Paper deep dive
Sheaf-theoretic Signal Processing on Graphs: Spectral Theory, Filtering, and Sampling
Gabriele D'Acunto, Leonardo Di Nino, Paolo Di Lorenzo, Sergio Barbarossa
Intelligence
Status: not_run | Model: - | Prompt: - | Confidence: 0%
Entities (0)
Relation Signals (0)
No relation signals yet.
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:Modern sensing, communication, and learning systems generate heterogeneous network signals, with local data differing in dimension, modality, and geometric structure. Processing such data requires a mathematical framework capable of simultaneously modeling heterogeneous local signal spaces and the transformations relating them. Network sheaves provide such a framework by associating local vector spaces with network entities and linear restriction maps with their interactions. This is the first paper to develop a unified sheaf signal processing (SSP) framework on network sheaves, extending the fundamental operations of signal processing, namely spectral analysis, filtering, and sampling, to heterogeneous local spaces. Unlike graph and topological signal processing, where signals are modeled over a common vector space, SSP jointly models heterogeneous local signal spaces and the linear transformations relating neighboring spaces through restriction maps. We define the Sheaf Fourier Transform (SFT), whose frequencies quantify signal inconsistency induced by the network topology, the restriction maps, and the local geometry. Building on this representation, we develop polynomial sheaf filters and formulate sampling as the joint selection of network nodes and intra-node components. We derive perfect recovery conditions for bandlimited sheaf signals and propose a greedy sampling-set design algorithm. To incorporate application-dependent signal models, including different bases, dictionaries, and learned embeddings, we introduce representation sheaves and characterize the natural transformations that preserve spectral properties and guarantee interoperability across representations. Experiments on synthetic, motion-capture, and financial datasets validate the proposed framework and demonstrate consistent improvements over canonical graph signal processing baselines.
Tags
Links
- Source: https://arxiv.org/abs/2608.01318v1
- Canonical: https://arxiv.org/abs/2608.01318v1
Trouble viewing inline? Open PDF directly →
Full Text
114,443 characters extracted from source content.
Expand or collapse full text
Sheaf-theoretic Signal Processing on Graphs: Spectral Theory, Filtering, and Sampling Gabriele D’Acunto , Leonardo Di Nino , Paolo Di Lorenzo , Sergio Barbarossa The Authors are with the Department of Information Engineering, Electronics, and Telecommunications, Sapienza University of Rome, 00184 Rome, Italy (e-mails: gabriele.dacunto, leonardo.dinino, paolo.dilorenzo, sergio.barbarossa@uniroma1.it). All authors are also with the National Inter-University Consortium for Telecommunications (CNIT), Parma, Italy. The work was supported by the SNS JU project 6G-GOALS [1] under the EU’s Horizon program Grant Agreement No 101139232, and by Huawei Technology France SASU under Grant N. Tg20250616041. Abstract Modern sensing, communication, and learning systems generate heterogeneous network signals, with local data differing in dimension, modality, and geometric structure. Processing such data requires a mathematical framework capable of simultaneously modeling heterogeneous local signal spaces and the transformations relating them. Network sheaves provide such a framework by associating local vector spaces with network entities and linear restriction maps with their interactions. This is the first paper to develop a unified sheaf signal processing (SSP) framework on network sheaves, extending the fundamental operations of signal processing, namely spectral analysis, filtering, and sampling, to heterogeneous local spaces. Unlike graph and topological signal processing, where signals are modeled over a common vector space, SSP jointly models heterogeneous local signal spaces and the linear transformations relating neighboring spaces through restriction maps. We define the Sheaf Fourier Transform (SFT), whose frequencies quantify signal inconsistency induced by the network topology, the restriction maps, and the local geometry. Building on this representation, we develop polynomial sheaf filters and formulate sampling as the joint selection of network nodes and intra-node components. We derive perfect recovery conditions for bandlimited sheaf signals and propose a greedy sampling-set design algorithm. To incorporate application-dependent signal models, including different bases, dictionaries, and learned embeddings, we introduce representation sheaves and characterize the natural transformations that preserve spectral properties and guarantee interoperability across representations. Experiments on synthetic, motion-capture, and financial datasets validate the proposed framework and demonstrate consistent improvements over canonical graph signal processing baselines. I Introduction In recent years, Graph Signal Processing (GSP) has emerged as a powerful extension of classical SP to irregular domains, where signals are supported on the vertices of a graph rather than on regular Euclidean grids [2, 3]. By encoding the connectivity of the domain into graph shift operators, such as the adjacency matrix or the graph Laplacian, GSP provides a principled framework to define graph Fourier transforms (GFTs), localized filters, sampling strategies, and learning methods for networked data [4, 2, 3, 5, 6, 7, 8]. The success of GSP stems from the fact that the processing operators are intrinsically adapted to the relational structure of the data. However, graph models are inherently restricted to pairwise interactions and cannot directly represent relations involving multiple entities. This limitation has motivated the extension of GSP to higher-order relational domains, giving rise to Topological Signal Processing (TSP) over simplicial or cell complexes [9, 10, 11, 12, 13]. Here, in addition to nodes, signals are associated to edges, polygons, and higher-dimensional cells. Building on this framework, recent work has further developed the relation between TSP and learning over simplicial and cell complexes [14]. However, both GSP and TSP retain the fundamental assumption that signals attached to comparable cells belong to a common vector space, which is unrealistic in heterogeneous networks. Consider, for example, a common physical phenomenon observed by a multimodal sensing apparatus, composed of a microphone, an accelerometer, and a camera. Although the underlying context is shared, the individual signal spaces differ in dimension, coordinates, units, and semantics. Consequently, these signals cannot be directly compared or modeled through square matrix-valued weights alone. Instead, the signal model must explicitly encode how individual signals are transported, observed, projected, or constrained across heterogeneous spaces. This calls for a framework in which the geometry of local signal spaces and the maps between them are intrinsic to the signal model itself. Category theory [15] offers the natural formal language for this problem: it is axiomatically centered not on mathematical structures in isolation, but on the relations, or morphisms, that hold between them, together with the rules for composing such relations in a consistent manner. Network sheaves instantiate this language over a graph [16]. A network sheaf assigns a local data space, called a stalk, to each node and edge of a graph, together with restriction maps associated with node–edge incidences. These maps specify how data from distinct local spaces are transported on a common edge space. Network sheaves do not simply enrich the relational domain, as TSP does; they modify the signal model itself incorporating linear relations among stalks. A sheaf signal is simply an assignment of values to the stalks. This perspective has recently attracted growing interest in spectral theory, dynamical systems, learning, and neural architectures [17]. However, its development as a unified SP framework including principled notions of spectral representation, filtering, and sampling for heterogeneous stalk-valued signals remains underinvestigated. Related work. Most GSP works restrict signals to scalar or homogeneous vector spaces and encode inter-node relationships through scalar edge weights [3]. Several extensions consider vector-valued signals. Time-varying graph signals are naturally modeled within time-vertex SP, which jointly exploits graph and temporal structures through product-domain harmonic analysis and filtering [18]. More generally, multiway GSP and multidimensional graph forecasting extend GSP to tensor-valued data and vector-valued time series by leveraging product-graph constructions across multiple domains [19, 20]. Vector-valued graph trend filtering has also been developed using group-sparsity and mixed-norm regularization [21]. A related direction considers matrix-weighted graphs, where matrix-valued edge weights define Laplacian-like operators acting on vector-valued node states [22]. Despite their broader modeling capabilities, these approaches still assume that signals belong to a common coordinate space and interact through prescribed linear operators. The groundwork for applying sheaf-theoretic concepts to SP was initially suggested by Robinson, who recognized that several classical SP problems, such as filtering, sampling, and sensor integration, admit sheaf-theoretic reformulations based on sheaf-morphisms and cohomology [23, 24, 25, 26, 27]. In contrast, our work adopts a model-based SP perspective, generalizing GSP and TSP, in which the network sheaf is regarded as the underlying signal model. We exploit this model to systematically develop the fundamental SP tools, i.e., signal representation, spectral analysis, filtering, and sampling, from the sheaf structure. In addition, unlike previous works that fix the signal sheaf to a single representation level, our formalism introduces and explicitly builds a representation sheaf capturing the most appropriate low-dimensional local representation of the signal, enabling representation-aware SP while remaining fully consistent with the signal sheaf. A complementary line of work