Persistent Homology in TDA

• Persistent Homology is a topological data analysis technique that computes multiscale features by tracking the birth and death of homological structures through filtrations.
• It is designed for robustness and interpretability, with applications ranging from neural network analysis to structural phase investigations in disordered materials.
• State-of-the-art implementations leverage optimized matrix reductions and scalable algorithms to efficiently process high-dimensional data with persistent invariants.

Persistent homology (PH) is a foundational technique in topological data analysis (TDA) that systematically quantifies the multiscale topological features of discrete data through homology computed over filtrations. PH encodes the birth and death of $p$-dimensional homological features as the underlying geometric object evolves across a parameter (typically scale), yielding invariants as barcodes or persistence diagrams. PH is robust to perturbations, coordinate-free, and interpretable, leading to diverse applications in mathematics, computational sciences, and machine learning.

1. Algebraic and Computational Foundations

A $k$-simplex is an unordered $(k+1)$-tuple of distinct points, and a simplicial complex $K$ is a finite collection of simplices closed under taking faces. For a fixed field $\mathbb{F}$ (commonly $\mathbb{F}_2$), the $k$-chains $C_k(K)$ form a vector space with a boundary operator $\partial_k: C_k \to C_{k-1}$ defined by

$$\partial_k[v_0,\dots,v_k] = \sum_{i=0}^k (-1)^i [v_0,\dots,\widehat{v}_i, \dots, v_k].$$

Homology groups are $$H_k(K) = \ker(\partial_k) / \operatorname{im}(\partial_{k+1})$$ and their dimensions, the Betti numbers $\beta_k$, count independent $k$-dimensional holes.

A filtration is a nested sequence $$K_0 \subseteq K_1 \subseteq \cdots \subseteq K_N = K$$ indexed by a filtration parameter (distance, function value, etc.). Examples include Vietoris–Rips and Čech complexes on metric spaces, lower-star filtrations on grayscale images, and more general sublevel set filtrations.

The persistent homology workflow computes homology $H_k(K_i)$ at every filtration step, then tracks the creation ("birth") and annihilation ("death") of homological features across inclusions $K_i \hookrightarrow K_j$. The persistence module structure theorem asserts any such $k$-th persistence module decomposes into interval modules, whose intervals $[b,d)$ describe the lifespan of features. These intervals are visualized as barcodes or persistence diagrams (multisets of points $(b,d)$ in $\mathbb{R}^2$) (Otter et al., 2015).

2. Stability, Uniqueness, and Inverse Problems

PH is provably stable under input perturbations. If $f,g: K \to \mathbb{R}$ are tame functions on a triangulable space, their diagram bottleneck distance satisfies

$d_B(D_k(f), D_k(g)) \leq \|f-g\|_\infty,$

meaning that small changes in data or function values only move points in the persistence diagram by at most the perturbation size (Otter et al., 2015, Turkeš et al., 2022).

However, persistence diagrams are not generally injective invariants. Recent results establish that, for generic point clouds $X \subset \mathbb{R}^d$, the entire VR barcode determines $X$ up to isometry if and only if the associated critical graph is globally rigid; similarly, local identifiability depends on infinitesimal rigidity (Beers et al., 12 Nov 2024). The fiber of the PH map (the set of point clouds with identical barcode) has dimension bounded between $nd-k+1$ and $nd-n+1$ (where $n$ is the number of points and $k$ the number of essential barcode endpoints), linking PH geometry to rigidity theory.

3. Structural Variants: Filtrations and Persistent Path Homology

PH's power depends strongly on filtration choice.

• Point cloud data: The VR and Čech complexes, or α-complex for low dimensions, are used for geometric and topological signal.
• Images: Cubical complexes and sublevel filtrations on pixel intensity yield efficient PH computation.
• Graphs: Graphs can be turned into 1D simplicial complexes and filtered by vertex (attribute) or edge (weight/color) functions (Immonen et al., 2023), or by custom notions such as localized height or soft predictions in neural network outputs (Oner et al., 2021).

For directed networks, standard PH is insensitive to edge orientation. Persistent path homology (PPH) generalizes to digraphs via chain complexes of allowed directed paths. PPH encodes asymmetry, distinguishes edge- and vertex-level cycles, and is stable under bottleneck metric, retaining information lost by symmetrization (Chowdhury et al., 2017).

Non-isotropic persistent homology (NIPH) exploits metric-dependence by varying the metric (e.g., via linear transformations or anisotropic scaling) and tracking the resulting shifts in persistence diagrams, thereby recovering geometric parameters such as orientation and anisotropy not visible to standard PH (Grande et al., 2023).

