Persistent Laplacian-Based ML

• Persistent Laplacian-based Machine Learning is a framework that builds filtered topological complexes and utilizes harmonic and non-harmonic spectra as feature descriptors.
• It leverages Laplacian operators to capture persistent homological information alongside multiscale geometric, directional, and shape attributes for robust learning.
• PLML workflows integrate diverse operator families on complexes, enabling advanced feature construction for applications such as shape analysis, protein interactions, and network analysis.

Persistent Laplacian-based Machine Learning (PLML) denotes a class of methods that build a filtered topological complex from data, construct persistent Laplacians or related persistent topological Laplacians on that filtration, and use their harmonic and non-harmonic spectra as features in a machine learning model (Zia et al., 2024). Across this literature, the harmonic spectrum recovers persistent homological information through kernel multiplicities, whereas the positive spectrum encodes additional multiscale geometric, shape, or directional structure that persistent homology alone does not record (Chen et al., 2022). The term now covers operator families on simplicial complexes, directed flag complexes, path complexes, sheaves, local link complexes, manifolds in Eulerian representation, and $N$-chain complexes, together with downstream pipelines for clustering, regression, classification, and representation learning (Liu et al., 8 Mar 2026, Su et al., 2024).

1. Historical formation and scope

An early precursor to PLML in shape analysis used Laplacian eigenfunctions as scalar fields on triangulated 2-manifolds, built lower-star filtrations from those fields, computed persistence diagrams, and used diagram distances for shape similarity and learning tasks (Zhang et al., 2019). In that setting, the first non-trivial Laplacian eigenfunction, the Fiedler vector, served as an intrinsic scalar descriptor whose sublevel-set topology was summarized by persistent homology. Although this pipeline did not yet operate with the later operator-centric persistent Laplacian formalism, it established the now-standard PLML pattern in which spectral geometry generates a multiscale topological representation.

The subsequent operator-theoretic literature made the Laplacian itself persistent. In biomolecular modeling, persistent Laplacian was introduced as a refinement of persistent homology whose harmonic spectrum recovers Betti information while the non-harmonic spectrum captures shape and robustness; this perspective was then integrated with gradient boosting trees and deep neural networks for mutation-induced protein-protein binding free-energy prediction and variant infectivity forecasting (Chen et al., 2022). Related SARS-CoV-2 studies emphasized the broader umbrella of persistent topological Laplacians, including persistent Laplacian and persistent sheaf Laplacian, to analyze mutation-induced structural changes and binding-induced conformational effects that persistent Betti curves alone did not separate cleanly (Wei et al., 2023).

A defining characteristic of PLML is therefore not merely the use of topology in learning, but the replacement of barcode-only representations by Laplacian operators whose kernels encode topology and whose positive spectra encode additional geometry, chemistry, or directionality. This distinction is repeated in directed, sheaf-theoretic, local, and manifold formulations, where the same harmonic/non-harmonic split is retained while the underlying category of complexes is generalized (Zia et al., 2024).

2. Core mathematical constructions

For a filtration $K^a \subseteq K^b$ of simplicial complexes, the standard persistent topological Laplacian in degree $n$ is defined on $C_n^a$ by

$$\Delta_n^{a,b} = d_{n+1}^{a,b}\big(d_{n+1}^{a,b}\big)^* + (d_n^a)^* d_n^a,$$

where $d_{n+1}^{a,b}$ is the persistent boundary obtained by restricting $d_{n+1}^b$ to $(n+1)$-chains whose boundary lands in $C_n^a$ (Jones et al., 15 Aug 2025). This operator is self-adjoint, positive semidefinite, and satisfies

$\ker \Delta_n^{a,b} \cong H_n^{a,b}(K;\mathbb{R}),$

so the multiplicity of the zero eigenvalue equals the persistent Betti number. The same framework yields a persistent Hodge decomposition in which $K^a \subseteq K^b$0 splits into an image of $K^a \subseteq K^b$1, a harmonic part, and an image of $K^a \subseteq K^b$2 (Jones et al., 15 Aug 2025).

This operator viewpoint separates PLML from barcode-only pipelines. Persistent homology records when homological classes are born and die; persistent Laplacians additionally expose the full spectrum of $K^a \subseteq K^b$3. The zero eigenvalues are the harmonic spectrum, while the positive eigenvalues are the non-harmonic spectrum. In the literature, the latter are treated as quantitative descriptors of shape, connectivity, rigidity, diffusion-like structure, or directed interaction patterns, depending on the underlying complex (Chen et al., 2022).

