Persistent Homology for Multiscale Domains

• The PHD-MS framework unifies geometric PDEs, kernel density methods, and multiparameter persistent homology to capture multiscale topological features.
• Robust filtrations via Laplace–Beltrami flows and density models enhance noise resistance and enable efficient analysis in high-dimensional data.
• Parallel reduction techniques and distributed homology methods facilitate real-time multiscale analysis across applications like nanomaterials and spatial transcriptomics.

Persistent Homology for Domains at Multiple Scales (PHD-MS) is a family of mathematical frameworks and algorithmic methods for quantifying topological features of data across multiple geometric, density, or domain scales. These frameworks extend conventional persistent homology by designing filtrations and associated algebraic invariants that explicitly encode multiscale phenomena, enhance robustness to noise, and enable efficient computation and interpretation in high-dimensional, multi-resolution, and multi-domain settings. PHD-MS unifies approaches rooted in objective-oriented geometric PDEs, kernel-based density models, multiparameter and distributed cosheaf homology, spectral invariants, and multilevel parallel algorithms.

1. Objective-Oriented Differential Geometry and Geometric Flows

A core PHD-MS paradigm is the use of objective-oriented filtrations generated by minimizing geometric functionals, as exemplified by the Laplace–Beltrami flow approach. For a scalar immersion $S(x)$ defined on a computational domain $\Omega \subset \mathbb{R}^3$, the surface free-energy functional

$$E[S] = \int_\Omega \gamma \sqrt{1 + |\nabla S|^2} \, dx$$

gives rise to the (possibly nonlinear) Laplace–Beltrami operator

$$\Delta_{LB} S := \frac{1}{\sqrt{g}} \partial_i(\sqrt{g} \, g^{ij} \partial_j S), \quad g = 1 + |\nabla S|^2.$$

The mean curvature is tied to the stationarity of $E[S]$. The gradient flow

$\partial_t S = -\Delta_{LB} S$

serves as a parabolic smoothing (“scale”) parameter: as $t$ increases, $S(x,t)$ evolves towards lower surface area and mean curvature, denoising high-curvature (small-scale) features preferentially.

By thresholding $S(x,t)$ at fixed $c > 0$, one obtains nested domains

$$\Omega_{t, c} := \{ x \in \Omega \mid S(x, t) \geq c \}, \quad \Omega_{0,c} \supseteq \Omega_{t_1,c} \supseteq \dots.$$

Cubical complexes are constructed by discretizing $\Omega$ on a grid; persistent homology is then computed for the inclusions of these sublevel sets across $t$. The same methodology can be validated against Vietoris–Rips distance-based filtrations, demonstrating consistency in Betti number barcodes while evincing strong denoising and geometry preservation for large, low-curvature features such as cavities in fullerene molecules (Wang et al., 2014).

This approach has yielded quantitative geometric predictors, as seen in the near-linear correlation between the persistent Betti-2 bar length and the total curvature energy of fullerene isomers ($cc \geq 0.92$, $\sigma \approx 0.1-0.6$ eV across 10 families), representing the first use of PH in objective-oriented quantitative energetics (Wang et al., 2014).

2. Multiscale and Multiparameter Filtrations

PHD-MS generalizes to multiparameter and multiresolution settings via kernel-based density methods and multifiltration modules. Multiresolution persistent homology (MRPH) introduces a continuous scale parameter $\eta$ into the construction of a rigidity-density field

$$\rho(x; \eta) = \sum_{i=1}^N w_i \kappa_\eta(\|x - x_i\|),$$

where $\kappa_\eta$ is a localized kernel, and $\eta$ tunes geometric resolution (Xia et al., 2015). Varying $\eta$ allows PH to focus on topological features at atomic, molecular, domain, or complex scales.

Filtrations can be constructed as:

• Matrix-based: adjacency matrices $M_{ij}(\eta)$ give rise to Rips complexes with fixed thresholds;
• Density-based: sublevel sets $X_\alpha(\eta)$ form cubical complexes.

This filtration paradigm offers grid-based and two-parameter filtrations $(\eta, \epsilon)$ or $(\eta, \alpha)$. Each value of $\eta$ corresponds to a separate PH diagram, and feature bands in the $(\eta, \alpha)$- or $(\eta, \epsilon)$-plane decompose multiscale phenomena, as demonstrated on fractal, DNA/RNA, virus capsid, and protein domain data sets, with MRPH scaling efficiently to million-point datasets (Xia et al., 2015).

Multiparameter PH modules $M = \bigoplus_{u \in \mathbb{N}^r} H_*(K_u)$ over $K[x_1,\ldots,x_r]$ capture joint persistences for $r>1$ parameters (e.g., density and proximity); their Hilbert series and associated primes stratify the support in parameter space and enable calculation of generalized multiscale invariants (Harrington et al., 2017).

3. Multiparameter and Distributed Homology: Filtration Algebra and Stability

PHD-MS addresses stability with respect to both function and domain perturbations. The multidimensional matching distance on rank invariants is stable under Hausdorff, symmetric-difference, and $L^\infty$-distances:

$$D_{\text{match}}(\rho_{(D,\Phi_1),q}, \rho_{(D,\Phi_2),q}) \leq \max\{\|d_{K_1} - d_{K_2}\|_\infty, \|\phi_1 - \phi_2\|_\infty\}$$

guaranteeing robust invariants under set or density fluctuations (Frosini et al., 2010).

Distributed persistent homology implements partitioned or cosheaf-based decompositions, constructing local Vietoris–Rips complexes by subsets partitioned via auxiliary fields such as density. Mayer–Vietoris spectral sequences and associated cellular cosheaves provide a hierarchical algebraic structure to assemble global PH from local computations, facilitating detection of features localized to specific regions or densities (Yoon et al., 2020).