4. Algorithmic Methods and Software Ecosystem

Computation of PH reduces fundamentally to matrix reduction (column-wise over finite fields) of the boundary matrix, ordered compatibly with the filtration. Modern software—Ripser (cohomology-based reduction), GUDHI, Dionysus, PHAT/DIPHA (parallel chunk/spectral reduction)—implements these techniques with optimizations such as clearing, chunking, and streaming (Otter et al., 2015, Aggarwal et al., 2021).

Scalability innovations include:

• Paired-indexing for high-dimensional simplices (Aggarwal et al., 2021).
• Implicit cohomology reduction and batch streaming for memory efficiency in processing millions of simplices (Aggarwal et al., 2021).
• Tight representative cycle computation for localizing topological features in large data (Aggarwal et al., 2022). Greedy algebraic cycle-shortening and stochastic refinement yield minimal support cycles within the same homology class.

Privacy and security requirements have prompted the implementation of PH entirely over encrypted data, using homomorphic encryption to reduce boundary matrices in ciphertext space, with provable correctness and bounded noise growth (Gold et al., 2023).

5. Multiparameter Persistence and Stable Vectorization

Standard PH yields a one-parameter persistence module; several applications require multiparameter PH (MPH), i.e., filtrations indexed by tuples $(t_1,\dots,t_n)$. MPH does not generally admit complete discrete interval decompositions. Instead, descriptors such as the bigraded Betti tables, rank invariants, and signed barcodes are used (Scaramuccia et al., 2018, Loiseaux et al., 2023).

Discrete Morse theory enables reduction in MPH by reconstructing a smaller Morse complex compatible with the multi-filtration, supporting scalable MPH computation for large complexes and enabling parallelization (Scaramuccia et al., 2018).

Stable vectorization of MPH descriptors is achieved by interpreting signed barcodes as signed measures and embedding them into Hilbert spaces via convolution (persistence image-style) or sliced Wasserstein kernels, with stability proven in the Kantorovich–Rubinstein norm (Loiseaux et al., 2023).

6. Expressivity, Structure Learning, and Application Contexts

The discriminative power of PH depends crucially on the chosen filtration and preprocessing.

• On graphs: Vertex- and edge-based color filtrations each see different attribute patterns, with necessary and sufficient conditions for discrimination formalized via color-separating and color-disconnecting sets. Neither type subsumes the other; RePHINE combines both with additional vertex color annotations, strictly increasing expressive power and improving GNN integration and classification accuracy (Immonen et al., 2023).
• Graphical models: PH bar birth and death times arise as explicit competing exponential events (edge and simplex clocks) in latent-position graphical models, enabling Bayesian inference on the population-level origin of topological differences (e.g., in neuroimaging) (Wu et al., 15 Nov 2025).
• Structural phase analysis: In disordered material systems, PH captures both local and global order via unified descriptors, with custom metrics (e.g., the Separation Index) quantifying topological separation between phases and outperforming classical order parameters (Wang et al., 21 Nov 2024).
• Feature localization: New algorithms allow efficient recovery of tight representative cycles that bound significant topological features (voids, loops) on large data, enabling precise scientific interpretation (Aggarwal et al., 2022).

PH's effectiveness on tasks such as hole counting, convexity detection, and geometric regression is empirically superior to several neural architectures (e.g., PointNet), robust to affine transformations and noise, and efficiently computable for low-dimensional signatures (Turkeš et al., 2022).

7. Practical Implementations and Extensions

PH is widely accessible through open-source libraries (Ripser, GUDHI, Dionysus, PHAT, DIPHA), each specializing in different filtration types or optimization regimes (Otter et al., 2015).

Implementations must balance combinatorial blowup with dimensionality and sample size (worst-case VR size is exponential in the sample size and ambient dimension), motivating:

• Sparse and landmark-based complexes (α-complexes, witness complexes).
• Parallel and streaming architectures (Aggarwal et al., 2021).
• Integration with machine learning via differentiable or vectorized persistence descriptors (landscapes, images, and signed measures) for use with standard classifiers (Asaad et al., 2022, Loiseaux et al., 2023).

PH's future extensions include scalable MPH, improved localization and interpretability, neural architectures jointly leveraging topological features, privacy-aware analytics, and formalization of its inverse/regeneration properties (Beers et al., 12 Nov 2024, Gold et al., 2023, Aggarwal et al., 2022). The field continues to see theoretical and applied innovation at the intersection of algebraic topology, computational geometry, machine learning, and statistical inference.