Fibered Barcode in Multi-Parameter TDA

• Fibered barcode is a topological invariant that extracts one-dimensional barcodes from multi-parameter persistence modules via restrictions to non-negative affine lines.
• It leverages an augmented arrangement for efficient computation and real-time query support, ensuring stability and a bounded Lipschitz metric.
• Its sheaf-theoretic reinterpretation and equivalence to projected barcodes bridge algebraic and combinatorial methods for scalable data analysis.

The fibered barcode is a central invariant assigned to multi-parameter persistence modules, particularly bipersistence modules, arising in topological data analysis. It encapsulates the collection of one-dimensional barcodes obtained by restricting the module to affine lines of non-negative slope and thus quantifies the persistence of topological features across multiple scales and directions. Fibered barcodes enjoy desirable properties of computability, stability, and efficient query support via specialized data structures. Recent advances have established fibered barcodes as instances of projected barcodes in sheaf-theoretic frameworks and have enabled their scalable computation and real-time visualization.

1. Definition and Formal Structure

Let $M$ be a bipersistence module, defined as a covariant functor $$M: (\mathbb{R}^2, \leq) \to \mathrm{Vect}_\mathbb{k}$$, with $\mathbb{k}$ a field and $\leq$ the product order. For any non-negatively sloped affine line $\ell \subset \mathbb{R}^2$ (parametrized as $\ell = \{u + t w \mid t \in \mathbb{R}\}$, $u \in \mathbb{R}^2$, $w = (w_1, w_2)$, $w_1, w_2 \geq 0$), the restriction $M|_\ell$ becomes a one-parameter persistence module: $t \mapsto M(u + t w)$. The fibered barcode $\mathcal{F}(M)$ is defined as the assignment from $\ell$ to the barcode of $M|_\ell$,

$\mathcal{F}(M)(\ell) = \mathcal{B}(M|_\ell)$

where $\mathcal{B}(M|_\ell)$ is the multiset of intervals describing persistent features on $\ell$ (Lesnick et al., 8 Nov 2025). For $n$-parameter modules $$M \colon (\mathbb{R}^n, \leq) \to \mathrm{Vect}_\mathbb{k}$$, the fibered barcode generalizes by considering restrictions to affine lines of positive slope in $\mathbb{R}^n$.

Sheaf-theoretic reinterpretation positions $M$ as a sheaf $F$ on $(\mathbb{R}^n, \gamma\text{-topology})$ with microsupport in $V \times \gamma^\circ_a$, where $\gamma = (-\infty, 0]^n$ and $$\gamma^\circ = \{u \in (\mathbb{R}^n)^* \mid u_i \geq 0\, \forall i\}$$ (Berkouk et al., 2022). The restriction to a line $\ell$ is realized functorially via pullback and pushforward:

$i_\ell^{-1} M \cong j_\ell \circ p_{\ell *} M,$

where $p_\ell : \mathbb{R}^n \to \ell$ is projection and $j_\ell$ identifies $\ell$ with $\mathbb{R}$.

2. Properties and Theoretical Guarantees

Fibered barcodes exhibit several key properties:

• Stability: For modules $M, N$ with interleaving distance $\delta$, the bottleneck distance of barcodes satisfies

$$d_B(\mathcal{F}(M)(\ell), \mathcal{F}(N)(\ell)) \leq \delta$$

for every line $\ell$, so $\mathcal{F}$ is $1$-Lipschitz for these metrics (Lesnick et al., 8 Nov 2025).

• Finiteness of Combinatorial Types: Despite uncountably many lines, only finitely many combinatorial types of slice barcodes occur, formalized by a line arrangement in slope space partitioning the simplex of slopes into $O((m + r)^2)$ open cells, each constant up to affine reparameterization (Lesnick et al., 8 Nov 2025).
• Equivalence to Projected Barcodes: In the sheaf-theoretic setting, fibered barcodes are instances of projected barcodes, obtained via the derived pushforward $R(u)_*M$ for a linear functional $u$, thus connecting algebraic and sheaf-theoretic perspectives (Berkouk et al., 2022).

