A stability theorem for bigraded persistence barcodes

Technical Summary: A Stability Theorem for Bigraded Persistence Barcodes

Problem Statement
The paper addresses two fundamental limitations in the application of bigraded persistent homology to data science, specifically when derived from the moment-angle complexes $Z_K$ associated with the Vietoris–Rips filtration of a finite pseudo-metric space $X$.

First, the standard bigraded homology of moment-angle complexes, while theoretically rich, is computationally prohibitive. By the Hochster decomposition, calculating these groups requires computing the simplicial homology of all full subcomplexes of the underlying simplicial complex, a task that scales poorly with data size.

Second, and more critically for data science applications, the standard bigraded persistent homology lacks a stability property. In topological data analysis, stability ensures that small perturbations in the input data (measured by the Gromov–Hausdorff distance) result in only small changes in the resulting persistence barcodes (measured by the bottleneck distance). Without this property, the invariants are not robust to noise, limiting their utility. The authors note that while bigraded persistence can distinguish point clouds indistinguishable by ordinary persistence (as shown in Example 3.4), its lack of stability renders it impractical for real-world applications.

Methodology
To resolve these issues, the authors utilize bigraded double homology of moment-angle complexes, a concept introduced and studied in [LPSS].

1. Double Homology Construction: Instead of using the standard homology $H_*(Z_K)$, the authors define a chain complex $CH_*(Z_K) = (H_*(Z_K), \partial')$ by endowing the bigraded homology groups with a second differential $\partial'$. The homology of this new complex, denoted $HH_*(Z_K)$, is the double homology.
2. Bigraded Persistent Double Homology: The authors define the persistent double homology module $PHHZ(X)$ by applying the double homology functor to the moment-angle complexes $Z_{R(X,t)}$ associated with the Vietoris–Rips filtration $\{R(X,t)\}_{t \geq 0}$.
3. The Doubling Operation: A central technical tool in the proof is the "doubling" operation on simplicial complexes and pseudo-metric spaces. Doubling a vertex $i$ in a complex $K$ (or a point $x$ in a space $X$) creates a new complex $K'$ (or space $X'$) by adding a duplicate vertex $i'$ (or point $x'$) with zero distance to the original.
   • Crucially, while doubling alters the standard homology of the moment-angle complex (Example 4.18), it preserves the double homology (Proposition 4.19).
   • Furthermore, any correspondence between two finite pseudo-metric spaces can be realized as a bijection between their iterated doublings (Proposition 4.16).

Key Contributions and Results

• Definition of Bigraded Persistent Double Homology: The paper formally defines the bigraded persistent double homology modules $PHHZ(X)$ and their corresponding barcodes $BB(X)$. These modules are shown to be computationally more accessible than their standard bigraded counterparts due to the smaller size of the double homology groups.
• Stability for Ordinary Bigraded Persistence (Modified): The authors first establish a stability theorem for the standard bigraded persistent homology $PHZ(X)$. However, this stability holds only with respect to a modified Gromov–Hausdorff distance ($d'_{GH}$), which restricts the infimum in the distance definition to bijective correspondences only (Theorem 4.21). The paper demonstrates that the standard Gromov–Hausdorff distance ($d_{GH}$) is insufficient for the stability of $PHZ(X)$, as doubling a point changes the barcode while leaving $d_{GH}$ unchanged (Example 4.18).
• Stability Theorem for Double Homology: The main result (Theorem 4.23) proves that the bigraded persistent double homology modules satisfy a stability theorem with respect to the standard Gromov–Hausdorff distance. Specifically, for any two finite pseudo-metric spaces $X$ and $Y$:
  $$d_{IL}(PHHZ(X), PHHZ(Y)) \leq 2d_{GH}(X, Y)$$
  where $d_{IL}$ is the interleaving distance.
• Isometry Theorem Application: By invoking the Isometry Theorem (Theorem 4.12), the authors derive a corresponding stability result for the bigraded barcodes $BB(X)$ and $BB(Y)$ in terms of the $\infty$-Wasserstein (bottleneck) distance (Corollary 4.24):
  $$W_\infty(BB(X), BB(Y)) \leq 2d_{GH}(X, Y)$$

Significance and Claims
The paper claims that the introduction of bigraded persistent double homology resolves the two primary drawbacks of the original bigraded approach.

1. Computational Efficiency: The double homology modules are smaller than the standard bigraded homology modules, making them more accessible from a computational viewpoint.
2. Robustness: Most importantly, the double homology modules possess the stability property essential for data science applications. The authors state that this property "says roughly that the bigraded barcode is robust to changes in the input data."

The authors emphasize that while bigraded persistence offers richer algebraic structures and the ability to distinguish data sets that ordinary persistence cannot (Example 3.4), it was previously unusable in practice due to instability. The stability theorem for double homology "opens a way for a more efficient application in data analysis" by combining the discriminatory power of bigraded invariants with the robustness required for noisy data. The paper does not propose new experimental applications but establishes the theoretical foundation necessary for such applications to be viable.