Persistent Topological Laplacians

• Persistent topological Laplacians are Laplace-type operators defined on filtered complexes that recover persistent homology while encoding multiscale geometric features.
• They extend classical Hodge Laplacians by incorporating non-harmonic spectra that reveal diffusion, stiffness, and connectivity beyond topological barcodes.
• Recent advances integrate algebraic stability, perturbation theory, and efficient computational algorithms, enabling applications in data analysis, protein engineering, and machine learning.

Searching arXiv for recent and foundational papers on persistent topological Laplacians. Persistent topological Laplacians, also called persistent Laplacians, are Laplace-type operators defined on filtered topological objects so that both topological information and geometric information are tracked across scales. In the simplicial setting, they extend the combinatorial Hodge Laplacian from a single simplicial complex to a filtration or to a pair of complexes $K\subseteq L$, with the multiplicity of the zero eigenvalue recovering persistent Betti numbers and the nonzero spectrum encoding additional multiscale geometric information (Jones et al., 15 Aug 2025). More generally, recent work places them in a broad operator-theoretic framework of Hilbert complexes, encompassing discrete combinatorial settings, cellular and cosheaf constructions, and de Rham complexes of smooth manifolds (Wolf et al., 24 Sep 2025). Across this literature, persistent topological Laplacians function as spectral refinements of persistent homology: they retain the harmonic information corresponding to persistence while adding non-harmonic spectral data that reflect diffusion, stiffness, connectivity, and homotopic shape evolution (Liu et al., 2023).

1. Definitions and mathematical setting

A standard starting point is a finite simplicial complex $K$ on a vertex set $V$, together with a filtration

$$\{K(r)\}_{r\ge 0},\qquad K(r)\subseteq K(s)\text{ for }r\le s,$$

with finitely many strict inclusions. Vietoris–Rips and Čech filtrations from point clouds are standard examples (Anh et al., 26 Jun 2025). On each $K(r)$, one forms chain groups over $\mathbb R$,

$C_k(K(r)) := \mathbb R^{\{\text{%%%%0%%%%-simplices of }K(r)\}},$

and boundary operators

$\partial_k : C_k(K(r))\to C_{k-1}(K(r)),$

given on an oriented $k$-simplex by

$$\partial_k [v_0\cdots v_k] = \sum_{i=0}^k (-1)^i [v_0\cdots \widehat{v_i}\cdots v_k].$$

For a single complex, the combinatorial $K$0-Hodge Laplacian is

$K$1

with the up and down parts

$K$2

(Anh et al., 26 Jun 2025).

One widely used persistent construction for a filtration pair $K$3 defines

$K$4

restricts the boundary by

$K$5

and sets

$K$6

(Jones et al., 15 Aug 2025). In this formulation, $K$7, so the multiplicity of the zero eigenvalue equals the persistent Betti number $K$8 (Jones et al., 15 Aug 2025).

Another influential formulation emphasizes a morphism of chain complexes $K$9 and defines a Laplacian associated to the morphism,

$V$0

where $V$1 is inclusion. For a persistence differential graded inner product space $V$2, the $V$3-persistent Laplacian is

$V$4

and its kernel is naturally isomorphic to the $V$5-persistent harmonic space and hence to $V$6-persistent homology (Liu et al., 2023).

A more recent operator-theoretic generalization begins with a Hilbert complex $V$7 and defines up-, down-, and full chain Laplacians by quadratic forms. For an inclusion $V$8, it introduces a persistent chain space

$V$9

a persistent differential

$$\{K(r)\}_{r\ge 0},\qquad K(r)\subseteq K(s)\text{ for }r\le s,$$0

and persistent Laplacians

$$\{K(r)\}_{r\ge 0},\qquad K(r)\subseteq K(s)\text{ for }r\le s,$$1

together with a full persistent Laplacian induced by the quadratic form

$$\{K(r)\}_{r\ge 0},\qquad K(r)\subseteq K(s)\text{ for }r\le s,$$2

(Wolf et al., 24 Sep 2025). This framework covers finite simplicial complexes, cellular or cosheaf chain complexes, and de Rham complexes of manifolds (Wolf et al., 24 Sep 2025).

