Persistent Sheaf Laplacians

• Persistent Sheaf Laplacians are multiscale spectral operators that extend sheaf cohomology by tracking local coefficient systems and geometric invariants across filtrations.
• They compute harmonic kernels to recover sheaf cohomology while using non-harmonic spectra and Rayleigh quotients to capture application-specific constraints.
• PSL are applied across diverse fields—from protein mutation to image analysis—offering robust, perturbation-aware features that enhance topological and geometric analyses.

Persistent Sheaf Laplacians (PSL) are multiscale spectral operators that combine cellular sheaf theory with persistent topological Laplacians. In this setting, a sheaf assigns local coefficient data to cells of a graph or simplicial complex and encodes compatibility through restriction maps; the resulting sheaf Laplacians recover sheaf cohomology through their harmonic kernels, while their non-harmonic spectra retain geometric, weighted, and application-specific constraint information. Along filtrations, PSL tracks these operators across scale, so that persistent sheaf Betti numbers, spectral trajectories, Rayleigh quotients, and related invariants become available within a single framework (Wei et al., 2021, Wei et al., 2023).

1. Algebraic definition

A cellular sheaf on a finite simplicial or cellular complex assigns to each cell $\sigma$ a finite-dimensional vector space and to each face relation a linear restriction map satisfying functoriality. In the PSL literature, the corresponding $k$-cochain space is the direct sum of stalks over $k$-cells,

$C^k(X;F)=\bigoplus_{\sigma\in X_k} F(\sigma),$

and the sheaf coboundary is assembled from signed incidences and restriction maps. One formulation writes, for a $(k+1)$-simplex $\tau$,

$$(\delta_F^k c)(\tau)=\sum_{\sigma\prec\tau,\ \sigma\in K_k} o(\sigma,\tau)\,F(\tau\to\sigma)c(\sigma),$$

while equivalent block-matrix descriptions are used throughout the broader sheaf-Laplacian literature (Girish et al., 9 Dec 2025).

With inner products on stalks, the adjoint $\delta_F^{k*}$ is defined in the usual way, and the sheaf Hodge Laplacians are

$$L_F^{k,\mathrm{up}}=\delta_F^{k*}\delta_F^k,\qquad L_F^{k,\mathrm{down}}=\delta_F^{k-1}\delta_F^{(k-1)*},\qquad L_F^k=L_F^{k,\mathrm{up}}+L_F^{k,\mathrm{down}}.$$

Under the standard discrete Hodge correspondence,

$\ker L_F^k \cong H^k(K;F),$

so harmonic sheaf $k$0-cochains represent sheaf cohomology classes (Girish et al., 9 Dec 2025).

The foundational 2021 paper formulates PSL as a persistent extension of these sheaf Hodge Laplacians. For an inclusion $k$1 with consistent sheaves, it defines a mixed persistent operator whose kernel computes the persistent sheaf cohomology image, thereby extending persistent combinatorial Laplacians from constant coefficients to sheaf-valued cochains (Wei et al., 2021). The survey literature places this construction within the more general framework of persistent Laplacians on differential graded inner-product spaces, emphasizing that the sheaf case is a cohomological specialization rather than an unrelated operator family (Wei et al., 2023).

2. Persistence, harmonic structure, and spectral trajectories

PSL becomes genuinely persistent when a filtration $k$2 of complexes is paired with a compatible family of sheaves $k$3. In that setting, inclusions $k$4 induce maps on sheaf cohomology, forming a persistence module of sheaf cohomologies. Tracking the scale-dependent operators $k$5 yields spectral families $k$6, described in one paper as spectral vineyards, and supports Rayleigh-quotient observables

$k$7

Birth and death of harmonic components $k$8 produce a PSL barcode; more generally, sub-threshold eigenvalues define near-harmonic features whose lifetimes are recorded in spectral diagrams (Girish et al., 9 Dec 2025).

A complementary persistent definition appears in the protein-mutation literature. For sheaves $k$9 on $k$0 and $k$1 on $k$2 with $k$3, the $k$4-th persistent sheaf Laplacian is

$k$5

and the harmonic and nonharmonic spectra are both used as features (Ren et al., 18 Jan 2026). PETLS adopts an equivalent filtration-pair formalism with

$k$6

together with the identification of $k$7 with the persistent sheaf Betti number $k$8 (Jones et al., 15 Aug 2025).

