Ephemeral Proof Chains in Topological Data Analysis

• Ephemeral proof chains are zero-length homological features in filtered chain complexes that offer a finer invariant than classical barcodes.
• They refine stability theorems and metric discriminability in topological data analysis by retaining transient chain-level information from Vietoris–Rips complexes.
• Their efficient computability enables enhanced shape differentiation and supports practical integration into machine learning pipelines.

Ephemeral proof chains, in the sense of the Usher–Zhang filtered-chain-complex framework, designate homological features in persistent homology that arise and vanish instantly at the chain complex level, i.e., elements whose birth and death coincide. While these “ephemeral” features are invisible to classical persistence modules and the standard barcode, they are key invariants at the filtered chain complex level. Their formalization leads to the notion of the “verbose barcode,” a strictly finer invariant than the conventional barcode. The theory of ephemeral proof chains refines both stability theorems and metric discriminability for multidimensional topological data analysis, particularly within the context of Vietoris–Rips complexes, with implications for both computability and practical shape analysis (Mémoli et al., 2022).

1. Filtered Chain Complexes and Ephemeral Generators

A filtered chain complex (FCC) is a triple $(C_*,\partial,\ell)$, where

• $C_* = \bigoplus_{k\geq0}C_k$ is a finite-dimensional chain complex over a field $F$,
• $\partial$ is the boundary operator,
• $\ell: C_* \to R \cup \{-\infty\}$ is a non-Archimedean filtration function.

The non-Archimedean property requires that for all $x, y \in C_*$ and $\lambda \in F\setminus\{0\}$,

$$\ell(x) = -\infty \iff x = 0, \quad \ell(\lambda x) = \ell(x), \quad \ell(x+y) \leq \max\{\ell(x), \ell(y)\}, \quad \ell(\partial x) \leq \ell(x).$$

An ascending family of subcomplexes is given by $C_*^t = \{x \in C_* \mid \ell(x) \leq t\}$, and homology at each filtration level recovers the classical persistence module.

Any FCC admits a (non-canonical) decomposition into elementary complexes $E(a,a+L,k)$, built from pairs $C_* = \bigoplus_{k\geq0}C_k$0 with $C_* = \bigoplus_{k\geq0}C_k$1 in degree $C_* = \bigoplus_{k\geq0}C_k$2, $C_* = \bigoplus_{k\geq0}C_k$3 in degree $C_* = \bigoplus_{k\geq0}C_k$4, $C_* = \bigoplus_{k\geq0}C_k$5, $C_* = \bigoplus_{k\geq0}C_k$6, $C_* = \bigoplus_{k\geq0}C_k$7. The case $C_* = \bigoplus_{k\geq0}C_k$8 yields a bar $C_* = \bigoplus_{k\geq0}C_k$9 of zero length—an ephemeral feature. Such ephemeral generators are invisible at the homology level, but present at the chain level.

2. Construction of the Verbose Barcode

Given an FCC $F$0, the verbose barcode refines the concise barcode by also recording zero-length (ephemeral) bars.

For each degree $F$1, one performs a singular-value decomposition of the boundary map $F$2 in the setting of non-Archimedean normed spaces. Orthogonal bases $F$3 of $F$4 and $F$5 of $F$6 are constructed such that $F$7 for $F$8 (with the rest mapping to zero). The filtration-drops $F$9 are ordered nonincreasingly.

The degree-$\partial$0 verbose barcode is: $\partial$1 where each bar $\partial$2 has length $\partial$3, possibly zero. Classical barcodes are recovered by discarding zero-length bars.

The following pseudocode expresses extraction of $\partial$4:

$C_*^t = \{x \in C_* \mid \ell(x) \leq t\}$3

3. Metrics on Verbose Barcodes and Chain Complexes

Let $\partial$5, with the $\partial$6-norm: $\partial$7 and extended to infinite death times as $\partial$8.

Given multisets $\partial$9 (of equal size), the matching distance is defined as

$\ell: C_* \to R \cup \{-\infty\}$0

For concise (classical) barcodes, matching to the diagonal $\ell: C_* \to R \cup \{-\infty\}$1 recovers the bottleneck distance $\ell: C_* \to R \cup \{-\infty\}$2, but in matching distances for verbose barcodes, ephemeral (zero-length) bars cannot be matched off the barcode.

At the chain level, the interleaving distance $\ell: C_* \to R \cup \{-\infty\}$3 between FCCs is defined by the existence of chain maps $\ell: C_* \to R \cup \{-\infty\}$4, $\ell: C_* \to R \cup \{-\infty\}$5 shifting filtrations by $\ell: C_* \to R \cup \{-\infty\}$6 and commuting up to inclusions. The main isometry theorem states: $\ell: C_* \to R \cup \{-\infty\}$7 which refines the classical stability result, since $\ell: C_* \to R \cup \{-\infty\}$8 on concise barcodes.

4. Stability Theorems for Verbose Vietoris–Rips Barcodes

For finite metric spaces $\ell: C_* \to R \cup \{-\infty\}$9 and $x, y \in C_*$0, let $x, y \in C_*$1 and $x, y \in C_*$2 denote their Vietoris–Rips FCCs. The pullback bottleneck distance in degree $x, y \in C_*$3 is

$x, y \in C_*$4

where the infimum ranges over common pullbacks $x, y \in C_*$5 of $x, y \in C_*$6 and $x, y \in C_*$7.

