Topological Data Analysis in Finsler Space

• Topological Data Analysis in Finsler space is defined by tracking homological features through a persistent module structure over Z_p, with interval decompositions (barcodes) representing birth and death times.
• The barcode theorem ensures a unique decomposition of graded Z_p[t]-modules, providing clear insights into feature evolution and stability in complex spaces.
• Efficient algorithms, including Smith normal form and boundary matrix reduction, are used to compute persistent invariants, ensuring robustness and computational efficiency in TDA.

A persistent homology module over $\mathbb{Z}_p$ is a functorial structure capturing the evolution of homological features across a filtered topological space, where all linear algebra is conducted over the finite field $\mathbb{Z}_p$. This object encodes the birth and death of topological features (such as connected components, holes, and higher-dimensional cycles) through parameter values, and enjoys a complete classification by interval decompositions—the so-called barcode theorem. Algebraically, such modules are finitely generated graded modules over $\mathbb{Z}_p[t]$, a principal ideal domain, so their structure admits a one-to-one correspondence with finite multisets of intervals, which are the backbone of topological data analysis (TDA) (Carlsson, 2020, Corbet et al., 2017, Skraba et al., 2013, Ranoa, 2024, Cavalcanti, 26 Dec 2025).

1. Categorical Definition and Algebraic Model

A (single-parameter) persistence module $M$ over $\mathbb{Z}_p$ is defined as a functor

$$M : (\mathbb{R}_+, \leq) \to \mathrm{Vect}(\mathbb{Z}_p)$$

assigning to each filtration value $r\in\mathbb{R}_+$ a finite-dimensional $\mathbb{Z}_p$-vector space $M(r)$, with structure maps $M(r\leq s):M(r)\to M(s)$ satisfying functoriality conditions. For computational purposes, it suffices to restrict to a discrete filtration (often indexed by $\mathbb{N}$), yielding equivalence with the category of $\mathbb{N}$-graded modules over the polynomial ring $\mathbb{Z}_p[t]$ where $t$ acts as the shift (progressing the filtration parameter by one).

Explicitly, one assembles

$M_* = \bigoplus_{i\in\mathbb{N}} M(i)$

with $t\cdot : M(i) \to M(i+1)$ given by the structure maps. The abelian group structure and grading are preserved by the module operations (Carlsson, 2020, Corbet et al., 2017, Ranoa, 2024).

2. Structure Theorem and Interval Decomposition

Given that $\mathbb{Z}_p[t]$ is a principal ideal domain, the classification of finitely presented graded $\mathbb{Z}_p[t]$-modules is governed by the classical structure theorem:

Every such module decomposes uniquely (up to reordering) as

$$M \cong \bigoplus_{j=1}^m \mathbb{Z}_p[t](-a_j)\;\;\oplus\;\;\bigoplus_{k=1}^n \big(\mathbb{Z}_p[t]/(t^{d_k})\big)(-c_k)$$

where the notation $(-a)$ denotes a grading shift by $a$, and $d_k\geq 1$ the length of the corresponding torsion block. Each summand corresponds precisely to an interval module (see below), so that persistence modules over $\mathbb{Z}_p$ are classified by their collection of intervals (barcodes) (Carlsson, 2020, Corbet et al., 2017, Skraba et al., 2013, Ranoa, 2024).

This correspondence is established functorially: every pointwise-finite module decomposes as

$M \cong \bigoplus_{i=1}^N I[a_i,b_i)$

with $I[a,b)$ the interval module supported on $[a,b)$ (Ranoa, 2024).

3. Interval Modules and the Barcode Theorem

The interval module $I[a,b)$ over $\mathbb{Z}_p$ is defined by

$$I[a,b)(r) = \begin{cases} \mathbb{Z}_p, & a \leq r < b \ 0, & \text{otherwise} \end{cases}$$

with all structure maps inside $[a,b)$ the identity and zero otherwise. This module encodes a homological feature born at $a$ and dying at $b$. Infinite bars, $[a,\infty)$, correspond to free summands, while finite bars, $[c, c+b)$, correspond to torsion summands of the module.

The barcode theorem asserts the uniqueness (up to ordering) of such decompositions: each persistence module over $\mathbb{Z}_p$ is isomorphic to a direct sum of such intervals. The collection of intervals $\mathcal{B}(M)$—the barcode—is a complete discrete invariant for the isomorphism class of $M$ (Carlsson, 2020, Corbet et al., 2017, Ranoa, 2024).

4. Algorithms and Computation

Computational approaches proceed by forming the corresponding chain complexes with boundary matrices, either directly over $\mathbb{Z}_p$ (standard in TDA software) or over $\mathbb{Z}_p[t]$ for the graded approach.