An important technical point is that persistence need not be represented only by classical birth–death barcodes. In some applications, persistence is encoded as spectral curves, minimum non-zero eigenvalue traces, or aggregated statistics across scales rather than explicit interval decompositions. This is especially visible in protein mutation analysis, where harmonic multiplicities and nonharmonic spectral summaries are tracked over filtrations without plotting classical barcodes (Ren et al., 18 Jan 2026).

3. Relation to graph Laplacians, Hodge Laplacians, and persistent homology

PSL is frequently described by comparison with three adjacent constructions. First, graph Laplacians act on scalar $k$9-cochains and encode only scalar connectivity. In the constant-sheaf limit with identity restrictions, sheaf Laplacians reduce to graph or simplicial Hodge Laplacians, but PSL allows multi-field, signed, directional, and context-dependent constraints to enter directly through restriction maps and stalk metrics (Girish et al., 9 Dec 2025).

Second, standard simplicial or Hodge Laplacians act on $C^k(X;F)=\bigoplus_{\sigma\in X_k} F(\sigma),$0-cochains using boundary or coboundary operators that do not themselves encode application-specific local models. Sheaf Hodge Laplacians generalize them by replacing plain incidence blocks with restriction maps, thereby modulating how information flows up and down the complex. The survey literature treats this as a coefficient-system generalization of combinatorial Hodge theory rather than a separate topological invariant (Wei et al., 2023).

Third, PSL differs from persistent homology. Persistent homology tracks births and deaths of homology classes across filtrations but is insensitive to many heterogeneous annotations. Several application papers state this contrast explicitly: persistent homology is described as purely topological, whereas PSL augments topology with spectral information and sheaf compatibility, so that harmonic spectra recover persistent sheaf cohomology while positive eigenvalues encode graded stability, stiffness, or compatibility strength (Ren et al., 18 Jan 2026). In the constant-sheaf limit with trivial restrictions, one paper states that PSL reduces to spectral persistence of standard Hodge Laplacians and kernel births corresponding to persistent homology barcodes (Girish et al., 9 Dec 2025).

A recurring misconception is to identify PSL with persistent homology plus weights. The literature instead presents it as a sheaf-valued spectral theory in which local algebraic data are part of the operator itself. This suggests that the main distinction is not only weighted topology, but the replacement of uniform coefficients by structured local coefficient systems.

4. Computation and implementation

PSL computation usually begins by building a filtration, such as Vietoris–Rips, alpha, clique, or thresholded graph complexes, and then assigning a sheaf with either scalar or low-dimensional stalks. The practical pipeline described in the brain-inspired representation paper is representative: build $C^k(X;F)=\bigoplus_{\sigma\in X_k} F(\sigma),$1, define $C^k(X;F)=\bigoplus_{\sigma\in X_k} F(\sigma),$2, assemble $C^k(X;F)=\bigoplus_{\sigma\in X_k} F(\sigma),$3 as a sparse block matrix with signed restriction-map blocks, form $C^k(X;F)=\bigoplus_{\sigma\in X_k} F(\sigma),$4, compute eigenpairs, and extract invariants such as $C^k(X;F)=\bigoplus_{\sigma\in X_k} F(\sigma),$5, births and deaths of kernel components, persistence of sub-threshold eigenvalues, and Rayleigh-quotient trajectories (Girish et al., 9 Dec 2025).

For sparse problems, eigensolvers for the smallest $C^k(X;F)=\bigoplus_{\sigma\in X_k} F(\sigma),$6 eigenpairs are reported to use Lanczos or LOBPCG in $C^k(X;F)=\bigoplus_{\sigma\in X_k} F(\sigma),$7 per iteration, and vineyard-style tracking is described as a way to warm-start from previous eigenvectors. When only a few cells or restriction maps change, low-rank or Sherman–Morrison–Woodbury-type updates are proposed. Practical heuristics include choosing small stalk dimensions such as $C^k(X;F)=\bigoplus_{\sigma\in X_k} F(\sigma),$8–$C^k(X;F)=\bigoplus_{\sigma\in X_k} F(\sigma),$9, exploiting sparsity, and restricting attention to $(k+1)$0 in neural datasets (Girish et al., 9 Dec 2025).

The PETLS software paper systematizes these ideas in a C++ library with Python bindings. It supports simplicial, alpha, directed flag, Dowker, and cellular Sheaf complexes, and implements PSL assembly together with Schur-complement computation of the persistent up-Laplacian. PETLS emphasizes sparse matrix assembly, self-adjoint eigensolvers, optional iterative methods, and tolerance-based extraction of persistent sheaf Betti numbers from near-zero eigenvalues (Jones et al., 15 Aug 2025).