has investigated sheaves as mathematical models for heterogeneous relational data. The spectral foundations of cellular sheaves were established by generalizing key notions of spectral graph theory to the sheaf Laplacian [28]. This framework has since been applied to opinion dynamics, sheaf neural networks, and geometric deep learning, where sheaf-based diffusion and convolutions support heterogeneous and asymmetric relations [29, 30, 31, 32]. More recently, these ideas have been extended to infinite-dimensional stalks through Hilbert bundles [33], and have found applications in optimization, semantic communications, and causal abstraction [34, 35, 36, 37]. Collectively, these two lines of works demonstrate both the applicability of sheaf theory to concrete signal-processing case studies and the expressive power of sheaves for representing heterogeneous local information and structured relations. Nevertheless, neither provides a unified SP framework encompassing signal representation, spectral analysis, filtering, and sampling for heterogeneous stalk-valued signals. Contributions. To the best of our knowledge, this paper introduces the first unified framework for sheaf-theoretic signal processing on networks with heterogeneous local signal spaces. Our main contributions are summarized as follows: 1. Signal-model-aware sheaf representations. We start from a network signal sheaf, describing raw data and their relations, and build a lower dimensional representation sheaf accommodating local orthogonal bases, dictionaries, latent embeddings, and task-specific feature maps. We show that the consistent mappings between representation and signal sheaves are category-theoretic natural transformations, and derive the conditions under which they preserve the sheaf structure. This enables SP to be carried out directly in low-dimensional, application-adapted representation spaces rather than on raw observations. Furthermore, we equip representation stalks with metric tensors induced from the raw-signal geometry, providing a unified formulation for heterogeneous signal spaces with different dimensions, physical units, and scales. 2. Sheaf spectral representation and filtering. Building on the representation sheaf, we develop a spectral framework for heterogeneous sheaf signals. We define the Sheaf Fourier Transform (SFT) from the eigendecomposition of the representation sheaf Laplacian and show that its frequencies jointly capture the network topology, the local geometry, and the transport constraints encoded by the restriction maps. We establish the spectral intertwining between the representation and signal sheaf Laplacians and show that the intrinsic spectral objects are basis-invariant eigenspaces rather than individual eigenvectors. These results naturally lead to polynomial sheaf filters and guarantee interoperability across sheaf representations. 3. Sampling and recovery of bandlimited sheaf signals. We introduce a sampling operator that jointly selects network nodes and intra-stalk components. We derive necessary and sufficient conditions for perfect recovery of bandlimited sheaf signals and show that the resulting theory recovers graph-signal sampling as a special case. Importantly, the spectral intertwining induces an orthogonal decomposition of the band into two components, one aligned with the local models, and one orthogonal to it. This offers a more parsimonious and computationally cheaper recovery strategy, by targeting only the part of the band aligned with the signal models. Finally, we formulate sampling-set design as a rank-maximization problem and propose a greedy algorithm with an energy-based tie-breaking criterion. Numerical experiments on synthetic and real datasets assess the performance of the proposed framework and demonstrate consistent improvements over canonical GSP methods. I A primer on sheaves over graphs This section provides a selective overview of the mathematical framework of network sheaves which is key to our work. The discussion is kept at the general level. Secs. I and IV will specialize the tools for the SP domain. Category theory as the language of relations. As anticipated in Section I, sheaf-theoretic signal processing (SSP) adopts a relational viewpoint, where the emphasis is placed on the relationships among local mathematical structures rather than on the structures themselves. This viewpoint naturally leads to category theory, a mathematical framework describing structured objects and the transformations between them [15]. A category consists of a collection of objects together with structure-preserving transformations, called morphisms, between them. Typical examples include the category Set of sets and functions over sets, the category Graph of graphs and graph homomorphisms, and the category Hilb of Hilbert spaces and linear operators. Throughout this paper we shall work with the latter, although the concepts introduced below apply to arbitrary target categories. A graph can itself be viewed as a category, namely a partially-ordered set (poset), whose objects are nodes and edges, and whose morphisms encode the node-edge incidence relations. A functor is defined as a mapping between categories. A network sheaf is a functor assigning to every object of the poset an object of a target category, and to every incidence relation a corresponding morphism. The objects associated with the nodes and edges are called stalks, while the associated morphisms are the restriction maps. In the present work, every stalk is a finite-dimensional Hilbert space and every restriction map is a linear operator encoded as a matrix. Besides assigning local mathematical structures to the graph, a network sheaf possesses a fundamental gluing property: local data that are compatible through the restriction maps uniquely determine a global section over the network, and every global section restricts consistently to the corresponding local data. A natural transformation is a way of mapping one functor into another functor while completely preserving all structural relationships. In this work, we will show how to build a natural transformation from the signal sheaf, where the raw signals live, and the representation sheaf, a structure that exploits all possible lower-dimensional representations on each node/edge. We will show how the natural transformation allows us to transport fundamental SP operations, like spectral analysis and filtering, from one sheaf to the other, with substantial savings in terms of complexity. Network sheaves on graphs valued in ℝ Hilb_R. Let =(,ℰ)G=(N,E) be an undirected graph with node set N and edge set ℰE, where ||=N|N|=N and |ℰ|=E|E|=E, respectively. A network sheaf F on G valued in ℝ Hilb_R consists of: i. a finite-dimensional real Hilbert space F(i)F(i) for each node i∈i , called the node stalk at i; i. a finite-dimensional real Hilbert space F(e)F(e) for each edge e∈ℰe , called the edge stalk at e; i. a bounded linear map i⊴e:F(i)→F(e)F_i e:F(i)→ F(e) for each node-edge incident pair i⊴ei e, called the restriction map. In the finite-dimensional setting, stalks are Euclidean spaces F(i)≅ℝdiF(i) ^d_i and F(e)≅ℝdeF(e) ^d_e, and the restriction maps are matrices i⊴e∈ℝde×diF_i e ^d_e× d_i. Each stalk carries an inner product induced by a metric tensor i∈++diG_i _++^d_i at node i and e∈++deG_e _++^d_e at edge e, so that ⟨i,i⟩F(i)=i⊤ii _i,\,y_i _F(i)=x_i G_iy_i, with induced norm ‖i‖i\|x_i\|_G_i (cf. Supp. S1-A). The local inner products and the restriction maps play complementary roles: the former define the intrinsic geometry of each stalk, whereas the latter specify how data attached to adjacent nodes are compared through their common edge space, in its intrinsic geometry. Thus, heterogeneous stalks with different units, scales, or noise levels are handled without any ad hoc rescaling. The dimensions did_i and ded_e are in general independent. Usually de≥max(di,dj)d_e≥ (d_i,d_j), for an edge e=(i,j)e=(i,j). In this case, the edge stalk acts as an integration space: the restriction maps embed the node vectors into a richer common space, and consistency demands a global agreement between ix_i and jx_j. However, we can also have de≤min(di,dj)d_e≤ (d_i,d_j). In this case, the edge stalk acts as a compression bottleneck and consistency requires that the vectors ix_i and jx_j agree on a common low-dimensional projection, allowing partial agreement while tolerating discrepancies in the complementary directions. GSP can be seen as the special case F(i)≅ℝdF(i) ^d, F(e)≅ℝdF(e) ^d, and i⊴e=dF_i e=I_d, with typically d=1d=1 and e=wedG_e=w_eI_d for all i⊴ei e, where we∈ℝ+w_e _+ is the edge weight. Cochains, sections and consistency. The space of node vectors (0-cochains) is the direct sum of the node stalks, C0(;F)=⨁i∈F(i)C^0(G;F)= _i F(i), and a node vector is =[1⊤,…,i⊤,…,N⊤]⊤∈ℝDx=[x_1 ,…,x_i ,…,x_N ] ^D with D=∑idiD= _id_i, i.e. the concatenation of all node stalk vectors. A section of F over a subgraph ⊆U is a node vector that is locally consistent on every edge internal to U: i⊴ei=j⊴ej,∀e∈.F_i ex_i=F_j ex_j\,, ∀\,e \,. (1) A global section is an assignment of a signal to every stalk of the sheaf such that all restriction maps are simultaneously satisfied. The space of global sections is denoted Γ(;F) (G;F ). The hierarchy is therefore: 0-cochains (no constraint) ⊃ sections over U (local consistency) ⊃ global sections (full consistency). Consistency is a structural condition, not a regularity one. In GSP, smoothness quantifies how much a vector varies across edges. Here, local consistency means that the views of the vector at adjacent nodes are exactly compatible through the restriction maps. A global section lies in the kernel of the sheaf Laplacian, as we show next. The space of edge vectors (11-cochains) is C1(;F)=⨁e∈ℰF(e)C^1(G;F)= _e F(e), with elements =[e1⊤,…,eE⊤]⊤∈ℝ∑edex=[x_e_1 ,…,x_e_E ] _ed_e. Edge vectors generalize the 1-cochains of TSP: TSP corresponds to F(e)=ℝF(e)=R with trivial restriction maps, while here each edge carries a vector in its own space F(e)F(e). The metric tensors assemble into block-diagonal matrices 0∈++∑idiG_0 _++ _id_i and 1∈++∑edeG_1 _++ _ed_e, inducing inner products on C0C^0 and C1C^1. Total variation and the sheaf Laplacian. As with GSP, a fundamental descriptor of a graph is its incidence matrix. Given an orientation e=(i,j)e=(i,j) (head i, tail j), the coboundary operator on a sheaf :C0(;F)→C1(;F)B:C^0(G;F)→ C^1(G;F) maps a node vector to an edge vector that measures the local inconsistency: ()e=i⊴ei−j⊴ej.