Persistent Homology Calculations

• Persistent homology calculations are methods that extract topological features by computing algebraic invariants like Betti numbers and persistence diagrams from data filtrations.
• They leverage discrete Morse theory and critical simplex algorithms to minimize computational overhead while maintaining interpretability.
• Utilizing spanning-tree and critical-simplex techniques, these calculations offer practical speedups and memory efficiency in processing high-dimensional datasets.

Persistent homology calculations underpin a robust framework in computational topology for quantifying and extracting topological features of data across multiple scales. These calculations involve constructing algebraic invariants—such as Betti numbers and persistence diagrams—through systematic workflows that rely on chain complexes, boundary operators, and algorithmic optimizations for both efficiency and interpretability. The methodology has evolved from classical algebraic-topological reductions to sophisticated algorithms exploiting discrete Morse theory and topological spanning-tree structures to sharply minimize computational overhead, yielding practical advances especially for high-dimensional data sets (Shi et al., 2023).

1. Mathematical Foundations: Simplicial Complexes and Homology

A finite simplicial complex $K$ consists of a set of simplices (vertices, edges, triangles, etc) closed under taking faces. Each dimension $p$ yields a ℤ₂-vector space $C_p(K)$ whose basis is the set of all $p$-simplices, with dimension $m_p$. The boundary operator $\partial_p: C_p \to C_{p-1}$ acts as

$$\partial_p([v_0, \dots, v_p]) = \sum_{i=0}^p [v_0, \dots, v_{i-1}, v_{i+1}, \dots, v_p]$$

computed over ℤ₂. In matrix representation, $\partial_p$ corresponds to the boundary matrix $B_p$ of size $m_{p-1} \times m_p$.

The standard homology and Betti number extraction follows:

• Cycles $Z_p = \ker(\partial_p)$, boundaries $B_p = \operatorname{im}(\partial_{p+1})$, and
• Homology $H_p(K; \mathbb{Z}_2) = Z_p / B_p$, with $\beta_p = \dim H_p$.

A filtration is realized via a discrete Morse function $f: K \to \mathbb{R}$ whose sublevel sets

$K^t = \{\alpha \in K: f(\alpha) \leq t\}$

form an increasing sequence $$\emptyset = K^{t_0} \subset K^{t_1} \subset \cdots \subset K^{t_N} = K$$, functionally structuring the calculation of topological feature persistence as the underlying complex evolves (Shi et al., 2023).

2. Discrete Morse Functions, Critical Simplices, and Filtration Construction

A discrete Morse function, according to Forman's framework, assigns real values to all simplices in a way that at most one face or one coface shares the function value with a given simplex. The critical simplices—those with neither face nor coface sharing their $f$-value ($|U(\alpha)| = |V(\alpha)| = 0$)—represent the minima of topological complexity for the complex.

For each simplex dimension $p$, the number of critical simplices $C_p$ satisfies $C_p \geq \beta_p$, with the alternating sum

$$\chi(K) = \sum_{p=0}^d (-1)^p m_p = \sum_{p=0}^d (-1)^p C_p$$

coinciding with the Euler characteristic. Lowering $C_p$ to precisely $\beta_p$ in each $p$ yields the optimal Morse function that minimizes criticality and computation (Shi et al., 2023).

3. Spanning Tree and Critical Simplex Algorithm

The "spanning-tree and critical-simplex" algorithm by Shi–Chen–Ma–Chen introduces a practical scheme as follows:

Step 0: Enumerate all simplices up to maximum dimension $d$, compute boundary matrices $B_p$.

Step 1: For each $p$, perform row-reduction on $B_p$ to determine its rank $r_p$ and to extract a maximal basis subset $T_p$: the "p-order spanning tree."

Step 2: For each $p$, classify non-tree p-simplices as either "used" by the $(p+1)$-tree or as "p-order cavity-generating simplices" responsible for the creation of new cycles in dimension $p$.