Local-patch constructions are common when full-complex spectra would be too costly or too coarse. In single-cell RNA-seq, HSSE builds cell-centered local Vietoris–Rips patches up to maximal simplex dimension $(k+1)$1, assigns one-dimensional stalks, partitions local filtration ranges into intervals, computes persistent sheaf Laplacians on those intervals, and summarizes each spectrum by $(k+1)$2, $(k+1)$3, $(k+1)$4, $(k+1)$5, and $(k+1)$6 (Wang et al., 27 Mar 2026). In image analysis, MPSL applies the same local-patch principle to kNN-induced clique complexes constructed from PCA embeddings across multiple reduced dimensions, again using scalar stalks and distance-kernel restriction maps (Wang et al., 16 Feb 2026).

5. Application domains and empirical use

PSL has been used as both an explanatory mathematical formalism and a feature-generation mechanism in learning pipelines. In the brain-inspired representation-learning framework, PSL is presented as a natural extension of cochain-map dynamics from scalar cochains to sheaf cochains. A regularized energy

$(k+1)$7

leads to gradient flows or discrete-time regularizers in which the Laplacian term penalizes inconsistency with sheaf restrictions. The paper reports lower bottleneck and Wasserstein distances, with $(k+1)$8 reduced by approximately $(k+1)$9–$\tau$0 across noise levels, EEG classification accuracy of $\tau$1 versus $\tau$2–$\tau$3 for baselines, and spike-train topology reconstruction with $\tau$4 versus $\tau$5–$\tau$6 (Girish et al., 9 Dec 2025).

In protein mutation prediction, PSL is specialized to encode physicochemical heterogeneity. The paper "Persistent Sheaf Laplacian Analysis of Protein Stability and Solubility Changes upon Mutation" defines scalar stalks and vertex-to-edge restriction maps by multiplication with $\tau$7, where $\tau$8 are partial atomic charges and $\tau$9 are Euclidean distances. It uses Vietoris–Rips filtrations from $$(\delta_F^k c)(\tau)=\sum_{\sigma\prec\tau,\ \sigma\in K_k} o(\sigma,\tau)\,F(\tau\to\sigma)c(\sigma),$$0 Å to $$(\delta_F^k c)(\tau)=\sum_{\sigma\prec\tau,\ \sigma\in K_k} o(\sigma,\tau)\,F(\tau\to\sigma)c(\sigma),$$1 Å in steps of $$(\delta_F^k c)(\tau)=\sum_{\sigma\prec\tau,\ \sigma\in K_k} o(\sigma,\tau)\,F(\tau\to\sigma)c(\sigma),$$2 Å, alpha complexes in one dimension, a $$(\delta_F^k c)(\tau)=\sum_{\sigma\prec\tau,\ \sigma\in K_k} o(\sigma,\tau)\,F(\tau\to\sigma)c(\sigma),$$3 Å mutation neighborhood, and element-specific C/N/O atom pairings. The resulting PSL feature construction yields a $$(\delta_F^k c)(\tau)=\sum_{\sigma\prec\tau,\ \sigma\in K_k} o(\sigma,\tau)\,F(\tau\to\sigma)c(\sigma),$$4-dimensional feature vector per protein and is fused with ESM-2 embeddings and auxiliary physicochemical descriptors in SheafLapNet, a six-layer multilayer perceptron with $$(\delta_F^k c)(\tau)=\sum_{\sigma\prec\tau,\ \sigma\in K_k} o(\sigma,\tau)\,F(\tau\to\sigma)c(\sigma),$$5 neurons per hidden layer. The reported benchmarks are PCC $$(\delta_F^k c)(\tau)=\sum_{\sigma\prec\tau,\ \sigma\in K_k} o(\sigma,\tau)\,F(\tau\to\sigma)c(\sigma),$$6 and RMSE $$(\delta_F^k c)(\tau)=\sum_{\sigma\prec\tau,\ \sigma\in K_k} o(\sigma,\tau)\,F(\tau\to\sigma)c(\sigma),$$7 kcal/mol on S2648, PCC $$(\delta_F^k c)(\tau)=\sum_{\sigma\prec\tau,\ \sigma\in K_k} o(\sigma,\tau)\,F(\tau\to\sigma)c(\sigma),$$8 and RMSE $$(\delta_F^k c)(\tau)=\sum_{\sigma\prec\tau,\ \sigma\in K_k} o(\sigma,\tau)\,F(\tau\to\sigma)c(\sigma),$$9 kcal/mol on S350, and normalized CPR $\delta_F^{k*}$0, normalized $\delta_F^{k*}$1 on PON-Sol2 (Ren et al., 18 Jan 2026).