The inclusion-based definition has also been generalized to arbitrary weight-preserving simplicial maps $K^a \subseteq K^b$4. In that setting, the persistent Laplacian acts on $K^a \subseteq K^b$5 and is obtained through a Schur-complement construction that canonically restricts ordinary Laplacians to persistence-relevant subspaces (Gülen et al., 2023). This extension is important for coarsenings, sparsified representations, and towers of complexes that are related by simplicial maps rather than literal inclusions. Its nullity again recovers the persistent Betti number of the map, and the paper establishes eigenvalue monotonicity results along simplicial towers (Gülen et al., 2023).

3. Feature construction and learning workflow

PLML does not prescribe a single feature map. The common pattern is to construct a filtered complex, compute a persistent Laplacian or a related persistent topological Laplacian, and then transform harmonic and non-harmonic spectral information into descriptors suitable for unsupervised or supervised learning.

One family of pipelines remains diagram-centric. In mesh learning, a Laplacian eigenfunction on a triangulated manifold induces a lower-star filtration, and the resulting 0-dimensional persistence diagram is used directly as the shape descriptor; similarities are computed by bottleneck or Wasserstein distances, and the resulting pairwise distance matrix feeds an unsupervised embedding method such as t-SNE (Zhang et al., 2019). This is a spectral-topological descriptor in which Laplacian information is injected before persistence rather than through a persistent Laplacian operator.

Operator-centric PLML typically vectorizes spectra. In the Persistent Directed Flag Laplacian (PDFL) pipeline for protein-ligand binding, filtered directed flag complexes are built from directed, weighted bipartite atom-interaction graphs, and for each filtration interval the nonzero eigenvalues of $K^a \subseteq K^b$6 are summarized by 10 spectral statistics: minimum nonzero eigenvalue, maximum eigenvalue, sum of positive eigenvalues, mean of positive eigenvalues, median, variance, standard deviation, count of positive eigenvalues, sum of squares, and count of eigenvalues nearly equal to zero (Zia et al., 2024). These statistics are computed across 36 element-specific graph channels, five filtration intervals, and Betti-0 and Betti-1 regimes, yielding 3600 features per complex for a single-kernel model and larger vectors for multi-kernel variants (Zia et al., 2024).

In virus-like particle stoichiometry prediction, harmonic and non-harmonic components are explicitly separated. For each of six atom sets, the 0-dimensional Vietoris–Rips persistent Laplacian contributes 20 harmonic barcode-bin features and 140 non-harmonic spectral statistics, while 1-dimensional and 2-dimensional Alpha-complex harmonic summaries add 14 more features. The resulting descriptor has $K^a \subseteq K^b$7 features per sample and is used with gradient boosting trees (Liu et al., 29 Jul 2025).

Other implementations are deliberately compact. Persistent sheaf Laplacian analysis of protein flexibility computes, at radii $K^a \subseteq K^b$8, $K^a \subseteq K^b$9, and $n$0 Å, the number of zero eigenvalues and the maximum, minimum, mean, and standard deviation of nonzero eigenvalues, producing 15 PSL features that are concatenated with global and local protein descriptors for blind B-factor prediction (Hayes et al., 12 Feb 2025). Persistent Hodge Laplacian learning on manifolds and volumetric data similarly uses spectra of Eulerian-grid Hodge or BIG Laplacians as the learning representation, avoiding remeshing-induced inconsistency across scales (Su et al., 2024).

These examples show that PLML spans diagram distances, raw eigenvalue statistics, harmonic/non-harmonic separations, multi-kernel concatenations, and mixed topological-spectral feature blocks. The unifying element is the use of persistent Laplacian spectra as the primary representation.

4. Major operator families and generalizations

The PLML literature has expanded from persistent Laplacians on simplicial filtrations to a broader family of persistent topological Laplacians.

Variant	Underlying structure	Distinctive property
Persistent Directed Flag Laplacian	Filtered directed flag complexes	Incorporates directionality in higher-order simplices and spectra (Zia et al., 2024)
Persistent Path Laplacian	Path complexes of digraphs	Harmonic spectra recover persistent path homology; non-harmonic spectra reveal homotopic shape evolution (Wang et al., 2022)
Persistent Sheaf Laplacian	Cellular sheaves on complexes or graphs	Encodes heterogeneous local data through restriction maps (Hayes et al., 12 Feb 2025)
Persistent Local Laplacian	Relative/local complexes or link filtrations	Harmonic space is isomorphic to persistent local homology; computations decouple by vertex (Liu et al., 8 Mar 2026)
Persistent Mayer Laplacian	$n$1-chain complexes with $n$2	Introduces multi-$n$3 spectral channels beyond the $n$4 case (Shen et al., 2023)
Persistent de Rham-Hodge Laplacian	Manifolds or volumetric data on Cartesian grids	Eulerian formulation avoids remeshing inconsistency (Su et al., 2024)

Directed variants address a recurrent limitation of undirected persistent homology and persistent Laplacian. PDFL defines persistent Laplacians on filtered directed flag complexes generated from digraphs whose edge directions may represent electronegativity-driven polarization or other asymmetric interactions; in this setting, simplices are ordered tuples and the resulting spectra encode direction-dependent connectivity (Zia et al., 2024). Persistent Path Laplacian achieves an analogous goal through path homology, where chain groups are built from allowed directed paths rather than simplices; its harmonic spectrum recovers persistent path homology and its non-harmonic spectrum records shape evolution in directed networks (Wang et al., 2022).

