong law of large numbers and central limit theorem; other kernels are supported as well.Source: §1 · p. 3
- For a single point-cloud sample from each class, Gaussian perturbations yield pixelwise random contributions whose empirical means form heatmaps.Each heatmap’s pixel values sum to the empirical mean classifier value; the expected pixel values are Lipschitz under small point-cloud perturbations.Source: §1 · p. 4
1.1. Outline.
- The introduction outlines decomposition via representative cycles, distribution to simplices or pixels/voxels, random feature maps, theoretical results, and computational examples.The decomposition is nontrivial for the non-additive persistence landscape, motivating a labeled persistence landscape.Source: §1.1 · p. 4
1.2. Related work.
- Prior work includes TDA interpretation for biological data [28] and representative-cycle computation methods the authors expect could be combined with theirs [13] [26].The introduction notes that interpretation is desired by end users such as scientists and engineers.Source: §1 · p. 1, §1.2 · p. 5
2. Background
The background introduces simplicial complexes, homology, and persistence, then describes computational persistent homology and feature maps used to represent persistence diagrams.
2.1. Ordered simplicial complexes.
- An ordered simplicial complex is a finite simplicial complex equipped with a total order that respects inclusion.Source: §2.1 · p. 5
2.2. Simplicial chains and homology.
- Simplicial chains form a vector space; cycles are in the boundary operator’s kernel, boundaries are in its image, and homology is cycles modulo boundaries.Source: §2.2 · p. 5
2.3. Weighted simplicial complexes and persistent homology.
- An order-preserving weight assigns filtration values to simplices, yielding nested complexes whose homology maps form a persistence module.Source: §2.3 · p. 6
2.4. Computational persistent homology.
- A persistence algorithm reduces the boundary matrix to identify birth-death pairs; associated representative cycles and bounding chains provide homology-class representatives but depend on the algorithm.The birth-death matching itself is independent of the persistence algorithm.Source: §2.4 · p. 6
- A persistence diagram is a multiset of birth-death pairs, and the figure contrasts an example diagram with its corresponding persistence landscape.Source: §2.4 · p. 7, Figure 3 · p. 7
2.5. Feature maps.
- Feature maps send persistence diagrams to a separable real Banach space; examples include total persistence and the persistence landscape, which is injective.Source: §2.5 · p. 7
2.6. Kernels.
- Kernels are introduced as nonnegative integrable functions, with Gaussian, triangular, and Epanechnikov kernels given as examples.Source: §2.6 · p. 8
3. Using representative cycles
Representative cycles label persistence-diagram features, enabling feature maps to be decomposed into interpretable contributions. Linear representations then expose each chain’s contribution to a learned estimate.
3.1. The four simplicial chains of a birth-death pair.
- A birth-death pair can be associated with its birth simplex, death simplex, representative cycle, and bounding chain.These are all simplicial chains, and the results apply whichever of the four is chosen.Source: §3.1 · p. 8
3.2. Labeling persistence diagrams with representative cycles.
- The labeled persistence diagram records a label for each birth-death pair, while forgetting labels recovers the ordinary diagram.The ordinary diagram is independent of the chosen ordering and persistence algorithm; the labeled diagram is not.Source: §3.2 · p. 9
3.3. Decomposing feature maps using representative cycles.
- A labeled decomposition assigns each feature-map component to a label and sums those components to recover the original feature map.For the persistence landscape, each labeled component is the part of the ordinary landscape assigned to that label.Source: §3.3 · pp. 9–10
- Additive feature maps, including total persistence, can be decomposed by summing the birth-death contributions carrying each label.The persistence landscape also admits a labeled decomposition, although it is not additive.Source: §3.3 · p. 10
3.4. Feature maps and linear representations.
- Applying a linear representation to each labeled component decomposes the learned scalar estimate into contributions from individual chains.The resulting chain contributions indicate which parts of the simplicial complex are responsible for the estimate.Source: §3.4 · p. 11
4. Distributing feature maps to simplices and pixels/voxels
The section distributes a feature-map value first across simplices and then, when the complex is embedded in a grid, across pixels or voxels. These distributions preserve the total value.
4.1. Distribution to simplices.
- The simplex distribution assigns contributions from labeled chains to simplices while preserving the feature-map value when summed over the complex.The learned scalar estimate can likewise be written as the sum of its simplex-level contributions.Source: §4.1 · p. 12
4.2. Distribution to pixels/voxels.
- For a complex embedded in a pixel or voxel grid, simplex values are distributed to intersecting cells, preserving the total across the image.The grid cells are those intersecting the embedded complex; the classifier estimate can be represented as a sum of cell contributions.Source: §4.2 · pp. 12–13, Figure 5 · p. 13
5. Randomizing feature maps
Feature maps based on labeled persistence data can be discontinuous as weights change. The paper randomizes weights and studies the resulting feature maps and their expectations, which converge and can be distributed across chains, simplices, or pixels/voxels.