3. Computational Framework: The Augmented Arrangement

Efficient querying and computation of fibered barcodes leverage the augmented arrangement data structure:

• Critical-Slope Arrangement: Given a minimal presentation $\Phi: F^1 \to F^0$ (with $m$ generators and $r$ relations), construct an arrangement of $m r$ affine lines in the slope simplex $\Delta = \{w_1, w_2 > 0, w_1 + w_2 = 1\}$. Each line is defined by $(\beta_i - \alpha_j) \cdot w = 0$, where $\beta_i$ and $\alpha_j$ are grades of generators and relations, respectively.
• Barcode-Templates: For each open cell $C$ in $\Delta$ (resulting from critical lines), select an interior slope $w_C$, form the barcode of $M$ restricted to the corresponding line, and store pairs $(\beta_{i(k)}, \alpha_{j(k)})$—birth and death grades—in a barcode-template $\mathcal{T}_C$.
• Search Structure: Overlay the arrangement with barcode-templates and build a search structure (e.g., balanced binary search tree) for locating the corresponding template for any query slope $w$ in $O(\log L)$ time ($L = m r$).
• Complexity: Construction requires $O((m + r)^2 \log(m + r))$ time and space. Querying any slice barcode proceeds in $O(\log(m r) + |\mathcal{F}(M)(\ell)|)$ time (Lesnick et al., 8 Nov 2025).

4. Query Algorithm and Evaluation

To answer a barcode query for a line $\ell$:

1. Normalize the slope $w$ of $\ell$ to the simplex $\Delta$.
2. Locate the cell $C$ containing $w$ via the search structure.
3. Retrieve barcode-template $\mathcal{T}_C = \{(\beta_{i(k)}, \alpha_{j(k)})\}$.
4. For each pair, compute birth and death along $\ell$:

$$t_{\text{birth}}(k) = \langle \beta_{i(k)} - u, w \rangle / \|w\|^2, \quad t_{\text{death}}(k) = \langle \alpha_{j(k)} - u, w \rangle / \|w\|^2$$

1. Return $$\mathcal{F}(M)(\ell) = \{[t_{\text{birth}}(k), t_{\text{death}}(k))\}$$.

This procedure realizes logarithmic-time barcode evaluation, enabling real-time interactive applications such as the RIVET visualization framework for bipersistent homology.

5. Sheaf-Theoretic Connections and Metrics

Sheaf theory provides a categorical and analytical underpinning for fibered barcodes:

• Derived pushforward $R f_* F$ along a continuous $f : \mathbb{R}^n \to \mathbb{R}$ yields a one-parameter persistence sheaf whose barcode is the projected barcode along $f$.
• Fibered barcodes correspond, under equivalence, to projected barcodes $\mathcal{B}_u(M)$ for appropriate linear functionals $u$.
• Stability is ensured via convolution/interleaving distance $d_{\text{conv}}$, with integral sheaf metric (ISM) and sliced convolution distance (SCD) defined by

$$d_{\text{ISM}}(F, G) = \sup_{\|u\| = 1} d_B(\mathcal{B}_u(F), \mathcal{B}_u(G));$$

$$d_{\text{SCD}}^p(F, G) = \left( \int_{\|u\| = 1} d_B(\mathcal{B}_u(F), \mathcal{B}_u(G))^p du \right)^{1/p}$$

Both metrics are lower bounds for $d_{\text{conv}}$ and computable by reduction to standard 1D barcode algorithms (Berkouk et al., 2022).

6. Implementation and Example

Practical implementation proceeds by direct reduction to 1D persistence computations:

• Standard libraries (Ripser, Dionysus, GUDHI, giotto-tda) operate on slices via real-valued filtrations $f_{u^k}(x) = u^k \cdot f(x)$.
• For ISM/SCD, sample $m$ directions $u^k$ in $Q_\gamma$ and process each as an independent 1D persistence computation ($O(N^2)$ per run, with $N$ the complex size).
• A plausible implication is that only modest $m$ ($50$–$200$) suffices for accurate metric estimates.
• Example: For a module presented by two generators $(0,0), (1,0)$ and one relation $(0,1)$, the arrangement of critical lines in slope space yields two cells, each encoded with corresponding barcode-templates for rapid query evaluation (Lesnick et al., 8 Nov 2025).

7. Significance and Applications

Fibered barcodes unify algebraic, combinatorial, and sheaf-theoretic approaches to multi-parameter persistence:

• They provide a stable and computable invariant for data analysis in settings where multi-scale, multi-directional topology is central.
• The augmented arrangement data structure enables interactive visualization and exploration in software such as RIVET (Lesnick et al., 8 Nov 2025).
• New metrics, ISM/SCD, facilitate quantification of module similarity with computational efficiency and theoretical grounding (Berkouk et al., 2022).

A plausible implication is that advances in the computation and stability of the fibered barcode will enable broader adoption in high-dimensional TDA, scientific visualization, and applied mathematics. The categorical equivalence with projected barcodes bridges algebraic and analytical methods, potentially informing further theoretical developments in persistence theory.

Property	Fibered Barcode	Projected Barcode
Definition	Slice along non-negative affine lines; barcode of slice	Pushforward along linear
Computability	Augmented arrangement, $O((m+r)^2\log(m+r))$ construction	Standard 1D persistence
Stability	$d_B$ Lipschitz in module interleaving distance	Follows from pushforward