4. Multiscale Clustering Filtrations and Domain Discovery

PHD-MS realizes topology-informed clustering assessment using the Multiscale Clustering Filtration (MCF) and its bifiltration generalization (MCbiF). Given a sequence of partitions $\theta: [t_1, \infty) \to \Pi_X$, the MCF at scale $t$ includes all simplices fully supported inside clusters at any coarser scale. The dimension-zero PH (Betti-0) measures hierarchical consistency; gaps in persistence diagrams indicate refinements or conflicts. Nontrivial higher-dimensional cycles correspond to cluster assignment conflicts and persistences across scales (Schindler et al., 2023).

The MCbiF extends to two-parameter filtrations $(s, t)$ indexed by intervals of scales, with multiparameter Hilbert functions:

$\mathrm{HF}_k(s, t) = \dim H_k(K^{s,t})$

quantifying the topological autocorrelation of clustering structures. In dimension 0, deviation from minimal expected component counts signals refinement violations; in dimension 1, nontrivial cycles encode higher-order inconsistency in partition sequences (Schindler et al., 16 Oct 2025).

These methods enable new feature maps for downstream statistical or machine learning tasks, outperforming consensus or label-based metrics for non-hierarchical and temporally-evolving clustering data. Applications include both synthetic and real biological datasets (e.g., wild mice social groupings) (Schindler et al., 2023, Schindler et al., 16 Oct 2025).

5. Persistent Spectral Invariants and Hyperdigraph Approaches

Integral to some PHD-MS frameworks is spectral theory on chain complexes and hyperdigraphs. Persistent combinatorial Laplacians $\mathcal{L}_q^{t,s}$ provide spectral decompositions whose zero (“harmonic”) eigenvalues recover persistent Betti numbers, while non-harmonic spectra encode geometric, connectivity, and stability features—for example, the algebraic connectivity or higher-frequency shape oscillations (Wang et al., 2019).

Persistent hyperdigraph homology extends topological invariants to directed hyperrelationships, leveraging Laplacians on chain complexes of directed hyperedges. The persistent Laplacian's zero eigenspace encodes persistent homology, while non-zero eigenvalues quantify fine geometric organization or asymmetry in the underlying data, enabling characterization across directed, multi-relational domains (Chen et al., 2023).

6. Parallel and Efficient Algorithms for Large-Scale PH

PHD-MS includes algorithmic advancements for scalable computation. Parallel multi-scale boundary matrix reduction uses new dependencies ($\beta_j$-vector, local injections, and compression lemmas) to expose columns whose reductions are immediate. This enables embarrassingly parallel pivot-finding and column add operations, dramatically reducing the practical number of iterations for full barcode computation. For Vietoris–Rips complexes with $\sim 10^4$ simplices, 95% essential barcode elements can be computed in two iterations, with convergence to $<1\%$ error in about ten—representing several orders of magnitude speedup over classical sequential reduction (Mendoza-Smith et al., 2017). The framework adapts agnostically to any filtered chain complex, enabling real-time and distributed persistent homology for truly large domains.

7. Applications and Quantitative Domain Analysis

PHD-MS frameworks have been validated in diverse domains:

• Nanostructure geometry and energetics (fullerene Betti-2 bar persistence predicting curvature energies (Wang et al., 2014));
• Biomolecular structural fingerprinting and dynamics, including million-atom viral capsids and domain segmentation in proteins and nucleic acids (Xia et al., 2015, Wang et al., 2019);
• Spatial transcriptomics, where persistent cluster-graph filtrations define multiscale spatial domains outperforming single-scale clustering in segmentation accuracy metrics such as normalized mutual information and Wasserstein distance (Beamer et al., 11 Nov 2025);
• Multiresolution and multiparameter feature extraction in large-scale data, including distributed algorithms for density-based or locally-supported PH for big point clouds (Yoon et al., 2020).

Table: Key PHD-MS Methodological Paradigms

Approach	Filtration Principle	Domain/Application
Laplace–Beltrami flow PH	Differential-geometric, mean curvature	Nanomaterials, biopolymers
Multiresolution PH (MRPH)	Kernel density, resolution parameter	High-throughput structure
Multiscale Clustering Filtration (MCF)	Simplicial, cluster cover across scales	Network clustering, bio-data
Persistent Combinatorial Laplacians	Spectral, chain complexes	Structure prediction, graphs
Multiparameter/Distributed Cosheaf PH	Scalar fields, patch covers	Localized feature detection
Parallel boundary reduction	Pivot and beta-vector dependencies	Efficient large-scale PH

• “Objective-oriented Persistent Homology” (Wang et al., 2014)
• “Persistent hyperdigraph homology and persistent hyperdigraph Laplacians” (Chen et al., 2023)
• “Multiresolution topological simplification” (Xia et al., 2015)
• “Stratifying multiparameter persistent homology” (Harrington et al., 2017)
• “Stability of multidimensional persistent homology with respect to domain perturbations” (Frosini et al., 2010)
• “PHD-MS: Multiscale Domain Identification for Spatial Transcriptomics via Persistent Homology” (Beamer et al., 11 Nov 2025)
• “MCbiF: Measuring Topological Autocorrelation in Multiscale Clusterings via 2-Parameter Persistent Homology” (Schindler et al., 16 Oct 2025)
• “Analysing Multiscale Clusterings with Persistent Homology” (Schindler et al., 2023)
• “Persistent spectral graph” (Wang et al., 2019)
• “Persistence by Parts: Multiscale Feature Detection via Distributed Persistent Homology” (Yoon et al., 2020)
• “Parallel multi-scale reduction of persistent homology filtrations” (Mendoza-Smith et al., 2017)