💻 computer science

Unreduced Persistence Diagrams for Topological Machine Learning

This paper demonstrates that machine learning pipelines utilizing topological features derived from unreduced persistence diagrams can achieve performance comparable to or better than those using fully-reduced diagrams, while simultaneously offering significant advantages in computational cost and memory efficiency.

Original authors: Nicole Abreu, Parker B. Edwards, Francis Motta

Published 2026-06-19

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Nicole Abreu, Parker B. Edwards, Francis Motta

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a computer to recognize shapes, like telling the difference between a circle, a sphere, and a donut (torus). To do this, mathematicians use a tool called Persistent Homology. Think of this tool as a "topological scanner" that looks at a cloud of points and asks, "Where are the holes? Where are the loops?"

The scanner produces a report called a Persistence Diagram (PD). This report is a list of dots, where each dot represents a feature (like a hole) and how long it "persisted" before disappearing as the shape grew.

The Problem: The Expensive Report

Traditionally, to get this report, the computer has to do a massive amount of heavy lifting called "reduction." It's like a librarian trying to organize a library by checking every single book against every other book to remove duplicates and find the perfect summary. This process is:

Slow: It takes a lot of time.
Memory Hungry: It requires a huge amount of computer RAM (memory).
Wasteful: The authors noticed something strange. When they fed these detailed reports into machine learning models, the models often ignored most of the information. It was as if the librarian spent hours writing a 500-page summary, but the reader only needed the first three sentences to understand the story.

The Solution: The "Unreduced" Sketch

The authors asked a simple question: What if we skip the heavy editing process entirely?

Instead of doing the full "reduction" to get the perfect, final report, they propose using Unreduced Persistence Diagrams.

The Analogy: Imagine you are sketching a face. The "reduced" method is like a professional artist spending hours refining every line, erasing mistakes, and perfecting the shading before showing you the drawing. The "unreduced" method is like quickly sketching the main features (eyes, nose, mouth) directly from the raw data without erasing or refining.
The Result: Surprisingly, the computer (the machine learning model) can often recognize the face just as well from the quick sketch as it can from the polished masterpiece.

What They Did

The team built a new, faster version of the software (based on a popular tool called Ripser) that skips the heavy editing. Instead of the full report, it generates these "sketches" (which they call Unreduced Diagrams or specific types like Low-Ones and Quasi-Apparent Pairs).

They tested this on three different challenges:

Shape Recognition: Distinguishing between circles, spheres, and donuts in noisy data.
Image Classification: Identifying clothing items (like sandals vs. sneakers) in the Fashion-MNIST dataset.
Brain Scan Regression: Predicting a person's age based on the structure of their brain's blood vessels.

The Findings

Performance: In almost every test, the models trained on the "sketches" (unreduced diagrams) performed just as well as, or sometimes even better than, the models trained on the "polished reports" (fully reduced diagrams).
Speed & Memory: This is the biggest win. Because they skipped the heavy editing:
- The new method used significantly less memory (sometimes up to 13 times less).
- It was much faster, especially when using multiple computer cores at once (parallel processing).
- In one extreme case, the old method ran out of memory and crashed, while the new method successfully finished the job.

The Catch (Stability)

The authors admit that these "sketches" are a bit more sensitive to noise than the "polished reports." If you shake the data too hard, the sketch might change its shape more drastically than the polished report would. However, in their experiments with realistic amounts of noise, the sketches remained stable enough to be useful.

The Bottom Line

The paper suggests that in the world of Topological Machine Learning, we might be wasting a lot of time and computer power trying to make our data reports "perfect." By using Unreduced Persistence Diagrams, we can get a "good enough" summary much faster and with less memory, and the computer learns just as effectively from it. It's a trade-off: a slightly rougher sketch for a massive gain in speed and efficiency.

Technical Summary: Unreduced Persistence Diagrams for Topological Machine Learning

Problem Statement

Topological Data Analysis (TDA) combined with supervised machine learning (ML) has become a standard approach for analyzing complex data across diverse domains, including biology, neuroscience, and material science. The standard pipeline involves computing Persistent Homology (PH) to generate Persistence Diagrams (PDs), vectorizing these diagrams, and training ML models.

However, two critical inefficiencies have been observed in this pipeline:

Information Redundancy: Empirical evidence suggests that ML models often ignore significant portions of the information contained in vectorized PDs. For instance, regression tasks have shown "near-optimal" performance using only the persistence of the 28th most persistent point, and benchmarks indicate that naive statistical vectorizations often perform as well as complex ones.
Computational Cost: Computing fully reduced PDs is often the most computationally demanding step in the pipeline, requiring substantial memory and time, particularly for large datasets.

The central hypothesis of this work is that if ML models can perform well with vectorizations that discard information, then the full computation of PH (specifically the reduction of boundary matrices) may be an unnecessary computational expense. The authors propose avoiding boundary matrix reduction entirely by utilizing unreduced persistence diagrams derived directly from unreduced boundary matrices.