Protein flexibility prediction supplies a second biomolecular line of work. One study extracts PSL features at radii $\delta_F^{k*}$2, $\delta_F^{k*}$3, and $\delta_F^{k*}$4 Å, using the count of zero eigenvalues together with the maximum, minimum, mean, and standard deviation of the positive eigenvalues, for a total of $\delta_F^{k*}$5 PSL features. Combined with global and local protein descriptors, these features support B-factor prediction on 346 proteins and are reported to produce an increase in accuracy of $\delta_F^{k*}$6 compared to the classical Gaussian network model (Hayes et al., 12 Feb 2025). A related paper on protein–nucleic acid complexes builds local alpha-complex filtrations at $\delta_F^{k*}$7, $\delta_F^{k*}$8, and $\delta_F^{k*}$9 Å, localizes the sheaf by setting the target atom label to $$L_F^{k,\mathrm{up}}=\delta_F^{k*}\delta_F^k,\qquad L_F^{k,\mathrm{down}}=\delta_F^{k-1}\delta_F^{(k-1)*},\qquad L_F^k=L_F^{k,\mathrm{up}}+L_F^{k,\mathrm{down}}.$$0 and neighbor labels to $$L_F^{k,\mathrm{up}}=\delta_F^{k*}\delta_F^k,\qquad L_F^{k,\mathrm{down}}=\delta_F^{k-1}\delta_F^{(k-1)*},\qquad L_F^k=L_F^{k,\mathrm{up}}+L_F^{k,\mathrm{down}}.$$1, and uses $$L_F^{k,\mathrm{up}}=\delta_F^{k*}\delta_F^k,\qquad L_F^{k,\mathrm{down}}=\delta_F^{k-1}\delta_F^{(k-1)*},\qquad L_F^k=L_F^{k,\mathrm{up}}+L_F^{k,\mathrm{down}}.$$2 spectral features per atom for linear regression. Across three datasets, it reports that PSL consistently outperforms GNM and mFRI, with up to a $$L_F^{k,\mathrm{up}}=\delta_F^{k*}\delta_F^k,\qquad L_F^{k,\mathrm{down}}=\delta_F^{k-1}\delta_F^{(k-1)*},\qquad L_F^k=L_F^{k,\mathrm{up}}+L_F^{k,\mathrm{down}}.$$3 improvement in Pearson correlation coefficient (Hayes et al., 23 Oct 2025).

In transcriptomics, PSL appears in both local representation learning and population-scale network analysis. HSSE for single-cell RNA-seq constructs multiscale local patches, uses one-dimensional stalks and kernel-based restriction maps modulated by center–neighbor labels, and aggregates PSL eigenvalue summaries into cell-level feature vectors without additional model training. On twelve benchmark datasets, it reports average Macro-F1 $$L_F^{k,\mathrm{up}}=\delta_F^{k*}\delta_F^k,\qquad L_F^{k,\mathrm{down}}=\delta_F^{k-1}\delta_F^{(k-1)*},\qquad L_F^k=L_F^{k,\mathrm{up}}+L_F^{k,\mathrm{down}}.$$4, Macro-Recall $$L_F^{k,\mathrm{up}}=\delta_F^{k*}\delta_F^k,\qquad L_F^{k,\mathrm{down}}=\delta_F^{k-1}\delta_F^{(k-1)*},\qquad L_F^k=L_F^{k,\mathrm{up}}+L_F^{k,\mathrm{down}}.$$5, and Macro-AUC $$L_F^{k,\mathrm{up}}=\delta_F^{k*}\delta_F^k,\qquad L_F^{k,\mathrm{down}}=\delta_F^{k-1}\delta_F^{(k-1)*},\qquad L_F^k=L_F^{k,\mathrm{up}}+L_F^{k,\mathrm{down}}.$$6, compared with $$L_F^{k,\mathrm{up}}=\delta_F^{k*}\delta_F^k,\qquad L_F^{k,\mathrm{down}}=\delta_F^{k-1}\delta_F^{(k-1)*},\qquad L_F^k=L_F^{k,\mathrm{up}}+L_F^{k,\mathrm{down}}.$$7, $$L_F^{k,\mathrm{up}}=\delta_F^{k*}\delta_F^k,\qquad L_F^{k,\mathrm{down}}=\delta_F^{k-1}\delta_F^{(k-1)*},\qquad L_F^k=L_F^{k,\mathrm{up}}+L_F^{k,\mathrm{down}}.$$8, and $$L_F^{k,\mathrm{up}}=\delta_F^{k*}\delta_F^k,\qquad L_F^{k,\mathrm{down}}=\delta_F^{k-1}\delta_F^{(k-1)*},\qquad L_F^k=L_F^{k,\mathrm{up}}+L_F^{k,\mathrm{down}}.$$9 for MDG (Wang et al., 27 Mar 2026). In Alzheimer’s disease snRNA-seq analysis, PSL is applied to STRING-thresholded PPI clique complexes, with node labels $\ker L_F^k \cong H^k(K;F),$0, graph-level sheaf Laplacians $\ker L_F^k \cong H^k(K;F),$1 and $\ker L_F^k \cong H^k(K;F),$2, and Wasserstein distances between original and gene-perturbed spectra used for target prioritization. The reported analysis identifies convergent vascular and microglial modules and supports downstream target–drug selection (Cottrell et al., 29 Sep 2025).