(Bx)_e=F_i ex_i-F_j ex_j\,. (2) B is a block matrix with blocks (e,i)≔i⊴eif i is head,−i⊴eif i is tail,de×diotherwise.B(e,i) casesF_i e&if $i$ is head,\\ -F_i e&if $i$ is tail,\\ 0_d_e× d_i&otherwise. cases (3) The total variation of a vector x is defined as: TV()=‖12=∑e∈ℰ‖i⊴ei−j⊴ej‖e2;TV(x)=\|Bx\|_G_1^2= _e \|F_i ex_i-F_j ex_j\|_G_e^2\,; (4) which reduces to ∑e∈ℰwe(xi−xj)2 _e w_e(x_i-x_j)^2 in the GSP case. Introducing the adjoint of B as ⋆=(0)−1⊤1B =(G_0)^-1B G_1 (cf. Supp. S1-B), the sheaf Laplacian is F=⋆,L_F=B B\,, (5) so that TV()=⟨,F⟩C0TV(x)= ,\,L_Fx _C^0. The operator FL_F is symmetric positive semidefinite on C0(;F)C^0(G;F) by construction (cf. Supp. S1-B), and ker(F)=Γ(;F) (L_F)= (G;F ). The sheaf Laplacian depends on both the restriction maps and the metric tensors eG_e through the adjoint ⋆B . Since ker(F)=ker() (L_F)= (B), the kernel depends only on the restriction maps and is unaffected by the choice of the metric. The metric tensors instead shape the non-zero spectrum: with a general anisotropic eG_e, the non-zero eigenvectors depend on the interplay between the restriction maps and the metric. The dimension of the kernel, and hence the number of zero eigenvalues, depends jointly on the graph topology and the restriction maps. Remark 1 (Block structure of the sheaf Laplacian). The sheaf Laplacian F∈ℝD×DL_F ^D× D has a block structure inherited from the direct sum decomposition of C0(;F)C^0(G;F). The (i,j)(i,j)-th block of size di×djd_i× d_j is [F]ij=i−1∑e∈⋆(i)i⊴e⊤ei⊴eif i=j,−i−1i⊴e⊤ej⊴eif e∈ℰ,di×djotherwise;[L_F]_ij= casesG_i^-1 _e∈ (i )F_i e G_e\,F_i e&if i=j,\\ -G_i^-1F_i e G_e\,F_j e&if e ,\\ 0_d_i× d_j&otherwise; cases (6) where ⋆(i)=e∈ℰ∣i⊴e (i )=\e i e\ denotes the set of edges incident to i. The diagonal blocks are positive semidefinite and encode the local geometry and the restriction maps at each node; the off-diagonal blocks encode the coupling between adjacent nodes through the edge metric. In the scalar GSP case, di=1d_i=1, i⊴e=1F_i e=1, and e=weG_e=w_e, so the off-diagonal block reduces to −we-w_e and FL_F recovers the weighted combinatorial Laplacian. Note that FL_F is not symmetric in the standard sense. Rather, FL_F is 0G_0-selfadjoint (cf. Supp. S1-B), i.e., ⟨,F⟩C0=⟨F,⟩C0 ,\,L_Fy _C^0= _Fx,\,y _C^0 for all ,∈C0(;F)x,y∈ C^0(G;F). The symmetrized version ~F=01/2F0−1/2 L_F=G_0^1/2L_FG_0^-1/2 is symmetric in the standard sense and shares the same spectrum as FL_F. Remark 2 (Unweighted case). When 0=G_0=I and 1=G_1=I, we have ⋆=⊤B =B and F=⊤L_F=B B. Remark 3 (Duality: cosheaves and extension maps). Every network sheaf F has a dual structure, the network cosheaf, obtained by reversing the direction of the restriction maps: the extension map on an incident pair i⊴ei e is the adjoint ¯i⊴e=i⊴e⊤:F(e)→F(i) F_i e=F_i e :F(e)→ F(i). The cosheaf governs the boundary operator ⋆B and the Laplacian on edge vectors. Refer to [16] for a comprehensive discussion. I From Signal Sheaf to Representation Sheaf via Natural Transformations When the stalks of the network sheaf F in Sec. I correspond to observed signal spaces, and the restriction maps correspond to transport operators among these spaces, we term F a signal sheaf. In this work, we assume that F is either given or constructed from prior knowledge, leaving its data-driven estimation for future work. In many SP applications, the vectors associated with the node stalks admit parsimonious representations over suitable dictionaries. Rather than processing the original high-dimensional signals, it is therefore desirable to operate directly on their low-dimensional representation vectors. This naturally raises the following question: under which conditions does a collection of local dictionary representations induce a consistent low-dimensional network sheaf? Specifically, let the signal attached to node i admit the representation i=ii,i∈ℝci,ci≤di,x_i=D_is_i, _i ^c_i, c_i≤ d_i, (7) where i∈ℝdi×ciD_i ^d_i× c_i is a dictionary associated with node i. Without loss of generality, we assume that the dictionaries belong to the Stiefel manifold, St(n,m)=∈ℝn×m|⊤=m,m<n.St(n,m)= \V ^n× m\, |\,V V=I_m,\;m<n \. (8) The representation vector =[1⊤,…,N⊤]⊤∈ℝCs=[s_1 ,…,s_N ] ^C with C=∑iciC= _ic_i, naturally define the node stalks of a representation sheaf, denoted by J, with J(i)≅ℝci,i∈.J(i) ^c_i\,, i . (9) Moreover, the metric tensor iG_i of the original stalk induces the metric i=i⊤ii∈++ci,H_i=D_i G_iD_i _++^c_i\,, (10) and similarly, for each edge stalk, e=e⊤ee,H_e=D_e G_eD_e\,, (11) where eD_e denotes the dictionary over the edge e. These local metrics assemble into the block-diagonal matrices ℍ0H_0 and ℍ1H_1, which replace 0G_0 and 1G_1 in the representation sheaf. iijjeei⊴ei ej⊴ej eJ(i)J(i)J(e)J(e)J(j)J(j)i⊴eJ_i ej⊴eJ_j eF(i)F(i)F(e)F(e)F(j)F(j)i⊴eF_i ej⊴eF_j eiD_ieD_ejD_j Figure 1: Dictionaries as components of the natural transformation δ [15] between the representation sheaf J and the signal sheaf F over the edge e=(i,j)e=(i,j). Unlike the node dictionaries, however, the edge dictionaries ee∈ℰ\D_e\_e and the representation restriction maps i⊴eJ_i e are not known a priori. Their existence is constrained by the requirement that the representation preserves the local consistency relations of the signal sheaf. This requirement is expressed by the commutative diagrams illustrated in Fig. 1, namely i⊴ei=ei⊴e,j⊴ej=ej⊴e,F_i eD_i=D_eJ_i e, _j eD_j=D_eJ_j e, (12) for every edge e=(i,j)e=(i,j). Equation (12) naturally leads to the following question: given the node dictionaries ii∈\D_i\_i , under which conditions does there exist a consistent representation sheaf J? The next proposition provides the answer. Proposition I.1 (Existence of local representation maps). Let e=(i,j)e=(i,j) be an edge of G, and assume that the node dictionaries i∈St(di,ci)D_i (d_i,c_i) and j∈St(dj,cj)D_j (d_j,c_j) are given. Define e≔[i⊴eij⊴ej].A_e bmatrixF_i eD_i&F_j eD_j bmatrix. (13) Then there exist an edge dictionary e∈St(de,ce)D_e (d_e,c_e) and representation restriction maps i⊴eJ_i e, j⊴eJ_j e satisfying (12) if and only if rank(e)≤ce.rank(A_e)≤ c_e. (14) Equivalently, the images of i⊴eiF_i eD_i and j⊴ejF_j eD_j must lie in a common subspace of F(e)F(e) of dimension at most cec_e. Proof. See Supp. S2. ∎ Having established the necessary and sufficient condition for the existence of a consistent local representation sheaf, we now provide an explicit construction of the corresponding edge dictionaries and consistent representation restriction maps. Proposition I.2 (Construction of local representation maps). Under the assumptions of Prop. I.1, suppose that rank(e)≤cerank(A_e)≤ c_e. Let re=rank(e)r_e=rank(A_e), and let e∈ℝde×reU_e ^d_e× r_e be an orthonormal basis of Im(e)Im(A_e). Complete eU_e to a matrix e∈St(de,ce)D_e (d_e,c_e) by adding ce−rec_e-r_e orthonormal columns. Then the representation restriction maps can be chosen as i⊴e=e⊤i⊴ei,j⊴e=e⊤j⊴ej.J_i e=D_e F_i eD_i, _j e=D_e F_j eD_j. (15) With this choice, the commutativity conditions (12) hold true. Proof. See Supp. S2. ∎ The previous propositions provide local conditions for constructing the representation over a single edge. If these conditions hold for every edge, the collection of dictionaries δ=oo∈∪ℰδ=\D_o\_o represents a natural transformation [15] from the representation sheaf J to the signal sheaf F δ:J⇒F.δ:J F\,. (16) The identification of a natural transformation δ from J to F is useful not only to reconstruct x from s, but also to transport consistency properties from one sheaf to the other, as it will be extensively shown in the following. Let us represent the node and edge components of δ as block-diagonal matrices 0≔blkdiag(ii∈),1≔blkdiag(ee∈ℰ),D_0 (\D_i\_i )\,,\;\;D_1 (\D_e\_e )\,, (17) acting respectively on 0- and 11-cochains, 0:0(;J)→0(;F)D_0:C^0 (G;J ) ^0 (G;F ) and 1:1(;J)→1(;F)D_1:C^1 (G;J ) ^1 (G;F ). Then, it is not difficult to prove the following. Proposition I.3 (Global section correspondence). Let δ as in Eq. 16 be a natural transformation with node component 0D_0. Then: (i) If ∈Γ(;J)s∈ (G;J ), then =0x=D_0s satisfies ∈Γ(;F)x∈ (G;F ). (i) Conversely, if ∈Γ(;F)x∈ (G;F ) and ∈Im(0)x (D_0 ), then =0⊤s=D_0 x satisfies ∈Γ(;J)s∈ (G;J ) and =0x=D_0s. Consequently, 0D_0 restricts to a linear isomorphism Γ(;J)≅Γ(;F)∩Im(0),Im(0)≔⨁i∈Im(i). (G;J )\; \; (G;F ) (D_0 )\,, (D_0 ) _i Im (D_i )\,. (18) Proof. See Supp. S2. ∎ Remark 4 (Interpretation of image containment in SP). The image-containment condition Im(e)⊆Im(e)Im (A_e ) (D_e ) in Prop. I.1 has a natural interpretation in SP: the edge dictionary eD_e must be expressive enough to represent any signal that are transported to the edge stalk from node i and node j through i⊴eF_i e and j⊴eF_j e. In other words, i⊴eF_i e and j⊴eF_j e should not map the node signal models Im(i)Im (D_i ) and Im(j)Im (D_j ) outside the edge signal model Im(e)Im (D_e ). IV Spectral theory over a network sheaf In this section, we introduce the Sheaf Fourier Transform (SFT) as a generalization of the GFT. Since the signal sheaf F and the representation sheaf J are related through the natural transformation δ in Eq. 16, the SFT can be defined on either sheaf. Unless otherwise specified, we develop the spectral theory on the representation sheaf, whose lower-dimensional stalks provide a more compact representation while preserving the spectral structure of the signal sheaf (cf. Sec. IV-B). IV-A Sheaf Fourier Transform Consider the representation sheaf J over the graph G, where ||=N|N|=N and |ℰ|=E|E|=E. Generalizing the approach used in GSP, the spectral analysis of sheaf signals is based on the eigendecomposition of the sheaf Laplacian with respect to the inner product induced by the metric tensor ℍ0H_0. Since JL_J is self-adjoint with respect to ⟨⋅,⋅⟩C0 ·,\,· _C^0, the corresponding Euclidean symmetric operator (cf. Supp. S1-B) is ~J=ℍ01/2Jℍ0−1/2=~~⊤,~⊤~=. L_J=H_0^1/2L_JH_0^-1/2= U U , U U=I. (19) Defining =ℍ0−1/2~,U=H_0^-1/2 U\,, (20) we obtain the generalized eigendecomposition J=,⊤ℍ0=,L_JU=U , H_0U=I, (21) where =diag(λ1,…,λC) =diag( _1,…, _C), 0≤λ1≤⋯≤λC0≤ _1≤·s≤ _C. The columns ℓ=1C\u_ \_ =1^C of U are called the sheaf Fourier modes, while the corresponding eigenvalues λℓ\ _ \ are the sheaf frequencies. Each mode ℓ=[(ℓ1)⊤,…,(ℓN)⊤]⊤u_ =[(u_ ^1) ,…,(u_ ^N) ] contains one block ℓiu_ ^i for every node stalk. Definition 1 (Sheaf Fourier Transform). The Sheaf Fourier Transform (SFT) of ∈C0(;J)s∈ C^0(G;J) is the vector of coefficients ^=⊤ℍ0. s=U H_0s\,. (22) The ℓ -th Fourier coefficient is ^(λℓ)=⟨,ℓ⟩C0=∑i∈i⊤iℓi. s( _ )= ,\,u_ _C^0= _i s_i H_iu_ ^i\,. (23) The inverse SFT is =∑ℓ=1L^(λℓ)ℓ=^.s= _ =1^L s( _ )\,u_ =U s\,. (24) The SFT computes the coefficients of the expansion of a sheaf signal over the orthonormal basis formed by the eigenvectors of the sheaf Laplacian. In complete analogy with GSP, the eigenvalues of the Laplacian play the role of frequencies. However, unlike the graph Laplacian, whose spectrum reflects only the graph topology, the spectrum of the sheaf Laplacian depends jointly on the graph topology, the restriction maps, and the local metric tensors. In particular, the sheaf frequency λℓ _ measures the inconsistency of the corresponding Fourier mode over the whole sheaf: λℓ=TV(ℓ)=∑e=(i,j)∈ℰ‖i⊴eℓi−j⊴eℓj‖e2. _ =TV(u_ )= _e=(i,j) \|J_i eu_ ^i-J_j eu_ ^j\|_H_e^2\,. (25) From this perspective, the sheaf Fourier modes admit a variational interpretation. They constitute the sequence of ℍ0H_0-orthonormal sheaf signals of progressively increasing total inconsistency: the first mode minimizes the total inconsistency among all unit-norm signals, while each subsequent mode minimizes it subject to being orthogonal to all preceding modes. This interpretation extends the classical variational characterization of the GFT to network sheaves: while graph Fourier modes minimize variation over the graph topology, sheaf Fourier modes minimize inconsistency jointly induced by the graph topology, the restriction maps, and the local Hilbert-space geometry. IV-B Spectral correspondence between the signal and representation sheaves The SFT introduced in the previous subsection can be defined either on the signal sheaf F or on the representation sheaf J. Owing to the natural transformation δ relating the two sheaves, however, the two spectral domains are not independent. As shown next, the corresponding sheaf Laplacians are intertwined by the natural transformation, implying that their spectral decompositions are in one-to-one correspondence within the representation subspace. Consequently, the SFT computed on the representation sheaf faithfully captures the spectral properties of the original signal sheaf while operating on lower-dimensional, signal-model-aware, stalks. Theorem IV.1 (Intertwining of the sheaf Laplacians). Let δ as in Eq. 16 with node component 0D_0 and edge component 1D_1 as in Eq. 17, respectively. Assume that the signal sheaf and the representation sheaf are endowed with metric tensors i,e\G_i,G_e\ and i,e\H_i,H_e\ in Eqs. 10 and 11. Furthermore, consider that the restriction maps of the signal sheaf are the metric-compatible lifts of those of the representation sheaf: i⊴e=ei⊴ei−1i⊤i,∀i⊴e.F_i e=D_e\,J_i e\,H_i^-1D_i G_i, ∀\,i e. (26) Then the corresponding sheaf Laplacians satisfy the intertwining relation F0=0J.L_F\,\,D_0=D_0\,\,L_J\,. (27) Proof. See Supp. S2. ∎ This proposition has an immediate consequence on the spectral theory, as established next. Corollary IV.2 (Spectral correspondence). Let J=μL_Jv= \,. Then F0=μ0.L_FD_0v=μ\,D_0v\,. (28) Hence, every eigenvalue of the representation sheaf Laplacian is an eigenvalue of the signal sheaf Laplacian, and the natural transformation maps every eigenspace of JL_J into the eigenspace of FL_F associated with the same eigenvalue. Additionally, since ⟨0,0⟩0(;F)=⟨,⟩0(;J), _0v,D_0w _C^0(G;F)= ,w _C^0(G;J)\,, (29) every ℍ0H_0-orthonormal eigenbasis of JL_J is mapped into a 0G_0-orthonormal set of eigenvectors of FL_F. Proof. See Supp. S2. ∎ The previous corollary shows that the representation sheaf is not merely a low-dimensional description of the signal sheaf, but a spectrally faithful representation of it. The natural transformation preserves the eigenvalues and maps the Fourier modes of the representation sheaf into Fourier modes of the signal sheaf with identical frequencies. Consequently, all spectral quantities associated with the representation sheaf admit an equivalent interpretation on the signal sheaf. In the remainder of the paper, we therefore develop the spectral theory on the representation sheaf, without loss of information within the subspace generated by the dictionaries. IV-C Intrinsic spectral representation Unlike graph Laplacians, whose eigenvalues are generically simple, the spectrum of a sheaf Laplacian often exhibits eigenvalues with multiplicity greater than one. This phenomenon is not accidental, but reflects the geometric structure encoded by the restriction maps. Whenever the latter transport information coherently across the sheaf, the associated eigenspaces acquire additional degrees of freedom, leading to repeated eigenvalues. Supp. S3 illustrates two representative mechanisms generating spectral multiplicity, namely the connection graph [38] and the hierarchical Stiefel embedding [36]. Importantly, spectral multiplicity makes the individual SFT coefficients depending on the particular orthonormal basis selected within each eigenspace. Consequently, the Fourier coefficients associated with individual eigenvectors are not intrinsic properties of either the signal or the sheaf. The intrinsic spectral objects are instead the eigenspaces of the sheaf Laplacian. Let μkk=1K\ _k\_k=1^K denote the set of distinct eigenvalues of JL_J. The eigenspace associated with the frequency μk _k is ℰk=ker(J−μkC),E_k= \! (L_J- _kI_C )\,, (30) with dimension mkm_k. Let k∈ℝC×mkU_k ^C× m_k be any ℍ0H_0-orthonormal basis of ℰkE_k, namely k⊤ℍ0k=mkU_k H_0U_k=I_m_k. Although the basis kU_k is not unique, the orthogonal projector onto ℰkE_k, k=kk⊤ℍ0,P_k=U_kU_k H_0, (31) is uniquely determined by the eigenspace and is therefore independent of the particular basis. Then, the intrinsic spectral component of ∈C0(;J)s∈ C^0(G;J) at frequency μk _k is its orthogonal projection onto ℰkE_k, k=k∈ℰk.s_k=P_ks _k\,. (32) Since the eigenspaces are mutually ℍ0H_0-orthogonal and span C0(;J)C^0(G;J), the projectors satisfy kℓ=δkℓk,∑k=1Kk=L,J=∑k=1Kμkk,P_kP_ = _k P_k\,, _k=1^KP_k=I_L\,, _J= _k=1^K _kP_k\,, (33) which is the spectral decomposition of the sheaf Laplacian. The signal then admits the intrinsic spectral decomposition =∑k=1Kk,s= _k=1^Ks_k, (34) where each component ks_k in (32) contains the portion of the signal associated with the frequency μk _k. Unlike the classical SFT coefficients, the decomposition in Eq. 34 is invariant with respect to any orthogonal change of basis within the eigenspaces. Furthermore, the energy carried by the k-th spectral component is ‖k‖ℍ02=⊤ℍ0k,\|s_k\|_H_0^2=s H_0P_ks, (35) and Parseval’s identity becomes ‖ℍ02=∑k=1K‖k‖ℍ02.\|s\|_H_0^2= _k=1^K\|s_k\|_H_0^2. (36) In the absence of repeated eigenvalues, every eigenspace is one-dimensional and the proposed formulation therefore constitutes a natural extension of the GFT to network sheaves: it preserves the classical spectral representation whenever the spectrum is simple, while remaining well-defined and invariant under orthogonal changes of basis within the eigenspaces when repeated eigenvalues occur. V Sheaf filtering Filtering is a fundamental operation in SP. In GSP, linear shift-invariant filters are functions of the graph Laplacian, whose spectrum defines the graph frequency domain. This framework extends naturally to network sheaves by replacing the graph Laplacian with the sheaf Laplacian. Thanks to the spectral correspondence established previously, filtering can be developed directly on the representation sheaf, yielding lower-dimensional filters that preserve the spectral behavior of their counterparts on the signal sheaf. A linear sheaf filter is any matrix function of the sheaf Laplacian. Among the possible choices, polynomial filters are particularly appealing, since they provide a parsimonious parameterization requiring only a small number of coefficients while preserving locality on the graph. Accordingly, we consider the class of polynomial filters =∑q=0QaqJq,y= _q=0^Qa_qL_J^\,qs, (37) where aqq=0Q\a_q\_q=0^Q are the filter coefficients. Using the spectral decomposition of JL_J in Eq. 33, the polynomial filter (37) can be rewritten as =∑k=1Kh(μk)kwithh(μ)=∑q=0Qaqμq;y= _k=1^Kh( _k)\,P_ks h(μ)= _q=0^Qa_qμ^q\,; (38) where h(μ)h(μ) in (38) is the frequency response of the filter. Let gkg_k denote the desired gain at the distinct sheaf frequency μk _k. Since a polynomial filter satisfies hk≔h(μk)=∑q=0Qaqμkq,k=1,…,K;h_k h( _k)= _q=0^Qa_q _k^q, k=1,…,K\,; (39) the filter coefficients can be selected by solving minaqq=0Q∑k=1K|gk−∑q=0Qaqμkq|2. _\a_q\_q=0^Q _k=1^K |g_k- _q=0^Qa_q _k^q |^2. (40) Equivalently, defining the Vandermonde matrix ∈ℝK×(Q+1) ^K×(Q+1) with entries []kq=μkq[ ]_kq= _k^q, q=0,…,Qq=0,…,Q, and =[g1,…,gK]⊤g=[g_1,…,g_K] , one obtains ⋆=argmin∈ℝQ+1‖−‖22=†,a = *arg\,min_a ^Q+1\|g- a\|_2^2= g, (41) where (⋅)†(·) denotes the Moore–Penrose pseudoinverse. If Q≥K−1Q≥ K-1 and the frequencies μk\ _k\ are distinct, the mask can be interpolated exactly. Importantly, the filter design depends only on the distinct frequencies μk _k and the associated spectral projectors kP_k, without requiring a choice of eigenbasis within degenerate eigenspaces. In particular, h(J)=∑k=1Khkk,h(L_J)= _k=1^Kh_kP_k, (42) meaning that the same gain hkh_k is applied to the entire intrinsic spectral component kP_ks. Thus, filtering is intrinsically defined at the eigenspace level and is invariant to the particular orthonormal basis chosen within each eigenspace. The intertwining relation established in Thm. IV.1 naturally extends to every polynomial filter. Corollary V.1 (Spectral correspondence of polynomial filters). Let h(λ)=∑q=0Qaqλq.h(λ)= _q=0^Qa_qλ^q. (43) Then h(F)0=0h(J).h(L_F)\,D_0=D_0\,h(L_J). (44) Consequently, filtering and lifting commute: filtering the representation signal and subsequently lifting it to the signal sheaf produces exactly the same result as first lifting the signal and then filtering it on the signal sheaf. Proof. See Supp. S2. ∎ The previous corollary highlights one of the main practical advantages of the proposed framework. Since every function of the sheaf Laplacian commutes with the natural transformation, spectral processing can be performed directly on the lower-dimensional representation sheaf without loss of