5.1. Feature maps as functions on a space of weights.
- The persistence diagram varies continuously and piecewise linearly with weights under appropriate metrics, but labeled diagrams and the induced chain, simplex, and pixel/voxel feature maps are discontinuous.The feature maps are nevertheless piecewise continuous and Borel measurable.Source: §5.1 · pp. 14–15
- The paper defines chain, simplex, and pixel/voxel feature maps by composing labeled feature decompositions with the corresponding weight-to-diagram maps.For the pixel/voxel map, the complex is assumed to have a position function.Source: §5.1 · p. 15
5.2. Random chain, simplex, and pixel/voxel feature maps.
- Random weights induce random chain, simplex, and pixel/voxel feature maps; one example perturbs a weight by mean-zero Gaussian noise.The maps are compositions of the corresponding feature maps with a random weight.Source: §5.2 · pp. 15–16
5.3. Convergence of our random feature maps.
- With finite expected norm, averages of independent copies of a Banach-space-valued random feature map converge almost surely to its expectation.If the space has type 2 and the random feature map also has finite second moment, its centered, scaled averages converge in distribution to a Gaussian with matching covariance.Source: §5.3 · p. 16
5.4. Distribution of our random feature maps.
- The expected distributed feature maps retain the paper’s distribution relationships under the stated expectation assumptions.The result follows because the relevant random variables agree and expectation is linear.Source: §5.4 · p. 16
6. Stabilizing the chain, simplex, and pixel/voxel feature maps
Taking expectations of random feature maps yields mean chain, simplex, and pixel/voxel maps. Under suitable hypotheses these maps are Lipschitz, and empirical averages provide approximations to the means or to averages over a family of examples.
6.1. Means of random feature maps.
- The expected chain, simplex, and pixel/voxel maps are considered coordinatewise, with each coordinate given by the expectation of its corresponding scalar component.These expectations are the basis for the mean feature-map definitions.Source: §6.1 · p. 17, §6.3 · p. 17
6.3. Mean feature maps.
- The mean chain, simplex, and pixel/voxel feature maps are formed by convolving their scalar feature components with a kernel.The definitions assume the relevant convolution integrals exist.Source: §6.3 · p. 17, §6.2 · p. 17
6.4. Stability of the mean feature maps.
- Under suitable hypotheses, convolution makes the mean feature maps Lipschitz with respect to changes in weights.The stated conditions include boundedness of the underlying function for several kernel-specific bounds; other results give Lipschitz constants from integrability or kernel regularity.Source: §6.4 · p. 18
6.5. Stable feature maps in practice.
- The paper proposes approximating a mean feature map by averaging independent kernel-noise samples because its analytic convolution expression is not feasible to obtain.The empirical mean converges almost surely to the expected value.Source: §6.5 · p. 18
6.6. Averaging over a family of examples.
- Averaging explanations over independently sampled examples converges almost surely to the expected explanation for the sampled weight distribution.This provides a class-level alternative to a stable summary for a single test case.Source: §6.6 · p. 19
7. Computational examples
The computational examples apply standard TDA methods to classification problems and use mean feature maps to explain classifications.
7.1. Classification for uniform samples on topologically distinct spaces.
- In the annulus comparison, the smaller annulus unique to the double-annulus class is strongly colored for that class, while the shared larger annulus looks similar across classes.The visualizations use birth edges, death triangles, representative cycles, and bounding chains.Source: Figure 7 · p. 21, §7.1 · p. 22
7.2. Classification for samples from different distributions on a topologically trivial space.
- In the Ginibre-versus-uniform-disk comparison, the larger voids in uniform samples are most strongly colored for the uniform-distribution class.The classifier visualizations use homology in degree 1 and labeled persistence landscapes.Source: §7.2 · p. 22, Figure 10 · p. 23
7.3. Classification for 3-d data.
- For cup-versus-cup-with-handle samples, the shared cup is colored similarly across classes, while the handle is strongly colored for the cup-with-handle class.The handle loop contributes the degree-1 homology class distinguishing the classes.Source: §7.3 · pp. 23–24, Figure 12 · p. 25
7.4. Anomaly detection in a time series.
- The time-series example uses TDA both to detect an anomaly and to locate it in the series.Its visualization classifies normal versus anomalous series and uses birth-simplex-labeled persistence landscapes.Source: §7.4 · pp. 24–25, Figure 13 · p. 26
7.5. Classification of discrete dynamical systems.
- For five linked-twist-map parameter cases, linear SVMs compare each class against the others, and representative-cycle mean feature maps visualize the classifications.The cases use r = 2.5, 3.5, 4.0, 4.1, and 4.3.Source: §7.5 · pp. 25–26, Figure 14 · p. 26