Sheaf and local generalizations extend the state space rather than only the filtration. Persistent Sheaf Laplacian replaces scalar simplex coefficients with stalks and restriction maps, enabling charge- or attribute-aware Laplacians on graphs and complexes (Hayes et al., 12 Feb 2025). Persistent local Laplacian uses relative chain complexes around a vertex and proves a unitary equivalence

$n$5

so persistent local spectra can be computed through link filtrations with a dimension shift (Liu et al., 8 Mar 2026). This local-link equivalence underlies the method’s decoupled and parallelizable structure.

Two further extensions enlarge PLML’s algebraic scope. Persistent Mayer homology and persistent Mayer Laplacians replace the usual condition $n$6 by $n$7, yielding $n$8 homological and spectral channels indexed by $n$9 and thereby enriching feature spaces beyond the ordinary Hodge setting (Shen et al., 2023). Meanwhile, the simplicial-map generalization of persistent Laplacian replaces inclusions by arbitrary weight-preserving simplicial maps and uses Schur complements as functorial restrictions of positive semidefinite operators to persistence-relevant subspaces (Gülen et al., 2023). A plausible implication is that PLML can be made intrinsic not only to filtrations but also to coarsening, alignment, and multi-resolution correspondences between complexes.

5. Applications and empirical record

In shape analysis, the mesh-learning pipeline based on Laplacian eigenfunctions and persistence diagrams was evaluated on 60 triangulated meshes divided into six semantic categories—cat, elephant, face, head, horse, and lion, with 10 meshes per category. A pairwise bottleneck-distance matrix built from 0-dimensional persistence diagrams of the Fiedler-vector lower-star filtration produced a t-SNE embedding with distinct clusters matching the six categories, and the authors reported better classification power than the metric-based descriptor of Chazal et al. (2009) (Zhang et al., 2019).

In protein-ligand binding, PDFL instantiated a fully directed PLML pipeline. On PDBbind v2007, v2013, and v2016, the consensus PDFL-plus-Transformer model achieved $C_n^a$0 with RMSE $C_n^a$1, $C_n^a$2 with RMSE $C_n^a$3, and $C_n^a$4 with RMSE $C_n^a$5, respectively. The same study reported that two-kernel models generally outperformed single-kernel models and that Lorentz-containing models tended to exceed exponential-only models (Zia et al., 2024).

In protein-protein interaction learning and SARS-CoV-2 forecasting, persistent Laplacian-based deep learning was used to model mutation-induced binding free-energy changes and relative infectivity. One study predicted that Omicron BA.4 and BA.5 were about $C_n^a$6 more infectious than BA.2 and projected them to become new dominating variants, while also reporting strong performance on three major benchmark datasets for mutation-induced protein-protein binding free-energy changes (Chen et al., 2022). A related persistent topological Laplacian analysis compared experimental and computational BA.2 structures in PTL-based learning and reported Pearson correlations of $C_n^a$7 versus $C_n^a$8 for TopLapGBT and $C_n^a$9 versus $$\Delta_n^{a,b} = d_{n+1}^{a,b}\big(d_{n+1}^{a,b}\big)^* + (d_n^a)^* d_n^a,$$0 for TopLapNet, showing that PTL-based models remained effective even when the structure was computationally generated (Wei et al., 2023).