Defining the pullback interleaving distance similarly,

$x, y \in C_*$8

the following holds: $x, y \in C_*$9 with $\lambda \in F\setminus\{0\}$0 denoting the Gromov–Hausdorff distance. Furthermore, classical bottleneck distances are bounded by the new distances: $\lambda \in F\setminus\{0\}$1 Thus, $\lambda \in F\setminus\{0\}$2 and $\lambda \in F\setminus\{0\}$3 provide strictly stronger lower bounds for $\lambda \in F\setminus\{0\}$4 than previously possible with concise barcodes.

5. Computability and Explicit Formulae

A combinatorial formula for pullback barcodes (Proposition 3.13) expresses $\lambda \in F\setminus\{0\}$5 for a pullback $\lambda \in F\setminus\{0\}$6 duplicating $\lambda \in F\setminus\{0\}$7 copies of points $\lambda \in F\setminus\{0\}$8 as

$\lambda \in F\setminus\{0\}$9

where $$\ell(x) = -\infty \iff x = 0, \quad \ell(\lambda x) = \ell(x), \quad \ell(x+y) \leq \max\{\ell(x), \ell(y)\}, \quad \ell(\partial x) \leq \ell(x).$$0 is an explicit combinatorial coefficient.

The pullback-vector reduction (Proposition 5.1) reformulates the minimization as a finite search over integer vectors $$\ell(x) = -\infty \iff x = 0, \quad \ell(\lambda x) = \ell(x), \quad \ell(x+y) \leq \max\{\ell(x), \ell(y)\}, \quad \ell(\partial x) \leq \ell(x).$$1, greatly reducing the complexity compared to all surjective correspondences (though still exponential in the worst case).

Degree 0 barcodes admit a closed-form (Proposition 5.2): for $$\ell(x) = -\infty \iff x = 0, \quad \ell(\lambda x) = \ell(x), \quad \ell(x+y) \leq \max\{\ell(x), \ell(y)\}, \quad \ell(\partial x) \leq \ell(x).$$2 (finite-length death times for $$\ell(x) = -\infty \iff x = 0, \quad \ell(\lambda x) = \ell(x), \quad \ell(x+y) \leq \max\{\ell(x), \ell(y)\}, \quad \ell(\partial x) \leq \ell(x).$$3), and $$\ell(x) = -\infty \iff x = 0, \quad \ell(\lambda x) = \ell(x), \quad \ell(x+y) \leq \max\{\ell(x), \ell(y)\}, \quad \ell(\partial x) \leq \ell(x).$$4 for $$\ell(x) = -\infty \iff x = 0, \quad \ell(\lambda x) = \ell(x), \quad \ell(x+y) \leq \max\{\ell(x), \ell(y)\}, \quad \ell(\partial x) \leq \ell(x).$$5,

$$\ell(x) = -\infty \iff x = 0, \quad \ell(\lambda x) = \ell(x), \quad \ell(x+y) \leq \max\{\ell(x), \ell(y)\}, \quad \ell(\partial x) \leq \ell(x).$$6

computable in $$\ell(x) = -\infty \iff x = 0, \quad \ell(\lambda x) = \ell(x), \quad \ell(x+y) \leq \max\{\ell(x), \ell(y)\}, \quad \ell(\partial x) \leq \ell(x).$$7 time after sorting.

Concrete examples confirm that $$\ell(x) = -\infty \iff x = 0, \quad \ell(\lambda x) = \ell(x), \quad \ell(x+y) \leq \max\{\ell(x), \ell(y)\}, \quad \ell(\partial x) \leq \ell(x).$$8 recovers the exact $$\ell(x) = -\infty \iff x = 0, \quad \ell(\lambda x) = \ell(x), \quad \ell(x+y) \leq \max\{\ell(x), \ell(y)\}, \quad \ell(\partial x) \leq \ell(x).$$9 distance in one- and two-point cases, and that in ultrametric examples, pairs with identical concise barcodes may have $C_*^t = \{x \in C_* \mid \ell(x) \leq t\}$0, manifesting the discriminating advantage of the verbose barcode.

6. Significance, Applications, and Prospects

The identification and retention of ephemeral bars meaningfully expand the discriminating capability of persistent homology without increasing the underlying homological complexity. In practice, matrix-reduction algorithms for persistence already compute zero-length pivot pairs, so extraction of verbose barcodes incurs negligible additional computational cost.

The robust and finer pullback distances $C_*^t = \{x \in C_* \mid \ell(x) \leq t\}$1 and $C_*^t = \{x \in C_* \mid \ell(x) \leq t\}$2 enable a stable matching across all homological dimensions, improving lower bounds on Gromov–Hausdorff distance, sharpening shape differentiation, and refining kernel or signed-measure methods in topological data analysis. A plausible implication is enhanced sensitivity to subtle shape perturbations and potential for more accurate integration into statistical machine learning pipelines, where zero-persistence features may now be exploited rather than discarded.

Potential directions for subsequent work include the development of fast approximation algorithms for verbose barcode distances, analysis of stability under alternative perturbations (such as noise perturbations in point clouds), and systematic exploration of ephemeral information in advanced data analytic or learning-theoretic contexts (Mémoli et al., 2022).