Methodology

Theoretical Framework

The authors define several types of unreduced persistence diagrams (PDs) constructed from a boundary matrix $M$ without performing the standard column-wise reduction algorithm. These are defined based on the properties of the matrix columns:

Apparent Pair (AP) PD: Pairs $(low(M_j), j)$ where the lowest non-zero entry in column $j$ is the only non-zero entry in that row to the left (a specific pattern indicating a persistence pair without reduction).
Lower Quasi-Apparent Pair (qAP↓) PD: Pairs $(low(M_j), j)$ where $\beta(M_j) \neq -1$ .
Upper Quasi-Apparent Pair (qAP↑) PD: Pairs $(\beta(M_j), j)$ where $\beta(M_j) \neq -1$ . Here, $\beta(M_j)$ is the highest row index of a 1 in column $j$ that cannot be eliminated by reduction.
Low-Ones (L1) PD: The multiset of pairs $(low(M_j), j)$ for all columns where $low(M_j) \neq -1$ .

The L1 and qAP↑ constructions are particularly emphasized for Vietoris-Rips (VR) complexes, as other unreduced types may yield only ephemeral pairs (zero persistence) in higher homology dimensions for these filtrations.

Stability Analysis

The paper investigates the Lipschitz stability of these transformations with respect to the bottleneck distance ( $d_B$ ).

Theoretical Results: The fully reduced (FR) PD transformation is globally stable with a Lipschitz constant $C=1$ . In contrast, unreduced PDs (AP, qAP) are generally not globally stable. Small perturbations in the filtration function can reorder the boundary matrix columns, causing the appearance or disappearance of apparent pairs, leading to combinatorial instability.
Local Stability: The authors prove that unreduced PDs are locally Lipschitz stable. If the perturbation in the filtration function is small enough to preserve the ordering of simplices (specifically, smaller than half the minimum gap between adjacent filtration values), the transformations remain stable.
Experimental Stability: Computational experiments on perturbed point clouds show that while large instability spikes (where the number of points in the diagram changes) occur, they are rare. The majority of perturbations result in stability ratios comparable to or better than the fully reduced case.

Computational Implementation

To benchmark the computational advantages, the authors implemented a modified version of the Ripser algorithm to compute qAP↑ diagrams for VR complexes.

Parallelization: Unlike the standard reduction algorithm, which processes columns serially (left-to-right), the computation of qAP↑ diagrams can be parallelized. The algorithm searches across columns of the filtration coboundary matrix independently.
Memory Efficiency: The implementation avoids storing the full boundary matrix and working boundaries required for reduction. It utilizes a hash map to store pairs and skips "zero apparent pairs" (ephemeral pairs) early in the process.

Key Results

Computational Performance

Benchmarking on a 32-core AMD Ryzen Threadripper PRO system against standard Ripser revealed significant gains:

Wall Time: Parallelized qAP↑ computation achieved an average 20x speed-up in wall time compared to full persistence computation.
Memory: The unreduced approach required approximately 1/13th of the memory of Ripser for non-thresholded examples and 1/1.86 for thresholded examples.
Scalability: The authors successfully computed unreduced diagrams for the torus4 dataset (50,000 points), a task where Ripser failed due to memory constraints (requiring >300GB RAM). For the cyclo-octane dataset, qAP↑ computation used 454 times less memory and 59.5 times less wall time than Ripser.

Machine Learning Performance

The authors evaluated the performance of Random Forest classifiers and regressors trained on vectorizations of unreduced PDs (L1, qAP, AP) versus fully reduced PDs (FR) across three tasks:

Synthetic Shape Classification: Classifying point clouds (circle, sphere, torus, etc.) with varying noise levels.
Fashion-MNIST Classification: Classifying clothing images using cubical complexes.
Brain Artery Tree Regression: Predicting subject age from 3D brain scan data.

Findings:

Parity and Improvement: In all experiments, at least one type of unreduced PD performed as well as or better than the fully reduced PD.
L1 Superiority: The L1 diagrams consistently performed on par with or exceeded FR diagrams. In the Fashion-MNIST task, L1 diagrams outperformed FR diagrams by at least 10 percentage points in average precision.
Robustness: These results were robust across different filtration types (Alpha, Rips), vectorization methods (Persistence Images, Adcock-Carlsson coordinates), and noise levels.
Feature Importance: Analysis of the Fashion-MNIST task revealed that L1 and FR models relied on distinct regions of the persistence image space. The most important features for L1 models were concentrated in $H_0$ (connected components), whereas FR models utilized features from both $H_0$ and $H_1$ more evenly. This suggests unreduced diagrams encode different, potentially useful information that is discarded during full reduction.

Significance and Claims

The paper concludes that unreduced persistence diagrams are effective substitutes for fully reduced diagrams in topological machine learning pipelines.

Computational Efficiency: By avoiding the expensive reduction step, practitioners can achieve significant reductions in memory usage and computation time, enabling the analysis of datasets that are currently intractable for standard PH methods.
Performance: Contrary to the assumption that reduction is necessary for quality, the authors demonstrate that the information discarded by reduction is often not critical for downstream ML performance and that unreduced diagrams may even capture distinct, beneficial features.
Future Directions: The authors note that while unreduced diagrams save computation time, vectorizing them can be slower due to the larger number of points. They suggest that future work should focus on designing algorithms that compute vectorizations directly from unreduced matrices without explicitly storing all pairs, further optimizing the pipeline.

The study modestly claims that these results suggest a shift in how PH-ML pipelines are constructed, potentially prioritizing computational efficiency and the retention of "raw" topological features over the strict mathematical completeness of fully reduced diagrams.