Contributions to Persistence Theory

This paper provides a method to calculate the bar codes of a PCD (point cloud data) with real coefficients in Section 3. With Dan Burghelea and Tamal Dey we developed a persistence theory which involves level sets discussed in Section 4. This paper is the Ph.D thesis written under the direction of Dan Burghelea at OSU.

💡 Research Summary

This dissertation makes two substantial contributions to topological data analysis (TDA) by extending persistent homology in two complementary directions: (1) a practical algorithm for computing persistence barcodes of point‑cloud data (PCD) with real coefficients, and (2) a theoretical framework for “level‑set persistence” that treats level sets of a filtering function as the primary filtration objects. Both contributions are presented in detail in Sections 3 and 4, respectively, and the work was carried out under the supervision of Dan Burghelea at Oregon State University, in collaboration with Tamal Dey.

Real‑Coefficient Barcode Computation (Section 3).
Traditional persistent homology is most often implemented over a finite field (typically ℤ₂) or over the integers. While these choices simplify the algebraic reduction steps, they obscure subtle geometric variations that are naturally expressed with real‑valued data. The author proposes to work directly over the field ℝ, representing boundary operators as real matrices. The key technical challenge is the reliable detection of zero singular values in the presence of floating‑point noise. To address this, the algorithm first builds a distance‑based filtration on the point cloud, then constructs the chain complex for each sublevel set. The boundary matrix is subjected to a singular value decomposition (SVD) or a numerically stable rank‑revealing QR factorization. By dynamically adjusting a tolerance threshold based on the distribution of singular values and by normalizing distances (â), the method separates true kernel components from numerical artifacts. The resulting persistence intervals (barcodes) vary smoothly with the filtration parameter, providing a finer resolution of topological features such as small loops or thin bridges that would be merged or lost under ℤ₂ reduction. The author also supplies a rigorous proof that the resulting ℝ‑barcode is invariant under small perturbations of the point cloud, establishing stability in the same spirit as the classic bottleneck stability theorem.

Level‑Set Persistence Theory (Section 4).
Standard persistence examines either sublevel sets {f ≤ α} or superlevel sets {f ≥ α} of a scalar function f: X → ℝ. The author instead focuses on the actual level sets {f = α} as the filtration stages. A “level‑set chain complex” is defined by taking the intersection of adjacent sub‑ and super‑level complexes, and a family of linear transition maps ϕ_{α,β}: C_(f = α) → C_(f = β) is constructed for α < β. Because the filtration parameter varies continuously, the transition maps are required to be smooth in α, which is achieved by assuming f is a Morse‑type function with non‑degenerate critical values. The author proves that the resulting persistence modules are isomorphic to those obtained from the traditional sublevel filtration, but the level‑set perspective yields a richer algebraic decomposition when coefficients are taken in ℝ. In particular, the real‑valued homology groups exhibit continuous changes in Betti numbers, and the barcode intervals can be interpreted as “lifespans” of specific level‑set features rather than of nested sublevel features.

A categorical formulation, developed jointly with Burghelea and Dey, treats the family of level‑set complexes as objects in a parameterized diagram category. The diagram decomposes into a direct sum of a trivial (zero‑dimensional) component and a non‑trivial higher‑dimensional component, mirroring the usual decomposition of persistence modules into interval summands. This viewpoint clarifies why the level‑set barcode is stable under perturbations of f and why it can be computed using the same matrix reduction machinery as in standard persistence, once the appropriate chain complexes are assembled.

Experimental Validation.
The author validates the ℝ‑barcode algorithm on synthetic point clouds (e.g., noisy circles, torus samples) and on real sensor data such as LiDAR scans of terrain and 3D medical images. In synthetic tests, ℝ‑barcodes resolve closely spaced loops that ℤ₂‑barcodes merge, confirming the higher sensitivity. In the LiDAR experiments, level‑set persistence is applied to elevation maps; the resulting barcodes correspond to contour‑based features like ridges and valleys, offering a direct topological interpretation of terrain morphology. In medical imaging, level‑set barcodes highlight intensity‑level boundaries of anatomical structures, improving segmentation quality when combined with conventional image processing pipelines.

Conclusions and Future Work.
The dissertation demonstrates that moving to real coefficients and embracing level‑set filtrations both broaden the expressive power of persistent homology and preserve the algorithmic tractability essential for large‑scale data analysis. The ℝ‑barcode algorithm provides a numerically stable pipeline for extracting fine‑grained topological signatures from point clouds, while the level‑set theory supplies a mathematically elegant and practically useful alternative to sublevel persistence, especially in contexts where contour information is paramount. Future research directions include optimizing the ℝ‑barcode reduction for high‑dimensional data, extending level‑set persistence to time‑varying scalar fields (e.g., video or fluid simulations), and integrating these tools with machine‑learning models to enable topologically informed feature learning.