information. Consequently, the computational complexity is governed by the dimensions of the representation stalks rather than those of the original signal stalks. More generally, the representation sheaf provides the natural computational domain for spectral processing over network sheaves, combining reduced dimensionality with complete preservation of the spectral structure. VI Sampling over a network sheaf Sampling concerns the recovery of a signal from a subset of its observed values. In GSP, perfect recovery from samples collected at a subset of the nodes requires the signal to be bandlimited with respect to the graph Laplacian [5]. We extend this framework to network sheaves. Unlike GSP, where each sample is a scalar associated to a node, in our case a sheaf associates a vector i∈F(i)x_i∈ F(i) with each node. Consequently, sampling selects a subset of the overall D-dimensional 0-cochain x and might involve the selection (or not) of some nodes and the observation of a subset of entries in each local vector i∈F(i)x_i∈ F(i). Given a sampling set ⊆1,…,DS \1,…,D\, we define the binary sampling matrix ∈0,1||×D, _S∈\0,1\^|S|× D, (45) whose rows are the canonical vectors corresponding to the sampled coordinates. The observed signal is therefore =.x_S= _Sx\,. (46) Equivalently, sampling can be represented through the orthogonal projector =⊤,M_S= _S _S, (47) while ⟂=D−M_S =I_D-M_S projects onto the unobserved coordinates. VI-A Sampling and recovery of sheaf signals Let μkk=1K\ _k\_k=1^K denote the distinct eigenvalues of the signal sheaf Laplacian FL_F, with associated eigenspaces ℱkk=1K\F_k\_k=1^K and 0G_0-orthogonal projectors kk=1K\P_k\_k=1^K. For a frequency index set ⊆1,…,KK \1,…,K\, define the bandlimiting projector =∑k∈k.B_K= _k P_k\,. (48) The corresponding bandlimited subspace is ℬ=Im(),B_K=Im(B_K)\,, (49) whose dimension is b=rank()=∑k∈mk,b_K=rank(B_K)= _k m_k, where mk=dim(ℱk)m_k= (F_k). A signal ∈C0(;F)x∈ C^0(G;F) is said to be bandlimited to the frequency set K if =.B_Kx=x\,. (50) For computational purposes, let ∈ℝD×bV_K ^D× b_K contain any 0G_0-orthonormal basis of ℬB_K. Then =⊤0,B_K=V_KV_K G_0\,, (51) and every bandlimited signal admits the representation =,x=V_K α, (52) for a unique coefficient vector ∈ℝb α ^b_K. Notice that the notion of bandlimitedness depends only on the intrinsic spectral subspace ℬB_K, not on the particular orthonormal basis V_K used to represent it. Theorem VI.1 (Sampling theorem for sheaf signals). Let ∈ℬx _K and let ∈ℝD×bV_K ^D× b_K be any 0G_0-orthonormal basis of ℬB_K, where b=dim(ℬ)b_K= (B_K). Then, x can be uniquely recovered from the samples =x_S= _Sx (53) if and only if rank()=b,rank\! ( _SV_K )=b_K, (54) or, equivalently, ℬ∩ker()=.B_K∩ ( _S)=\0\. When these conditions hold, x can be uniquely recovered as =()†.x=V_K ( _SV_K ) x_S. (55) Proof. See Supp. S2. ∎ Remark 5 (Geometric interpretation). The rank condition in Eq. 54 holds for arbitrary metric tensors. When the sampling projector =⊤M_S= _S _S is orthogonal with respect to the Hilbert-space inner product induced by 0G_0 (e.g., when 0=DG_0=I_D, or more generally whenever 0=0M_SG_0=G_0M_S), the recovery condition admits the equivalent characterization ‖⟂‖0<1. \|B_KM_S \|_G_0<1. (56) The operator ⟂B_KM_S measures the largest bandlimited component that may remain hidden in the unsampled coordinates. Thus, the above condition guarantees that no nonzero bandlimited signal is invisible to the sampling operator. The proof follows [5], replacing the Euclidean inner product with the 0G_0-induced Hilbert product. Corollary VI.2 (Minimum number of samples). A necessary condition for the perfect recovery of a sheaf signal ∈ℬx _K is ||≥b,|S|≥ b_K, (57) where b=dim(ℬ)b_K= (B_K) is the bandwidth of the signal. Proof. See Supp. S2. ∎ It is worth to point out that the condition in Eq. 57 is necessary but not sufficient: the sampled coordinates must also be suitably located so that _SV_K has full column rank. This distinction is especially important for sheaf signals: two sampling sets with the same cardinality may have very different recovery properties because they may select different nodes and different entries within the corresponding stalks. VI-B Band splitting induced by the natural transformation δ When the signal sheaf is induced by a representation sheaf through a natural transformation, the local signal model induces a natural decomposition of every spectral band. This decomposition reveals that only the model-aligned component needs to be recovered whenever the signal is known to belong to the representation subspace. Let δ:=0ℍ0−10⊤0 _δ:=D_0H_0^-1D_0 G_0 denote the 0G_0-orthogonal projector onto Im(0)Im (D_0 ). The projector is well defined whenever F arises, via δ:J⇒Fδ:J F, as the lift of the representation sheaf J (Thm. IV.1). As we show next, δ _δ then reveals a natural, model-aligned splitting of every band. Corollary VI.3 (δ-informed splitting of the band). Every band ℬB_K splits 0G_0-orthogonally as ℬ=Im(∥)⊕Im(⟂), _K=Im (B_K ) (B_K )\,, (58) where ∥:=δ,⟂:=(−δ); _K = _δB_K\,, _K =(I- _δ)B_K\,; into a component aligned with the local signal model i=iix_i=D_is_i, and one orthogonal to it. The aligned component admits the explicit form Im(∥)=Im(0),Im(B_K )=Im(D_0U_K)\,, (59) where U_K collects ℍ0H_0-orthonormal bases kU_k of the eigenspaces ℰkE_k of JL_J, k∈k , with mk′≔dim(ℰk)≤mkm _k (E_k)≤ m_k. Proof. See Supp. S2. ∎ Interestingly, Cor. VI.3 shows that, whenever the signal of interest is known to obey the local model, i.e., to lie in Im(0)Im (D_0 ), exact recovery does not require resolving the whole band ℬB_K, but only its aligned component Im(∥)Im(B_K ): since mk′≤mkm _k≤ m_k for every k∈k (often strictly, and possibly mk′=0m _k=0) the aligned part can have dimension considerably smaller than the nominal bandwidth b_K. In practice, Thm. VI.1 can then be applied with =0U=D_0U_K in place of V_K, targeting Im(∥)Im(B_K ): the effective bandwidth to be resolved shrinks from b_K to ∑k∈mk′ _k m _k, and so does the number of samples in Cor. VI.2 sufficient for perfect recovery. Sec. VII-C illustrates this reduction empirically. Before concluding this section, it is worth to point out that, unlike GSP, where recoverability depends only on the graph topology, in network sheaves it is jointly determined by the graph topology, the restriction maps, and the metric tensors defining the sheaf geometry. Thm. VI.1 therefore establishes a direct connection between the algebraic structure of the sheaf and the recoverability of bandlimited signals. VI-C Sampling Strategy The recovery condition of Cor. VI.2 characterizes when a given S enables perfect recovery, but does not prescribe how to choose it. Let ∈ℝD×bV ^D× b denote any 0G_0-orthonormal basis of a target subspace ℬ⊆C0(;F)B C^0(G;F), b=dimℬb= . By default =V=V_K and b=b=b_K, the full band of Thm. VI.1, but V may equally be taken as the δ-aligned basis 0D_0U_K of Cor. VI.3, with b=b∥:=∑k∈mk′=dimIm(∥)b=b_K := _k m_k = Im (B_K ), whenever the signal is known to lie in Im(0)Im (D_0 ). The design problem is to find a minimum-size sampling set S such that rank()=brank( _SV)=b, which is NP-hard in general [39]. Greedy algorithm. Given as minimum sampling budget the signal bandwidth b (cf. Cor. VI.2), we seek the sampling set that maximizes the rank of the sampling operator restricted to the bandlimited subspace (cf. Eq. 54): max⊆ℐ _S rank() ( _SV) (60) s.t. ||=b; |S|=b\,; where ℐI denotes the set of all sheaf signal entries. Defining f()=rank()f(S)=rank( _SV), the objective is the rank function of a linear matroid, and is therefore monotone and submodular [40]. Consequently, a greedy algorithmic approach provides a (1−1/e)(1-1/e)-approximation to the optimal solution. Tie-breaking. Identify each global coordinate in 1,…,D\1,…,D\ with a pair (i,k)(i,k), i∈i , k=1,…,dik=1,…,d_i, via the stalk offsets, and let ~:=01/2 V:=G_0^1/2V, which is Euclidean-orthonormal since ~⊤~=⊤0=b V V=V G_0V=I_b. Among all pairs that increase the rank, ties are broken by the row norm ηi,k=‖k⊤~[stalki,:]‖2, _i,k= \|e_k V[stalk_i,:] \|_2, (61) which measures how much of the target subspace’s energy raw coordinate k at node i captures. Crucially, the atoms of selection are always raw stalk coordinates, never dictionary atoms: _S remains the plain binary sampling matrix of Eqs. 46 and 47 throughout, regardless of which target basis V is used. Alg. 1 summarizes the greedy sampling strategy. Algorithm 1 Greedy Sampling for Bandlimited Sheaf Signals 0: 0G_0-orthonormal basis ∈ℝD×bV ^D× b of the target subspace ℬB (e.g. V_K, Thm. VI.1, or 0D_0U_K, Cor. VI.3) 0: Sampling set S, operator _S ←∅S← , r←0r← 0, ←(i,k):i∈,k=1,…,diC←\(i,k):i ,\,k=1,…,d_i\, ~←01/2 V _0^1/2V ηi,k←‖k⊤~[stalki,:]‖2 _i,k←\|e_k V[stalk_i,:]\|_2 for all (i,k)∈(i,k) while r<br<b do +←(i,k)∈∖:rank(∪(i,k))>rC^+←\(i,k) :rank( _S∪\(i,k)\V)>r\ if +=∅C^+= then break end if (i∗,k∗)←argmax(i,k)∈+ηi,k(i^*,k^*)← _(i,k) ^+ _i,k ←∪(i∗,k∗)S ∪\(i^*,k^*)\, r←r+1r← r+1 end while return S, ∈0,1||×D _S∈\0,1\^|S|× D (Eq. 46) VII Empirical assessment Our framework is validated through three numerical examples. The first investigates filtering on synthetic data, the second demonstrates the proposed framework on a real-world multi-view motion capture dataset, and the third shows an application of sampling for financial portfolio reconstruction. VII-A Filtering signals from correlated latent factors We consider the network sheaf topology shown in Fig. 2: two cliques of 55 nodes each, connected by 33 bridge edges. Each clique k∈1,2k∈\1,2\ is associated with a latent factor k=Ck+ky_k=C_km+ _k in ℝrR^r (r=10r=10), where ∼(,r)m ( 0,I_r) is a confounder shared across both cliques, inducing correlation between 1y_1 and 2y_2, and k∼(,r) _k ( 0,I_r) is an idiosyncratic component specific to clique k, independent of m and of k′ _k for k′≠k ≠ k. The scalars C1,C2∈[0,1]C_1,C_2∈[0,1] control the correlation between the two factors. For simplicity, we set C1=C2=C_1=C_2=C. Each node i carries a raw signal i=ik(i)x_i=H_iy_k(i), where k(i)∈1,2k(i)∈\1,2\ denotes the clique containing i (i.e., k(i)=1k(i)=1 for i∈0,…,4i∈\0,…,4\ and k(i)=2k(i)=2 for i∈5,…,9i∈\5,…,9\), and i∈ℝd×rH_i ^d× r (d=20d=20) is a random linear embedding, drawn independently across nodes with no correlation between them. On intra-clique edges, the consistency condition requires both endpoints to recover the same latent factor