Step 3: Assign Morse function values incrementally:

• Start at an arbitrary vertex, assigning $f(v_0)=0$.
• Traverse spanning trees by dimension and assign Morse values in traversal order.
• Assign subsequent values to cavity-generators.
• Proceed dimension by dimension, always assigning Morse values first to tree simplices, then cavity-generators.

This process produces a Morse function whose filtration sequence corresponds to a minimal-critical-simplices construction: precisely $\beta_p$ critical simplices for each $p$ (Shi et al., 2023).

4. Extraction of Persistence Diagrams and Betti Numbers

The method guarantees a direct interpretability of critical simplex events:

• Each critical $p$-simplex with Morse value $t$ gives birth to an $H_p$ class at filtration index $t$.
• Each critical $(p+1)$-simplex with value $t'$ kills exactly one $p$-class, thus closing the persistence interval.

If no killing occurs, the class persists indefinitely. The barcode is constructed as the collection of birth-death pairs from this pairing scheme. Betti numbers at each filtration sublevel $K^t$ are computed as the count of unpaired critical $p$-simplices up to $t$, minus those already paired by higher-dimensional critical simplices. This yields immediate access to the full persistence diagram (Shi et al., 2023).

5. Computational Analysis and Comparison with Matrix Reduction

The complexity is dominated by the row-reduction step for each boundary matrix, $O(m_p^3)$ per dimension, with total worst-case complexity $O(N^3)$ for $N = \sum_p m_p$ simplices. In practical datasets—where the boundary matrices are typically very sparse and the filtration dimension is small (often ≤3)—the method shows empirical speedups (2–5×) and memory reductions (to ≤50% of the standard requirement) over classical matrix-reduction persistence. Memory usage is further optimized since only individual boundary matrices are stored and discarded after processing, obviating the need for the cumulative block-matrix necessary in standard approaches (Shi et al., 2023).

Summary table:

Method	Critical Simplices	Theoretical Complexity	Empirical Performance
Spanning-Tree–Critical-Simplex	Exactly $\beta_p$	$O(N^3)$	2–5× faster, ≤50% memory
Classical Matrix Reduction	Arbitrary (≥$\beta_p$)	$O(N^3)$	Baseline (higher resource)

6. Illustrative Example

For a triangle with vertices $v_0,v_1,v_2$, edges $e_0=(v_0,v_1)$, $e_1=(v_1,v_2)$, $e_2=(v_2,v_0)$, and the triangle $\tau=(v_0,v_1,v_2)$:

• $m_0=3, m_1=3, m_2=1$
• Ranks: $r_1=2$, $r_2=1$
• Betti numbers: $\beta_0=1$, $\beta_1=0$, $\beta_2=0$
• Spanning-tree edges: $T_1 = \{e_0, e_1\}$
• Cavity-generator edges: $Cg_1 = \{e_2\}$

Assigning Morse values specifies the persistent pairing leading to the barcode:

• H₀: birth at 1, never dies ⇒ [1, ∞)
• H₁: birth at 6, death at 7 ⇒ [6, 7)

This concise event-based bookkeeping exemplifies the method's direct pathway to persistence diagrams (Shi et al., 2023).

7. Theoretical and Practical Context

This approach offers several significant advances relative to traditional persistence calculation schemes:

• Minimal criticality ensures only as many critical simplices as the true Betti numbers, reducing computation on redundancies that correspond to short-lived or “noise” bars present in classical reductions.
• The algorithm leverages discrete Morse theory to achieve optimal Morse matchings, refining the taxonomic granularity of persistent features.
• The evaluation on real datasets (C. elegans neural network, BA-model graphs, alpha-complexes of point clouds such as the Stanford Dragon) consistently demonstrates improved efficiency in both speed and memory consumption, with no loss of accuracy in computed persistence diagrams (Shi et al., 2023).

These characteristics position the spanning-tree–critical-simplex algorithm as a benchmark for persistent homology computation when scalability and interpretability are required. The alignment of the critical simplex set with Betti number invariants also facilitates theoretical analyses involving topological signal versus noise discrimination in high-dimensional TDA workflows.