2. Hodge-theoretic meaning and relation to persistence

Persistent topological Laplacians are designed so that their harmonic sector recovers persistence. In the simplicial pair construction, the persistent Hodge decomposition takes the form

$$\{K(r)\}_{r\ge 0},\qquad K(r)\subseteq K(s)\text{ for }r\le s,$$3

with

$$\{K(r)\}_{r\ge 0},\qquad K(r)\subseteq K(s)\text{ for }r\le s,$$4

(Jones et al., 15 Aug 2025). In the categorical formulation, there is a natural isomorphism between the harmonic-space functor and the homology functor, and the persistent Hodge decomposition identifies the kernel of $$\{K(r)\}_{r\ge 0},\qquad K(r)\subseteq K(s)\text{ for }r\le s,$$5 with persistent homology (Liu et al., 2023). In the operator-theoretic framework, a generalized Hodge theorem gives

$$\{K(r)\}_{r\ge 0},\qquad K(r)\subseteq K(s)\text{ for }r\le s,$$6

for a Hilbert complex, and under a closed-range hypothesis one obtains

$$\{K(r)\}_{r\ge 0},\qquad K(r)\subseteq K(s)\text{ for }r\le s,$$7

for persistent Laplacians (Wolf et al., 24 Sep 2025).

This spectral viewpoint refines classical persistent homology. Persistent homology tracks the birth and death of homology classes across a filtration and is summarized by barcodes or persistence diagrams (Jones et al., 15 Aug 2025). Persistent topological Laplacians recover that information through the multiplicity of zero eigenvalues, but their positive eigenvalues and eigenvectors encode additional geometric and combinatorial structure, including diffusion scales, higher-order connectivity, stiffness, and multiscale geometric information beyond persistent homology (Jones et al., 15 Aug 2025). The literature repeatedly describes this distinction as the difference between purely topological persistence and a topological–spectral description that also captures homotopic shape evolution (Wang et al., 2019, Wei et al., 2023).

A recurring misconception is that persistent Laplacians merely repackage barcodes. The theory does not support that reduction. The harmonic spectra reproduce persistent Betti numbers, but the non-harmonic spectra contain information not present in barcodes, including distinctions between geometrically different but homotopy-equivalent structures and quantitative sensitivity to local structural change (Wang et al., 2019, Wei et al., 2023). Another misconception is that the “full” persistent Laplacian is always the most natural object. Recent generalized theory shows that the classical full persistent Laplacian can fail to satisfy monotonicity and stability, whereas the up- and down-persistent Laplacians satisfy these properties individually, and their nonzero spectra fully determine the nonzero spectrum of the full Laplacian (Wolf et al., 24 Sep 2025).

3. Main constructions and variants

The literature now contains a family of persistent topological Laplacians tailored to different data modalities.

The foundational simplicial constructions include persistent combinatorial Laplacians and persistent spectral graph formulations on Vietoris–Rips or related filtrations of point clouds (Wang et al., 2019, Mémoli et al., 2020). In these models, a filtration produces a family of simplicial complexes and associated Laplacian matrices whose harmonic spectra recover persistent homology and whose non-harmonic spectra are used for data analysis, modeling, and prediction (Wang et al., 2019).

Persistent sheaf Laplacians replace ordinary chain or cochain complexes by sheaf cochain complexes, allowing local labels or physical quantities to be fused into the operator through stalks and restriction maps (Wei et al., 2021). In the point-cloud constructions described there, the spectra of persistent sheaf Laplacians encode both geometrical and non-geometrical information, and the theory is presented as a way to fuse different types of data (Wei et al., 2021). Persistent interaction Laplacians further localize the construction to specified interacting subsystems by replacing ordinary chains with interaction chains built from tuples of simplices with non-empty intersection (Liu et al., 2024).

