Universality of the Homotopy Interleaving Distance (1705.01690v2)

Abstract: As a step towards establishing homotopy-theoretic foundations for topological data analysis (TDA), we introduce and study homotopy interleavings between filtered topological spaces. These are homotopy-invariant analogues of interleavings, objects commonly used in TDA to articulate stability and inference theorems. Intuitively, whereas a strict interleaving between filtered spaces $X$ and $Y$ certifies that $X$ and $Y$ are approximately isomorphic, a homotopy interleaving between $X$ and $Y$ certifies that $X$ and $Y$ are approximately weakly equivalent. The main results of this paper are that homotopy interleavings induce an extended pseudometric $d_{HI}$ on filtered spaces, and that this is the universal pseudometric satisfying natural stability and homotopy invariance axioms. To motivate these axioms, we also observe that $d_{HI}$ (or more generally, any pseudometric satisfying these two axioms and an additional "homology bounding" axiom) can be used to formulate lifts of several fundamental TDA theorems from the algebraic (homological) level to the level of filtered spaces. Finally, we consider the problem of establishing a persistent Whitehead theorem in terms of homotopy interleavings. We provide a counterexample to a naive formulation of the result.

• The paper defines the homotopy interleaving distance, extending traditional interleavings to a homotopy-invariant measure for filtered spaces.
• It establishes universal pseudometric properties and novel stability axioms, ensuring reliable topological data analysis beyond algebraic methods.
• The research paves the way for advanced TDA algorithms, facilitating robust comparisons and improved inference in data-driven applications.

Universality of the Homotopy Interleaving Distance

The intricacies of topology intersect with practical applications in data analysis through the lens of the paper on "Universality of the Homotopy Interleaving Distance" by Andrew J. Blumberg and Michael Lesnick. This work explores the foundational aspects of Topological Data Analysis (TDA), aiming to formulate homotopy-theoretic grounds for stability and inference within the field of filtered topological spaces.

Overview and Main Contributions

The paper introduces the concept of "homotopy interleavings," extending the traditional notion of interleavings from being merely algebraic counterparts to homological analysis to also considering homotopy-theoretical aspects. Traditional interleavings declared filtered spaces approximately isomorphic; however, homotopy interleavings certify these spaces as approximately weakly equivalent. This distinction hinges on a more robust understanding of the structure and stability of spaces in TDA.

The principal achievement is the definition and characterization of the homotopy interleaving distance, $d_{HI}$, on filtered spaces. This distance satisfies universal pseudometric properties under the axioms of stability and homotopy invariance, harmonizing its applicability with theoretical constructs. The authors show that this distance can lift several foundational TDA theorems, including the Rips stability theorem, to the level of filtered spaces—moving beyond algebraic barcodes to spatial relationships.

Numerical and Theoretical Implications

The homotopy interleaving distance induces a metric that is not only robust in terms of theoretical formulation but practically vital for inducing stability in the persistence homology framework. For instance, the homology bounding axiom ensures that the bottleneck distance—a standard metric in comparing barcodes—aligns consistently with homotopy interleavings, thereby bridging algebraic topology with homotopical assessments.

Among the bold implications of this research is its ability to speculate on the general theory of approximate homotopy for filtered spaces. With a universal metric like $d_{HI}$, researchers might further refine methodologies for comparing and predicting topological changes in data sets that are subject to dynamic alterations.

Speculations on Future Developments

The findings open several pathways for future exploration in AI and TDA. Given the stability and universal characterization of the homotopy interleaving distance, there is potential for developing algorithms that leverage this metric for more efficient processing of complex data types, such as multi-parameter persistence models. Additionally, extending the principles of homotopy interleavings might provide novel insights into the coarse geometry of networks and high-dimensional data.

In summary, Blumberg and Lesnick’s paper paves the way for a deeper integration of topological methods in data science, fostering innovations in the algorithms that form the backbone of AI's interpretative frameworks. The universality of the introduced metrics marks a significant step forward in aligning theoretical homotopy with practical applications in data-driven technology.