For $\mathbb{Z}_p$ coefficients, the classical reduction/clearing algorithm operates on the (block lower-triangular) boundary matrix $B$ over $\mathbb{Z}_p$, ordered by filtration:

1. Each column represents a simplex, entries denote incidence data mod $p$.
2. Column additions are performed so that each column contains at most one lowest nonzero row (the "Low" index).
3. Pairings of simplices by their Low positions correspond to birth and death times of features; the resulting intervals constitute the barcode for homology in each degree.

Alternatively, algorithms based on computing the graded Smith normal form (GSNF) over $\mathbb{Z}_p[t]$ are employed when more refined module data is sought. This requires generalized Gaussian elimination respecting the grading, and remains cubic in the number of simplices, with additional polynomial arithmetic overhead (Skraba et al., 2013, Ranoa, 2024).

5. Geometric Considerations and Torsion Phenomena

The universal coefficient theorem implies that, for a complex with torsion in integral homology, working over $\mathbb{Z}_p$ with $p$ coprime to the torsion summands ensures that all homology groups are free and of dimension given by the Betti numbers. Smith normal form analysis of boundary matrices over $\mathbb{Z}$ identifies all “bad” primes (those dividing any torsion coefficients). For $p$ not in this finite set, the persistent homology over $\mathbb{Z}_p$ recovers exactly the rank invariant and barcodes coincide with those of rational coefficients (Cavalcanti, 26 Dec 2025):

Step	Description	Output
Build boundary matrices	For each filtration value and $k$	$A_k(\epsilon)$ over $\mathbb{Z}$
Compute Smith normal forms	Find elementary divisors at each scale	Set of “bad” primes $P$
Choose prime $p \not\in P$	Ensures torsion-free mod $p$ homology	“Torsion-free” barcode
Reduce matrices mod $p$ and compute	Standard persistence algorithm on $\mathbb{Z}_p$	Barcodes over $\mathbb{Z}_p$

This procedure eliminates spurious torsion features and reflects the free part of homology, which aligns with the geometric representability constraints from Finsler and metric geometry (Cavalcanti, 26 Dec 2025).

6. Stability and Metric Properties

Barcode stability is assured by the following theorems:

• For two finite metric spaces $X, Y$, the bottleneck distance $W_\infty$ between their Vietoris–Rips barcodes $B_k(X)$, $B_k(Y)$ satisfies

$W_\infty(B_k(X), B_k(Y)) \leq d_{GH}(X, Y)$

where $d_{GH}$ is the Gromov–Hausdorff distance.

• For tame real-valued functions $f, g$ on a triangulable domain, the barcode stability satisfies

$$W_\infty(\mathrm{Barcode}(f), \mathrm{Barcode}(g)) \leq \|f - g\|_\infty$$

All arguments and results apply to $\mathbb{Z}_p$ coefficients, showing that persistent homology over finite fields is robust to perturbations of data (Carlsson, 2020). This ensures the continuity of barcodes with respect to the input, making persistent homology over $\mathbb{Z}_p$ a principled tool for TDA.

7. Examples and Characteristic $p$ Aspects

Common illustrative filtrations, such as a noisy sampling of a circle, result in barcodes computed over $\mathbb{Z}_p$ (usually $p=2$ for computational speed), revealing infinite $H_0$ bars (connectedness), a single long $H_1$ bar (main loop of the circle), and short bars corresponding to noise and artifacts. Over $\mathbb{Z}_p$, all homology groups are vector spaces and torsion phenomena appear only when $p$ is a “bad” prime for the structure of the chain complex (Carlsson, 2020, Ranoa, 2024).

Characteristic $p$ phenomena are notable: over $\mathbb{Z}_2$, all reduction steps use XOR operations with no orientation signs, while for $p>2$ sign management is required, but there is never denominator blowup as in rational arithmetic. The algebraic structure is always clean—no higher torsion arises, and the decomposition into barcodes is unique (Ranoa, 2024).

Summary

Persistent homology modules over $\mathbb{Z}_p$ are precisely described by interval decompositions, with all algebraic, computational, and stability theory aligning with the structure theorem for graded modules over the PID $\mathbb{Z}_p[t]$. Torsion appears only for “bad” primes and can be detected and circumvented via Smith normal form analysis of boundary operators. These properties underlie almost all practical TDA computations and provide a bridge between algebraic theory, algorithmic implementation, and geometric interpretation (Carlsson, 2020, Corbet et al., 2017, Skraba et al., 2013, Ranoa, 2024, Cavalcanti, 26 Dec 2025).