ky_k, giving i⊴e=pinv(i)∈ℝr×dF_i e=pinv(H_i) ^r× d. On bridge edges, the consistency condition instead requires the two endpoints to agree on the MMSE estimate of the shared component CCm, yielding i⊴e=βpinv(i)F_i e=β\,pinv(H_i), with β=C2C2+1=Cov(C,i)/Var(i)β= C^2C^2+1=Cov(Cm,y_i)/Var(y_i) the optimal linear regression coefficient for estimating CCm from iy_i. The representation sheaf J is then obtained via i⊴e=i⊴eiJ_i e=F_i eD_i, where i∈St(d,c)D_i (d,c) (c=10c=10) is a local DCT dictionary, ensuring naturality by construction. From the restriction maps i⊴eJ_i e, we assemble the sheaf Laplacian JL_J and compute its eigendecomposition. Let μk\ _k\ and ℰk\E_k\ denote the distinct eigenvalues and eigenspaces of JL_J. A bandlimited test signal is synthetized from Ksig=15K_sig=15 selected eigenspaces using a decaying energy profile wk\w_k\: =∑k∈wkkk,k∼Uniform(|ℰk|−1),s= _k w_k\,U_k α_k\,, α_k (S^|E_k|-1 )\,, (62) where K is the index set of the KsigK_sig selected eigenspaces and |ℰk|−1S^|E_k|-1 denotes the unit sphere in ℝ|ℰk|R^|E_k|, ensuring basis-invariance within each eigenspace. By construction, s is bandlimited with respect to JL_J—its energy is supported on K—but is not in general a global section of J, as K need not be restricted to the kernel eigenspaces μk=0\ _k=0\. The signal is then lifted to the raw domain stalk-wise via i=iix_i=D_is_i. Figure 2: Sensor network topology of the filtering experiment. Once we generate a set of bandlimited signals, we corrupt them with addiditve white Gaussian noise at different signal-to-noise ratio (SNR) levels. We aim at assessing the filtering capabilities of the two sheaf representations JL_J and FL_F. Thus, we implement spectral filtering via hard-thresholding: we rank the eigenspaces by the signal’s energy in each, then project the signal onto the highest-ranked ones. The number of eigenspaces is chosen by exploring the tradeoff between bandwidth and reconstruction error, and reporting the optimal value. The filtering is implemented in the representation domain, so that the reconstruction error is computed after lifting to the signal domain. We compare the performance of filtering with respect to JL_J with (i) filtering in the raw domain with FL_F, (i) filters based only on local dictionaries projection, and (i) a GSP baseline using a graph Laplacian learned from additional noiseless training signals with the algorithm from [41]. For the latter, we consider two variants to ensure a fair comparison in terms of filtering-complexity: (i) GLSigRep-InterOnly, where intra-stalk edges are forbidden, and (i) GLSigRep-TopoMasked, where inter-stalk edges are also constrained to adhere to the topology in Fig. 2. Figure 3: Reconstruction error (left panel) and fraction of modes at the optimal bandwidth (top-right corner). Fig. 3 reports the NMSE against the SNR for all considered approaches, with the top-right inset showing each method’s optimal relative bandwidth (i.e., the fraction of eigenvectors retained). The representation sheaf JL_J strongly outperforms every baseline, both in NMSE and filter sparsity. The GSP baselines fail to localize the signal spectrally: they require a much larger bandwidth, yet their NMSE does not improve on simply projecting onto the local dictionaries. The signal sheaf FL_F likewise fails to capture the signal’s bandlimitedness, since the spectral intertwining of Thm. IV.1 does not hold here: the restriction maps of F are not derived from J, but the converse. Filtering with the two sheaves thus yields different results, with F performing significantly worse. VII-B Denoising inter-frame displacement in CMU Panoptic We consider the CMU Panoptic Studio dataset111http://domedb.perception.cs.cmu.edu, sequence 171204_pose1, which contains synchronized recordings of a human subject observed by a calibrated multi-camera system. For each frame, the dataset provides the 33D coordinates of the 1919 body joints (COCO19 format, world coordinates in cm) and the corresponding synchronized observations from 3131 calibrated HD cameras. Each camera c is described by its intrinsic calibration cK_c and extrinsic parameters (c,c)(R_c,t_c), defining the nonlinear projection πc(c,c,c):ℝ3→ℝ2 _c(K_c,R_c,t_c):R^3 ^2. The signal of interest is the inter-frame displacement of the body joints between two consecutive frames t and t+Δt+ . The graph is instantiated at each reference frame t and comprises two distinct sets of nodes. The first consists of one node for each skeleton joint j, with stalk F(j)≅ℝ3F(j) ^3 carrying the corresponding 33D inter-frame displacement j(t,Δ)=j(t+Δ)−j(t)d_j(t, )=x_j(t+ )-x_j(t). The second consists of one node for each visible joint-camera pair (c,j)(c,j), with stalk F(c,j)≅ℝ2F(c,j) ^2 carrying the corresponding image displacement cj=πc(j(t+Δ))−πc(j(t))u_cj= _c(x_j(t+ ))- _c(x_j(t)). Three families of edges are considered: bone edges, connecting adjacent joints in the kinematic skeleton; projection edges, linking each joint j to its image observations (c,j)(c,j); and view edges, connecting observations (c1,j)(c_1,j) and (c2,j)(c_2,j) of the same joint across different cameras. Bone edges eb=(j1,j2)e_b=(j_1,j_2) are associated with the edge stalk F(eb)≅ℝ3F(e_b) ^3 and restriction maps j1⊴eb=j2⊴eb=3F_j_1 e_b=F_j_2 e_b=I_3, so that consistency enforces locally translational motion between adjacent joints in the canonical coordinate system. Projection edges ep=(j,(c,j))e_p=(j,(c,j)) are instead frame-dependent. They are associated with the edge stalk F(ep)≅ℝ2F(e_p) ^2, the restriction map j⊴ep=cj=∂πc∂|j(t)∈ℝ2×3F_j e_p=T_cj= . ∂ _c |_x_j(t) ^2× 3 obtained by linearizing the camera projection around the reference joint position j(t)x_j(t), and (c,j)⊴ep=2F_(c,j) e_p=I_2. Since ker(j⊴ep) (F_j e_p ) coincides with the viewing ray cjr_cj of camera c passing through j(t)x_j(t), a single image observation determines the 33D displacement only up to its component along the viewing direction. To encode the complementary information provided by multiple cameras, view edges connect observations of the same joint across different cameras. These edges are associated with a one-dimensional stalk F(e)≅ℝF(e) and restriction maps (c1,j)⊴e=⊤c1jc2j†F_(c_1,j) e=m T_c_1jT_c_2j and (c2,j)⊴e=⊤F_(c_2,j) e=m , where ⟂c2=c2jc1jm _c_2=T_c_2jr_c_1j and ‖=1\|m\|=1. Up to a scaling factor, this is the unique linear relation between the two image displacements that is satisfied by every underlying 33D displacement, and it degenerates only when the two camera centers are collinear with the observed joint. The reduced edge dimension (de=1<di=dj=2d_e=1<d_i=d_j=2) also illustrates the interpretation of low-dimensional edge stalks as compression bottlenecks discussed in Sec. I. Finally, the 33D and image displacement signals are normalized independently to unit norm before constructing the sheaf signals. Figure 4: NMSE versus relative bandwidth, for different SNR and methods, over the Panoptic CMU experiment. We compare the sheaf Laplacian FL_F against an isotropic, GSP-like operator acting on the joint coordinates. This baseline is the same network used in the sheaf construction, restricted to the nodes representing the skeletal joints and stripped of the camera-view enrichment layer; the operator can only enforce local translational consistency. The comparison isolates denoising performance on the signal of interest: the joint displacement between consecutive frames. Fig. 4 shows the NMSE versus the relative bandwidth for top-k spectral filtering at different SNR levels when using the signal sheaf F and the GSP baseline, averaged over 88 noise realizations and 100100 frames. Since the sheaf depends on the reference frame, it is reconstructed at each frame by linearizing the camera projections around the current joint positions. F consistently achieves a lower NMSE than the GSP baseline across all bandwidth and SNR values, reflecting a more compact spectral representation of the signal. Indeed, it allows exploiting multi-view redundancy via the camera projections encoded in its restriction maps, resulting in more robust denoising. VII-C Recovering financial portfolios from sampled observations We consider a set U of financial stocks222The constituents of S&P 100 Index., with ||=101|U|=101. Five investors, labeled from A to E, buy and sell subsets of the stocks in U. Denote by i⊂U_i the subset corresponding to the i-th investor, consisting of did_i stocks, and let the k-th entry of a vector i∈ℝdix_i ^d_i be the fraction of the capital invested by investor i corresponding to the stock k∈ik _i. Further, the above subsets have heterogeneous size: dA=24,dB=30,dC=52,dD=71,dE=21d_A=24,\,d_B=30,\,d_C=52,\,d_D=71,\,d_E=21. We then define a graph G with five nodes =A,B,C,D,EN=\A,B,C,D,E\, and with line topology A−B−E−D−CA-B-E-D-C. For each edge e=(i,j)e=(i,j) of G, i∩j≠∅U_i _j≠ . In addition, AU_A and BU_B share no stocks with either CU_C or DU_D, thus EU_E acts as a bridge between these two clusters. Now, we build the sheaf F by defining the node stalks as F(i)≅ℝdiF(i) ^d_i, where the node signals are those vectors ix_i, i∈i . Thus, the 0-cochain is ∈ℝ198x ^198. Regarding the edge stalks, for each e=(i,j)e=(i,j) with di<djd_i<d_j, we let F(e)≡F(j)F(e)≡ F(j). For instance, for e=(C,D)e=(C,D), F(e)≡F(D)F(e)≡ F(D). The restriction maps compare two neighboring investors over the stocks they hold in common; we make this precise below, once we introduce the data-driven dictionaries, since that is how these maps are actually constructed. Each ix_i can be equivalently represented over a dictionary i∈St(di,c)D_i (d_i,c), whose columns represent principal components–in the financial domain interpreted as statistical risk factors–explaining most of the variance of the data (here we set c=5c=5). We build the dictionaries iD_i according to standard statistical factor models [42] using the time series of daily returns of the 101101 stocks over the period from Jan 1st1^st, 20222022 to Dec 31st31^st, 20242024, gathered from Yahoo Finance. The local signal models are thus i=iix_i=D_is_i, where is_i is the investor-specific representation of the fractions of invested money in ix_i over the 55 statistical risk factors. According to the local signal models and to the constructed signal sheaf F, the node stalks of the representation sheaf J are J(i)≅ℝcJ(i) ^c, i∈i , with valuations corresponding to is_i. Thus, the 0-cochain