Directed and asymmetric data motivate additional variants. Persistent path Laplacians are built from path homology on digraphs, with harmonic spectra recovering persistent path homology and non-harmonic spectra revealing the homotopic shape evolution of data during filtration (Wang et al., 2022). Persistent directed flag Laplacians extend the framework to directed flag complexes, where simplices correspond to directed cliques, yielding a persistent directed flag Laplacian suited to directed networks and asymmetric data (Jones et al., 2023). The algebraic-stability framework of Laplacian trees also applies to real-valued functions on digraphs via path homology (Liu et al., 2023).

For data on manifolds or volumetric domains, persistent de Rham–Hodge Laplacians, or persistent Hodge Laplacians, are defined in an Eulerian representation using structure-preserving Cartesian grids (Su et al., 2024). This approach avoids the numerical inconsistency caused by remeshing the involving manifolds in the Lagrangian representation and yields discrete Laplacians whose kernels approximate topological invariants while nonzero spectra track geometric evolution (Su et al., 2024).

A concise summary of major variants is useful:

Variant	Underlying object	Distinguishing feature
Persistent Laplacian / persistent combinatorial Laplacian	Filtered simplicial complexes	Kernel recovers persistent homology; nonzero spectrum captures multiscale geometry (Mémoli et al., 2020)
Persistent sheaf Laplacian	Cellular sheaves on filtered complexes	Encodes geometrical and non-geometrical information (Wei et al., 2021)
Persistent path Laplacian	Path complexes of digraphs	Harmonic spectra recover persistent path homology (Wang et al., 2022)
Persistent directed flag Laplacian	Directed flag complexes	Distinct analysis of directed higher-order structure (Jones et al., 2023)
Persistent interaction Laplacian	Interaction chain complexes	Emphasizes specified interacting elements or subsystems (Liu et al., 2024)
Persistent de Rham–Hodge Laplacian	Filtrations of manifolds	Eulerian, grid-based manifold topological learning (Su et al., 2024)

This range of constructions suggests that “persistent topological Laplacian” is best understood as an umbrella term for persistent Laplacian-type operators on filtered chain or cochain complexes, rather than a single matrix formula. A plausible implication is that the common structure is Hodge-theoretic rather than simplicial: persistence is encoded by restricting differentials across scales, and harmonic spaces recover persistent (co)homology.

4. Spectral properties, stability, and perturbation theory

A central line of work studies the robustness of persistent Laplacian spectra. Algebraic stability for persistent Laplacians was established by organizing all Laplacians persisting from a parameter into “Laplacian trees” and proving that the interleaving distance of persistence Laplacian trees equals the interleaving distance of the underlying persistence differential graded inner product spaces: $$\{K(r)\}_{r\ge 0},\qquad K(r)\subseteq K(s)\text{ for }r\le s,$$8 (Liu et al., 2023). For non-decreasing real-valued functions $$\{K(r)\}_{r\ge 0},\qquad K(r)\subseteq K(s)\text{ for }r\le s,$$9 on a fixed simplicial complex, this yields

$K(r)$0

(Liu et al., 2023). Similar bounds are given for weighted digraphs (Liu et al., 2023).

A different and more local notion of robustness appears in the recent one-simplex perturbation theorem for up-persistent Laplacians. If $K(r)$1, where $K(r)$2 is a single $K(r)$3-simplex, then for every radius $K(r)$4 and every index $K(r)$5,

$K(r)$6

This bound is uniform in the filtration scale and independent of the size of the complex (Anh et al., 26 Jun 2025). The paper describes the result as the first eigenvalue-level robustness guarantee for spectral topological data analysis and notes that the constant $K(r)$7 is optimal for rank-one Hermitian perturbations (Anh et al., 26 Jun 2025). The proof exploits a rank-one decomposition

$K(r)$8

with $K(r)$9, and combines matrix interlacing with Weyl’s inequality (Anh et al., 26 Jun 2025).