In virus-like particle learning, a PLML model trained on the VLP200 dataset achieved AUC $$\Delta_n^{a,b} = d_{n+1}^{a,b}\big(d_{n+1}^{a,b}\big)^* + (d_n^a)^* d_n^a,$$1, sensitivity $$\Delta_n^{a,b} = d_{n+1}^{a,b}\big(d_{n+1}^{a,b}\big)^* + (d_n^a)^* d_n^a,$$2, specificity $$\Delta_n^{a,b} = d_{n+1}^{a,b}\big(d_{n+1}^{a,b}\big)^* + (d_n^a)^* d_n^a,$$3, precision $$\Delta_n^{a,b} = d_{n+1}^{a,b}\big(d_{n+1}^{a,b}\big)^* + (d_n^a)^* d_n^a,$$4, and NPV $$\Delta_n^{a,b} = d_{n+1}^{a,b}\big(d_{n+1}^{a,b}\big)^* + (d_n^a)^* d_n^a,$$5. On the larger VLP706 dataset, binary AUC values were $$\Delta_n^{a,b} = d_{n+1}^{a,b}\big(d_{n+1}^{a,b}\big)^* + (d_n^a)^* d_n^a,$$6 for 60 versus 180, $$\Delta_n^{a,b} = d_{n+1}^{a,b}\big(d_{n+1}^{a,b}\big)^* + (d_n^a)^* d_n^a,$$7 for 60 versus 240, and $$\Delta_n^{a,b} = d_{n+1}^{a,b}\big(d_{n+1}^{a,b}\big)^* + (d_n^a)^* d_n^a,$$8 for 180 versus 240; the 4-class task over 60, 180, 240, and 420 achieved AUC $$\Delta_n^{a,b} = d_{n+1}^{a,b}\big(d_{n+1}^{a,b}\big)^* + (d_n^a)^* d_n^a,$$9 and accuracy $d_{n+1}^{a,b}$0. A perturbative sequence analysis further suggested the robustness ordering $d_{n+1}^{a,b}$1 (Liu et al., 29 Jul 2025).

In protein flexibility prediction, persistent sheaf Laplacian analysis reported an increase in accuracy of $d_{n+1}^{a,b}$2 compared to the classical Gaussian network model on a dataset of 364 proteins, and also proposed a blind machine-learning predictor that combines PSL spectral summaries with global and local protein features (Hayes et al., 12 Feb 2025). Persistent Hodge Laplacian learning on manifolds and volumetric data was demonstrated as a proof-of-principle method for protein-ligand binding affinity prediction on two benchmark datasets, indicating that PLML can operate directly on manifold representations rather than only on point-cloud or graph filtrations (Su et al., 2024).

6. Robustness, software, and open directions

A decisive recent development is eigenvalue-level robustness theory. For the up-persistent Laplacian, adding one simplex $d_{n+1}^{a,b}$3 to a complex changes every eigenvalue by at most

$d_{n+1}^{a,b}$4

uniformly in filtration scale and independently of complex size (Anh et al., 26 Jun 2025). The same work notes that this bound directly propagates to heat-kernel signatures, diffusion descriptors, spectral neural-network layers, and kernel methods that are Lipschitz in the spectrum. In the context of PLML, this is the first explicit eigenvalue-level robustness guarantee under local combinatorial updates.

Computational infrastructure has also matured. PETLS is an efficient and flexible C++ library with Python bindings for persistent topological Laplacians; it interfaces with simplicial, alpha, directed flag, Dowker, and cellular Sheaf complexes, implements existing and new algorithms for persistent Laplacians, and provides recommendations on algorithm selection and complex choice for data analysis in machine learning (Jones et al., 15 Aug 2025). Its default Schur-complement machinery is particularly aligned with the operator-theoretic view developed for simplicial maps, where Schur complements act as canonical restrictions of positive semidefinite operators to persistence-relevant subspaces (Gülen et al., 2023).

Scalability is being addressed structurally as well as numerically. Persistent local Laplacian introduces a block-decoupled architecture in which each vertex produces an independent local problem, making the construction inherently suited to massive parallelization and large-scale network analysis (Liu et al., 8 Mar 2026). Persistent de Rham-Hodge Laplacians on Cartesian grids pursue a complementary strategy: fixed-grid Eulerian discretization avoids remeshing inconsistency and supports learning directly on manifolds or volumetric data (Su et al., 2024).

Several limitations remain explicit in the literature. Directed flag complexes can become costly because the number of directed cliques grows quickly with graph density, parameter selection in multi-kernel pipelines still requires empirical exploration, and some feature spaces become very large—for example, the four-kernel PDFL representation retains about $d_{n+1}^{a,b}$5 non-zero features after pruning always-zero coordinates (Zia et al., 2024). Many biomolecular PLML systems also continue to treat persistent Laplacian spectra as precomputed features rather than end-to-end differentiable layers, so the Laplacian stage is not yet integrated into backpropagation through the underlying geometry (Chen et al., 2022).

The published agenda is correspondingly broad. Explicit directions include multi-eigenfunction mesh descriptors, vectorized persistent spectral features for classical ML and deep neural networks, directed filtrations for asymmetric systems, local/link-based parallel formulations, generalized morphism Laplacians, and $d_{n+1}^{a,b}$6-chain Mayer extensions with multi-$d_{n+1}^{a,b}$7 channels (Zhang et al., 2019, Shen et al., 2023, Liu et al., 8 Mar 2026). Taken together, these developments indicate that PLML is evolving from a single spectral-topological trick into a general methodology for learning on multiscale structures whose topology, geometry, asymmetry, and attributes must be represented simultaneously.