is ∈ℝ25s ^25. Similarly to F, for each edge, the edge stalk of J coincides with the representation stalk of the same reference node j already chosen for that edge in F, so that e=jD_e=D_j and j⊴e=cJ_j e=I_c. Since iD_i and jD_j correspond to different stock subsets, we compare two investors’ representations through the stocks they hold in common. Specifically, we pad each dictionary with zero rows to the full set U, obtaining ~i D_i and ~j D_j. Then, we set representation-level restriction map i⊴e=~j⊤~iJ_i e= D_j D_i. The restriction maps of F are obtained by lifting the representation-level restriction through the dictionaries via Eq. 26 (metric tensors here are the identity), thus guaranteeing exact spectral correspondence. The resulting sheaf Laplacian JL_J has kernel with dimension 55, and 2020 one-dimensional eigenspaces, for 2121 eigenspaces in total. By Cor. IV.2, every eigenvalue of JL_J is also an eigenvalue of FL_F, and the corresponding 2121 eigenspaces are embedded isometrically into those of FL_F. The kernel of FL_F, however, has dimension 178178: only 55 of these directions come from the lifted kernel of J, the remaining 173173 being orthogonal to every dictionary. A bandlimited test signal is then generated on J following the eigenspace-uniform procedure of Eq. 62, selecting Ksig=5K_sig=5 eigenspaces of JL_J (the 55-dimensional kernel plus four simple nonzero eigenvalues), for a b=9b_K=9-dimensional target band. By Cor. IV.2, lifting the generated representation to the signal sheaf F via the dictionaries yields a signal x with the same energy as s on the same KsigK_sig eigenspaces. Diagonalizing FL_F directly to isolate the band K would be misleading: its kernel mixes the 173173 dictionary-invisible directions with the 55 relevant ones. Instead, lifting the 99 eigenvectors of JL_J via δ yields, by Cor. IV.2, the relevant subspace of ker(F) (L_F) for the band K: this 99-dimensional lifted subspace is the target basis V used for sampling in Alg. 1. Given the target band K with b=9b_K=9, our goal is to reconstruct x from few of its own raw entries—i.e., individual portfolio-weight measurements i[k]x_i[k], the fraction invested by i in a specific stock k∈ik _i. Concretely, =Ψ⊤x_S= _S x according to Eq. 55, where the sampling operator Ψ _S is obtained through Alg. 1, whose atoms of selection are precisely these raw stock-level entries and never dictionary atoms: dictionaries only shape the 99-dimensional target subspace via the lift above, not which entries of x are queried. (a) (b) (c) Figure 5: (a) Reconstruction error (NMSE) vs number of selected samples for the tested methods. NMSE is clipped at 10−810^-8 level for readability. Concerning δ-driven sheaf sampling (cf. Cor. VI.3), we provide (b) the resulting greedy sample selection (white) with numbers indicating the sampling order, and (c) the energy decomposition across investors. The results are collected in Fig. 5(a)-(c). Specifically, Fig. 5(a) reports the NMSE of signal recovery versus the number of samples, comparing the proposed sheaf approach against the baselines of Sec. VII-A, each fed to the same greedy procedure but with its own target subspace and, consequently, its own minimum sampling budget. As we can see from Fig. 5(a), δ-driven reconstruction (using Cor. VI.3) is essentially exact: the NMSE of the 198198-dimensional sheaf signal x is on the order 10−2910^-29, consistent with Thm. VI.1 once S achieves rank(Ψ⊤)=b=9rank( _S V)=b_K=9, which Alg. 1 attains with exactly 99 raw measurements—the minimum possible by Cor. VI.2. Instead, diagonalizing FL_F directly and selecting, via Cor. IV.2, the eigenspaces overlapping the band ℬB_K (δ-agnostic F) requires b=182b=182 raw measurements for exact recovery, 20×20× more than the δ-driven approach. The reason is that the 55 dictionary-relevant directions from ker(J) (L_J) are conflated inside the 178178-dimensional kernel of FL_F, together with the 173173 dictionary-orthogonal ones. Any individual eigenvector returned by diagonalizing a degenerate eigenspace is arbitrary within it; recovering the true band exactly therefore requires retaining the eigenspace in full. Finally, the GSP baseline, is markedly worse in both the inter-only and the topology-masked variant. The NMSE remains order of magnitudes above, and exact reconstruction is only attained once every signal coordinate is sampled. Fig. 5(b) illustrates the sampled stock-level positions across investors by the δ-driven sheaf approach. As we can see from Fig. 5(b), the method selects 11 sample from A, 33 from B, and 55 from E. Conversely, C and D are never directly observed. This is consistent with the energy distribution for the generated signal at investor-level, illustrated in Fig. 5(c), where we can see how signal energy concentrates on A, B, and E, which is precisely why Alg. 1 targets them. Finally, the signal x is nearly locally consistent on E−D−CE-D-C, with TV(C,D)=0.089TV_(C,D)\!=\!0.089, TV(E,D)=0.017TV_(E,D)\!=\!0.017. Thus, the restriction maps alone tightly constrain the investments of C and D from their sampled neighbors, without requiring a direct measurement. In contrast, TV(A,B)=0.993TV_(A,B)\!=\!0.993 and TV(E,B)=0.748TV_(E,B)\!=\!0.748, justifying sampling from those investors. VIII Conclusions and future works We introduced a sheaf-theoretic framework for SP on graphs with heterogeneous local signal spaces. By modeling node and edge data as network sheaves valued in ℝ Hilb_R, the proposed framework extends GSP to settings where local signals differ in dimension, representation, and geometry. Building on this model, we developed a unified treatment of spectral representation, filtering, and sampling. A key ingredient of the framework is the use of natural transformations to relate signal and representation sheaves, enabling spectral processing in low-dimensional representation spaces while preserving the spectral structure of the original signals. These results establish a common foundation for SP over heterogeneous network data. Several directions deserve further investigations. An important theoretical question is the interplay between local and global bandlimitedness, where signals are simultaneously compressible within each stalk and bandlimited over the sheaf. Furthermore, in this paper we assumed the restriction maps and the graph topology to be known a priori. An interesting ongoing activity is how to learn both from data. On the learning side, a further interesting direction is task-oriented sheaf learning, where the sheaf structure is jointly optimized with downstream SP objectives, such as denoising, interpolation, or classification. Ultimately, extending the framework to cell complexes of arbitrary order would allow capturing multiway relations among heterogeneous signals. References [1] E. C. Strinati, P. Di Lorenzo et al., “Goal-oriented and semantic communication in 6G AI-native networks: The 6G-GOALS approach,” in 2024 Joint European Conference on Networks and Communications & 6G Summit. IEEE, 2024, p. 1–6. [2] D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Vandergheynst, “The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains,” IEEE Signal Processing Mag., vol. 30, no. 3, p. 83–98, 2013. [3] A. Ortega, P. Frossard, J. Kovačević, J. M. Moura, and P. Vandergheynst, “Graph signal processing: Overview, challenges, and applications,” Proceedings of the IEEE, vol. 106, no. 5, p. 808–828, 2018. [4] A. Sandryhaila and J. M. Moura, “Discrete signal processing on graphs,” IEEE Trans. on Signal Processing, vol. 61, no. 7, p. 1644–1656, 2013. [5] M. Tsitsvero, S. Barbarossa, and P. Di Lorenzo, “Signals on graphs: Uncertainty principle and sampling,” IEEE Trans. on Signal Processing, vol. 64, no. 18, p. 4845–4860, 2016. [6] G. Mateos, S. Segarra, A. G. Marques, and A. Ribeiro, “Connecting the dots: Identifying network structure via graph signal processing,” IEEE Signal Processing Mag., vol. 36, no. 3, p. 16–43, 2019. [7] S. Sardellitti, S. Barbarossa, and P. Di Lorenzo, “On the graph Fourier transform for directed graphs,” IEEE Journal of Selected Topics in Signal Processing, vol. 11, no. 6, p. 796–811, 2017. [8] P. Di Lorenzo, P. Banelli, E. Isufi, S. Barbarossa, and G. Leus, “Adaptive graph signal processing: Algorithms and optimal sampling strategies,” IEEE Trans. on Signal Processing, vol. 66, no. 13, p. 3584–3598, 2018. [9] S. Barbarossa and S. Sardellitti, “Topological signal processing over simplicial complexes,” IEEE Trans. on Signal Processing, vol. 68, p. 2992–3007, 2020. [10] M. T. Schaub, Y. Zhu, J.-B. Seby, T. M. Roddenberry, and S. Segarra, “Signal processing on higher-order networks: Livin’on the edge… and beyond,” Signal Processing, vol. 187, p. 108149, 2021. [11] M. Yang, E. Isufi, M. T. Schaub, and G. Leus, “Simplicial convolutional filters,” IEEE Trans. on Signal Processing, vol. 70, p. 4633–4648, 2022. [12] C. Battiloro, L. Testa, L. Giusti, S. Sardellitti, P. Di Lorenzo, and S. Barbarossa, “Generalized simplicial attention neural networks,” IEEE Trans. on Signal and Information Processing over Networks, vol. 10, p. 833–850, 2024. [13] E. Grimaldi, C. Battiloro, and P. Di Lorenzo, “Topological dictionary learning,” IEEE Trans. on Signal Processing, vol. 74, p. 200–214, 2026. [14] E. Isufi, G. Leus, B. Beferull-Lozano, S. Barbarossa, and P. Di Lorenzo, “Topological signal processing and learning: Recent advances and future challenges,” Signal Processing, vol. 233, p. 109930, 2025. [15] S. Mac Lane, Categories for the working mathematician. Springer, 1971, vol. 5. [16] J. M. Curry, Sheaves, cosheaves and applications. University of Pennsylvania, 2014. [17] A. Ayzenberg, T. Gebhart, G. Magai, and G. Solomadin, “Sheaf theory: From deep geometry to deep learning,” arXiv:2502.15476, 2025. [18] F. Grassi, A. Loukas, N. Perraudin, and B. Ricaud, “A time-vertex signal processing framework: Scalable processing and meaningful representations for time-series on graphs,” IEEE Trans. on Signal Processing, vol. 66, no. 3, p. 817–829, 2018. [19] J. S. Stanley, E. C. Chi, and G. Mishne, “Multiway graph signal processing on tensors: Integrative analysis of irregular geometries,” IEEE Signal Processing Mag., vol. 37, no. 6, p. 160–173, 2020. [20] A. Natali, E. Isufi, and G. Leus, “Forecasting multi-dimensional processes over graphs,” in Proc. of IEEE ICASSP 2020, p. 5575–5579. [21] R. Varma, H. Lee, J. Kovačević, and Y. Chi, “Vector-valued graph trend filtering with non-convex penalties,” IEEE Trans. on Signal and Information Processing over Networks, vol. 6, p. 48–62, 2020. [22] M. H. Trinh, C. V. Nguyen, Y. H. Lim, and H.