Generalized persistent Laplacian theory sharpens the discussion of spectral behavior. In that framework, the up- and down-persistent Laplacians satisfy monotonicity and stability individually, while the full persistent Laplacian may fail both properties (Wolf et al., 24 Sep 2025). The up- and down-persistent spectral counting functions are 1-Lipschitz with respect to the filtration interleaving distance: $\mathbb R$0 (Wolf et al., 24 Sep 2025). At the same time, the nonzero spectrum of the full Laplacian is the multiset union of the nonzero spectra of the up and down parts, both in finite-dimensional and infinite-dimensional settings (Wolf et al., 24 Sep 2025). This suggests that analyzing up and down components separately is not merely technically convenient but spectrally complete.

The generalized theory also clarifies a controversy sometimes implicit in earlier work: whether one should privilege the full persistent Laplacian because it resembles the ordinary Hodge Laplacian. The newer results indicate that full persistent Laplacians can fail monotonicity and stability, whereas up- and down-persistent Laplacians are individually preferable and sufficient for spectral analysis (Wolf et al., 24 Sep 2025). This does not invalidate earlier full-Laplacian constructions; it changes the theoretical emphasis.

5. Algorithms, software, and computational practice

Computing persistent topological Laplacians is dominated by constructing the persistent up-part. PETLS, a C++ library with Python bindings titled “PErsistent Topological Laplacian Software,” implements existing and new algorithms for these computations and interfaces with simplicial, alpha, directed flag, Dowker, and cellular Sheaf complexes (Jones et al., 15 Aug 2025). Its default pipeline stores a global boundary matrix $\mathbb R$1 for the final filtration level, realizes $\mathbb R$2 as submatrices, computes down-Laplacians as

$\mathbb R$3

and constructs $\mathbb R$4 using a chosen algorithm, with the Schur complement method as the default (Jones et al., 15 Aug 2025).

The Schur complement method begins with the full up-Laplacian at scale $\mathbb R$5,

$\mathbb R$6

block-partitions it as

$\mathbb R$7

and then computes the persistent up-Laplacian by the Schur complement

$\mathbb R$8

(Jones et al., 15 Aug 2025). Other reviewed approaches include a gauge or projection method, a reduction plus explicit restriction algorithm, and a specialized algorithm for non-branching complexes (Jones et al., 15 Aug 2025).

PETLS also introduces several implementation ideas motivated by spectral computation. In the top dimension, it can use a “flipped” Laplacian principle: since the nonzero eigenvalues of $\mathbb R$9 and $C_k(K(r)) := \mathbb R^{\{\text{%%%%0%%%%-simplices of }K(r)\}},$0 coincide, PETLS can compute the nonzero spectrum of a large top-dimensional down-Laplacian by diagonalizing a smaller up-Laplacian, which can give $C_k(K(r)) := \mathbb R^{\{\text{%%%%0%%%%-simplices of }K(r)\}},$1 speedups on the top-dimensional spectral computation (Jones et al., 15 Aug 2025). It also supports reduction to the orthogonal complement of the kernel, using a basis of persistent homology representatives to compress $C_k(K(r)) := \mathbb R^{\{\text{%%%%0%%%%-simplices of }K(r)\}},$2 to a smaller positive definite block (Jones et al., 15 Aug 2025).

The software literature emphasizes that PTL eigenvalue problems are atypical: they often have high multiplicities of zero, very sparse but structured matrices, and problem sizes where both dense and sparse solvers are plausible (Jones et al., 15 Aug 2025). PETLS therefore provides multiple backends, including Eigen’s SelfAdjointEigenSolver, the Spectra library, and SciPy/LAPACK via Python bindings (Jones et al., 15 Aug 2025). This suggests that algorithm selection is not secondary. A plausible implication is that the practical behavior of persistent Laplacians depends as much on spectral linear algebra choices as on the topological construction.