Image analysis provides a different operationalization. The MPSL framework builds local simplicial complexes from kNN neighborhoods in PCA-reduced spaces with target dimensions $\ker L_F^k \cong H^k(K;F),$3, uses one-dimensional stalks and exponential distance-kernel restrictions, computes $\ker L_F^k \cong H^k(K;F),$4- and $\ker L_F^k \cong H^k(K;F),$5-order sheaf Laplacian spectra, and aggregates seven statistics per spectrum across neighborhood sizes and dimensions. On COIL20, PCA averages Accuracy $\ker L_F^k \cong H^k(K;F),$6 and Macro F1 $\ker L_F^k \cong H^k(K;F),$7, whereas MPSL single-dimension averages Accuracy $\ker L_F^k \cong H^k(K;F),$8 and Macro F1 $\ker L_F^k \cong H^k(K;F),$9, with multi-dimension aggregation reaching Accuracy $k$00 and Macro F1 $k$01. On ETH80, PCA averages Accuracy $k$02 and Macro F1 $k$03, while MPSL single-dimension averages Accuracy $k$04 and Macro F1 $k$05, and multi-dimension aggregation reaches Accuracy $k$06 and Macro F1 $k$07 (Wang et al., 16 Feb 2026).

6. Stability, limitations, and open directions

Several papers emphasize that PSL is not only expressive but perturbation-aware. If $k$08 is continuous in operator norm, then $k$09 is continuous, and one paper gives the Weyl-type bound

$k$10

together with a Davis–Kahan bound for eigenvectors. The same paper argues that these imply Lipschitz continuity of Rayleigh quotients and stability of PSL spectral vineyards under small sheaf or filtration changes (Girish et al., 9 Dec 2025). The PETLS and survey papers likewise tie PSL robustness to the general algebraic-stability framework for persistent Laplacians, though they describe full PSL-specific stability theory as inherited analogically rather than fully developed in its own right (Jones et al., 15 Aug 2025, Wei et al., 2023).

At the same time, the current PSL literature is explicit about modeling constraints. Many practical systems use one-dimensional stalks $k$11 and simple scalar restriction maps because they are inexpensive and interpretable; higher-order interactions, directional hydrogen bonds, multi-body effects, or richer transport laws are then implicit rather than explicit (Ren et al., 18 Jan 2026). Local spectral constructions can be less stable on very small datasets, while on very large heterogeneous datasets global geometric structure may dominate and reduce the marginal benefit of local sheaf spectra (Wang et al., 27 Mar 2026). Computationally, persistent homology and local homology reductions remain expensive in the more elaborate sheaf-neural constructions, and GPU support is identified as limited (Cesa et al., 2023).

These limitations define a recognizable research agenda. Proposed extensions include persistent Path Laplacian, persistent hyperdigraph Laplacians, persistent Dirac operators, and sheaf/copresheaf neural networks for richer directed and higher-order relations (Ren et al., 18 Jan 2026). Other papers propose learned sheaf maps, connection-type restrictions, and end-to-end differentiable persistent local homology sheaves, suggesting that future PSL variants may move from hand-crafted restriction maps toward data-adaptive coefficient systems (Cesa et al., 2023, Hayes et al., 12 Feb 2025). A plausible implication is that the long-term development of PSL will depend less on the basic Hodge-theoretic core—which is already well established—and more on domain-specific sheaf design, scalable eigensolvers, and principled stability theory for nontrivial sheaf perturbations.