6. Applications, feature engineering, and current directions

Persistent topological Laplacians have been used in biology, physics, and machine learning (Anh et al., 26 Jun 2025). In protein and molecular applications, persistent Laplacian features have been used for SARS-CoV-2 variant analysis, protein engineering, protein–ligand binding, and virus-like particle stoichiometry (Wei et al., 2023, Qiu et al., 2023, Liu et al., 29 Jul 2025). In one SARS-CoV-2 study, persistent topological Laplacians were used to examine mutation-induced structural changes of receptor-binding domain complexes and to analyze binding-induced structural changes across variants, with the non-harmonic spectra detecting structural differences that persistent Betti numbers alone could not (Wei et al., 2023). In protein engineering, persistent Laplacians are described as making structure-based embeddings a superb option in AI-assisted directed evolution and AI-assisted protein engineering, alongside persistent path, sheaf, hypergraph, and hyperdigraph Laplacians (Qiu et al., 2023).

A large-scale recent application to virus-like particles constructs Vietoris–Rips and Alpha filtrations on element-specific atom sets from VLP asymmetric units, computes 0D persistent Laplacians and 1D/2D harmonic information, and assembles a 1044-dimensional feature vector from harmonic and non-harmonic summaries (Liu et al., 29 Jul 2025). That work reports performance on VLP200 and VLP706 datasets and attributes the predictive signal to the joint use of harmonic and non-harmonic spectra (Liu et al., 29 Jul 2025).

Machine-learning pipelines based on full spectra face a “varying length” problem and high dimensionality. A recent alternative distills persistent Laplacians into three mathematically grounded invariants: Betti numbers, the spectral gap, and analytic torsion (Grlj et al., 15 Jun 2026). For a persistent Laplacian $C_k(K(r)) := \mathbb R^{\{\text{%%%%0%%%%-simplices of }K(r)\}},$3 with persistent Betti number $C_k(K(r)) := \mathbb R^{\{\text{%%%%0%%%%-simplices of }K(r)\}},$4, the spectral gap is

$C_k(K(r)) := \mathbb R^{\{\text{%%%%0%%%%-simplices of }K(r)\}},$5

and the torsion-style summary uses

$C_k(K(r)) := \mathbb R^{\{\text{%%%%0%%%%-simplices of }K(r)\}},$6

(Grlj et al., 15 Jun 2026). On MNIST, QM-3D, and SKEMPI WT, this reduced feature space is reported to capture the essential predictive signal of the full spectrum, while reducing training time and memory (Grlj et al., 15 Jun 2026). This suggests that persistent Laplacian feature engineering may be moving toward principled low-dimensional spectral summaries rather than raw eigenvalue lists.

The broader software and theory landscape indicates several open directions. Weighted complexes, multiple simplex insertions, down-persistent versions in full detail, map-induced perturbations, and structure-aware norms are all identified as natural extensions of current perturbation theory (Anh et al., 26 Jun 2025). PETLS points toward integrated algorithms that combine persistent homology computation with PTL kernel reduction, scalable eigensolvers tailored to large nullspaces, and support for higher-dimensional sheaf stalks (Jones et al., 15 Aug 2025). Generalized persistent Laplacian theory suggests a shift in emphasis from full Laplacians to up- and down-components as the robust and sufficient spectral invariants (Wolf et al., 24 Sep 2025).

Taken together, the literature portrays persistent topological Laplacians as a family of Hodge-theoretic operators for filtrations, with kernels that recover persistence and positive spectra that encode multiscale geometry. Their development now spans simplicial complexes, digraphs, sheaves, interaction complexes, manifolds, and generalized Hilbert complexes; their computational realization is supported by dedicated software; and their current theoretical frontier concerns stability, perturbation bounds, and principled spectral summaries for learning tasks (Liu et al., 2023, Anh et al., 26 Jun 2025, Jones et al., 15 Aug 2025, Wolf et al., 24 Sep 2025).