-S. Ahn, “Matrix-weighted consensus and its applications,” Automatica, vol. 89, p. 415–419, 2018. [23] M. Robinson, “Asynchronous logic circuits and sheaf obstructions,” Electronic Notes in Theoretical Computer Science, vol. 283, p. 159–177, 2012. [24] —, “Understanding networks and their behaviors using sheaf theory,” in 2013 IEEE Global Conference on Signal and Information Processing. IEEE, 2013, p. 911–914. [25] —, “A sheaf-theoretic perspective on sampling,” in Sampling Theory, a Renaissance: Compressive Sensing and Other Developments. Springer, p. 361–399. [26] —, Topological signal processing. Springer, 2014, vol. 81. [27] —, “Sheaves are the canonical data structure for sensor integration,” Information Fusion, vol. 36, no. C, p. 208–224, 2017. [28] J. Hansen and R. Ghrist, “Toward a spectral theory of cellular sheaves,” Journal of Applied and Computational Topology, vol. 3, no. 4, p. 315–358, 2019. [29] —, “Opinion dynamics on discourse sheaves,” SIAM Journal on Applied Mathematics, vol. 81, no. 5, p. 2033–2060, 2021. [30] J. Hansen and T. Gebhart, “Sheaf neural networks,” in TDA & Beyond, 2020. [31] C. Bodnar, F. Di Giovanni, B. Chamberlain, P. Lio, and M. Bronstein, “Neural sheaf diffusion: A topological perspective on heterophily and oversmoothing in GNNs,” Advances in Neural Information Processing Systems, vol. 35, p. 18 527–18 541, 2022. [32] C. Battiloro, Z. Wang, H. Riess, P. Di Lorenzo, and A. Ribeiro, “Tangent bundle convolutional learning: From manifolds to cellular sheaves and back,” IEEE Trans. on Signal Processing, vol. 72, p. 1892–1909, 2024. [33] K. Tandon, J. Gould, T. Bhatia, F. Dominici, A. Ribeiro, and C. Battiloro, “Consistent geometric deep learning via Hilbert bundles and cellular sheaves,” arXiv preprint arXiv:2605.06395, 2026. [34] R. Ghrist and H. Riess, “Cellular sheaves of lattices and the Tarski Laplacian,” Homology, Homotopy and Applications, vol. 24, no. 1, 2022. [35] E. Grimaldi, M. E. Pandolfo, G. D’Acunto, S. Barbarossa, and P. Di Lorenzo, “Learning network sheaves for AI-native semantic communication,” in 2025 59th Asilomar Conference on Signals, Systems, and Computers. IEEE, 2025, p. 1692–1696. [36] G. D’Acunto, P. Di Lorenzo, and S. Barbarossa, “Networks of causal abstractions: A sheaf-theoretic framework,” arXiv preprint arXiv:2509.25236, 2026. [37] —, “Learning consistent causal abstraction networks,” in Proc. of IEEE ICASSP 2026, Barcelona, Spain. [38] F. Chung, W. Zhao, and M. Kempton, “Ranking and sparsifying a connection graph,” Internet Mathematics, vol. 10, no. 1-2, p. 87–115, 2014. [39] P. Di Lorenzo, S. Barbarossa, and P. Banelli, “Sampling and recovery of graph signals,” in Cooperative and Graph Signal Processing, 2018. [40] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher, “An analysis of approximations for maximizing submodular set functions,” Mathematical Programming, vol. 14, no. 1, p. 265–294, 1978. [41] X. Dong, D. Thanou, P. Frossard, and P. Vandergheynst, “Learning Laplacian matrix in smooth graph signal representations,” IEEE Trans. on Signal Processing, vol. 64, no. 23, p. 6160–6173, 2016. [42] G. Connor and R. A. Korajczyk, “Performance measurement with the arbitrage pricing theory: A new framework for analysis,” Journal of Financial Economics, vol. 15, no. 3, p. 373–394, 1986. Supplementary Material for “Sheaf-theoretic Signal Processing on Graphs: Spectral Theory, Filtering, and Sampling” Supplementary Section 1 Linear Algebra with Metric Tensors This appendix collects the algebraic identities involving metric tensors that are used throughout the paper. All results are standard in the theory of finite-dimensional inner product spaces; we gather them here for convenience and to fix notation. The material in this section is based on [1]. Supplementary Section 1-A Inner Products and Induced Norms Let ∈++dG _++^d be a symmetric positive definite matrix (a metric tensor) on ℝdR^d. The G-inner product and its induced norm are ⟨,⟩=⊤,‖=⊤,,∈ℝd. ,y _G=x Gy, \|x\|_G= x Gx, ,y ^d. (1) The pair (ℝd,⟨⋅,⋅⟩)(R^d, ·,· _G) is a finite-dimensional real Hilbert space. Since ∈++dG _++^d, it admits a unique symmetric positive definite square root 1/2∈++dG^1/2 _++^d satisfying 1/21/2=G^1/2G^1/2=G, and ⟨,⟩=⟨1/2,1/2⟩=(1/2)⊤(1/2), ,y _G= ^1/2x,G^1/2y _I=(G^1/2x) (G^1/2y), (2) so every G-inner product reduces to the standard Euclidean inner product after the change of variables ~=1/2 x=G^1/2x. For the space of 0-cochains C0(;F)≅ℝLC^0(G;F) ^L, L=∑idiL= _id_i, the global metric tensor is 0=blkdiag(ii∈)∈++LG_0=blkdiag(\G_i\_i ) _++^L, and the 0G_0-inner product is ⟨,⟩C0=⊤0=∑i∈i⊤ii, ,y _C^0=x G_0y= _i x_i G_iy_i, (3) where the second equality follows from the block-diagonal structure of 0G_0. Similarly, for 1-cochains C1(;F)≅ℝMC^1(G;F) ^M, M=∑edeM= _ed_e, the global metric is 1=blkdiag(ee∈ℰ)G_1=blkdiag(\G_e\_e ). Supplementary Section 1-B G-Adjoint of a Linear Map Let :(ℝd0,⟨⋅,⋅⟩0)→(ℝd1,⟨⋅,⋅⟩1)A:(R^d_0, ·,· _G_0)→(R^d_1, ·,· _G_1) be a linear map represented by a matrix ∈ℝd1×d0A ^d_1× d_0. Its (0,1)(G_0,G_1)-adjoint is the unique linear map ⋆:(ℝd1,⟨⋅,⋅⟩1)→(ℝd0,⟨⋅,⋅⟩0)A :(R^d_1, ·,· _G_1)→(R^d_0, ·,· _G_0) satisfying ⟨,⟩1=⟨,⋆⟩0∀∈ℝd0,∈ℝd1. ,y _G_1= ,A y _G_0 ∀\,x ^d_0,\,y ^d_1. (4) Expanding both sides using Eq. 1, ()⊤1=⊤⋆⊤0(Ax) G_1y=x A G_0y for all ,x,y, which gives ⋆=0−1⊤1.A =G_0^-1A G_1. (5) When 0=d0G_0=I_d_0 and 1=d1G_1=I_d_1, Eq. 5 reduces to the standard matrix transpose ⋆=⊤A =A . The sheaf coboundary adjoint ⋆=(0)−1⊤1B =(G_0)^-1B G_1 is the instance of Eq. 5 with =A=B, 0=0G_0=G_0, and 1=1G_1=G_1. A matrix ∈ℝd×dA ^d× d is G-symmetric (or G-selfadjoint) if ⋆=A =A, i.e., =⊤,GA=A G, (6) or equivalently, 1/2−1/2G^1/2AG^-1/2 is symmetric in the standard sense. The sheaf Laplacian F=⋆L_F=B B is 0G_0-symmetric by construction. Supplementary Section 1-C G-Orthogonal Projectors A matrix ∈ℝd×dM ^d× d is a G-orthogonal projector onto a subspace ⊆ℝdV ^d if: (i) Idempotence: 2=M^2=M; (i) G-symmetry: =⊤GM=M G. These two conditions together imply that M projects onto =ImV= M along the G-orthogonal complement ⟂=:⟨,⟩=0∀∈V _G=\x: ,v _G=0\,∀\,v \. Construction. Let ∈ℝd×rV ^d× r be a matrix whose columns form a G-orthonormal basis for V, i.e., ⊤=rV GV=I_r. Then =⊤M=VV G (7) is the G-orthogonal projector onto span()span(V). Regarding idempotence: 2=⊤⊤=(⊤)⊤=r⊤=M^2=VV GVV G=V(V GV)V G=VI_rV G=M. G-symmetry is also verified as follows: =⊤GM=GVV G and ⊤=(⊤)⊤=⊤M G=(VV G) G=GVV G. Supplementary Section 2 Proofs Proof of Prop. I.1. If i⊴ei=ei⊴e,j⊴ej=ej⊴eF_i eD_i=D_eJ_i e, _j eD_j=D_eJ_j e (8) then the columns of i⊴eiF_i eD_i and j⊴ejF_j eD_j belong to Im(e)Im(D_e). Hence, Im(e)⊆Im(e).Im(A_e) (D_e). (9) Since e∈St(de,ce)D_e (d_e,c_e), we have dimIm(e)=ce (D_e)=c_e, and therefore rank(e)≤cerank(A_e)≤ c_e. Conversely, if rank(e)≤cerank(A_e)≤ c_e, then Im(e)Im(A_e) can be embedded in a cec_e-dimensional subspace of F(e)F(e). Choosing eD_e as any orthonormal basis of such a subspace, the columns of i⊴eiF_i eD_i and j⊴ejF_j eD_j belong to Im(e)Im(D_e), so there exist matrices i⊴eJ_i e and j⊴eJ_j e satisfying (8). ∎ Proof of Prop. I.2. Since Im(e)⊆Im(e)Im(A_e) (D_e) by construction, and eD_e has orthonormal columns, we have ee⊤e=e.D_eD_e A_e=A_e\,. (10) Using i⊴e=e⊤i⊴ei,j⊴e=e⊤j⊴ej;J_i e=D_e F_i eD_i, _j e=D_e F_j eD_j\,; (11) we have ei⊴e=ee⊤i⊴ei=i⊴ei.D_eJ_i e=D_eD_e F_i eD_i=F_i eD_i\,. (12) The same argument gives ej⊴e=j⊴ej.D_eJ_j e=F_j eD_j\,. (13) Therefore, both commutativity conditions in Eq. 8 hold. ∎ Proof of Prop. I.3. Let ∈Γ(;J)s∈ (G;J ). For every edge e=(i,j)e=(i,j), i⊴ei=j⊴ej.J_i es_i=J_j es_j\,. (14) Using the naturality conditions, i⊴ei=ei⊴e,j⊴ej=ej⊴e,F_i eD_i=D_eJ_i e, _j eD_j=D_eJ_j e\,, (15) we obtain i⊴ei=ei⊴ei=ej⊴ej=j⊴ej.F_i ex_i=D_eJ_i es_i=D_eJ_j es_j=F_j ex_j\,. (16) Hence, ∈Γ(;F)x∈ (G;F ). Conversely, let ∈Γ(;F)x∈ (G;F ) with i∈Im(i)x_i (D_i ) for every node. Since i∈St(di,ci)D_i (d_i,c_i), the representations are uniquely recovered as i=i⊤i.s_i=D_i x_i\,. (17) For every edge e=(i,j)e=(i,j), naturality together with ∈Γ(;F)x∈ (G;F ) gives ei⊴ei=i⊴ei=j⊴ej=ej⊴ej,D_eJ_i es_i=F_i ex_i=F_j ex_j=D_eJ_j es_j\,, (18) and since eD_e has orthonormal, hence injective, columns, it can be cancelled on the left, yielding i⊴ei=j⊴ejJ_i es_i=J_j es_j, i.e., ∈Γ(;J)s∈ (G;J ). Therefore, 0D_0 is a linear isomorphism between Γ(;J) (G;J ) and Γ(;F)∩Im(0) (G;F ) (D_0 ). ∎ Proof of Theorem IV.1. Let 0≔blkdiag(ii∈),1≔blkdiag(ee∈ℰ).D_0 (\D_i\_i ), _1 (\D_e\_e )\,. (19) The natural transformation condition gives, at the coboundary level, F0=1J.B_FD_0=D_1B_J\,. (20) Indeed, for every edge e=(i,j)e=(i,j), (F0)e (B_FD_0s)_e =i⊴eii−j⊴ejj =F_i eD_is_i-F_j eD_js_j (21) =e(i⊴ei−j⊴ej) =D_e (J_i es_i-J_j es_j ) =(1J)e. =(D_1B_Js)_e\,. We now show that the corresponding adjoints also intertwine. By the metric-compatible lifting assumption, i⊴e=ei⊴ei−1i⊤i,F_i e=D_eJ_i eH_i^-1D_i G_i\,, (22) and, since i,iG_i,H_i are symmetric, transposing gives i⊴e⊤=iii−1(i⊴e)⊤e⊤.F_i e =G_iD_iH_i^-1 (J_i e ) D_e \,. (23) Using the adjoint formulas i⊴e⋆=i−1i⊴eTe,(i⊴e)⋆=i−1(i⊴e)⊤e,F_i e =G_i^-1F_i e^TG_e, (J_i e ) =H_i^-1 (J_i e ) H_e\,, (24) together with the induced-metric identity e=e⊤eeH_e=D_e G_eD_e, we obtain i⊴e⋆e _i e D_e =i−1i⊴eTee =G_i^-1F_i e^TG_eD_e (25) =ii−1(i⊴e)⊤e⊤ee⏟=e=i(i⊴e)⋆. =D_iH_i^-1 (J_i e ) D_e G_eD_e_=\,H_e=D_i (J_i e ) \,. Therefore, at the global level, F⋆1=0J⋆.B_F D_1=D_0B_J \,. (26) Using (20) and (26), we obtain the result ∀∈C0(;J)∀\,s∈ C^0(G;J) F0=F⋆F0=F⋆1J=0J⋆J=0J.L_FD_0=B_F B_FD_0=B_F D_1B_J=D_0B_J B_J=D_0L_J\,. (27) ∎ Proof of Cor. IV.2. If J=μL_Jv= , then F0=0(J)=μ0,L_F\,D_0v=D_0(L_Jv)=μ\,D_0v\,, (28) so 0D_0v is an eigenvector of FL_F associated with the same eigenvalue, provided 0≠D_0v≠ 0. Finally, using the definition i=i⊤iiH_i=D_i G_iD_i, we obtain ⟨0,0⟩0(;F) _0v,\,D_0w _C^0(G;F) =∑i∈(ii)⊤i(ii) = _i (D_iv_i) G_i(D_iw_i) (29) =⟨,⟩0(;J). = ,\,w _C^0(G;J)\,. Thus 0D_0 is an isometric embedding between the corresponding 0-cochain Hilbert spaces. ∎ Proof of Cor. V.1. Since F0=0J,L_FD_0=D_0L_J\,, (30) it follows by induction that Fq0=0Jq,q≥0.L_F^\,qD_0=D_0L_J^\,q, q≥ 0. (31) Multiplying by the coefficients aqa_q and summing over q=0,…,Qq=0,…,Q yields h(F)0=0h(J).h(L_F)\,D_0=D_0\,h(L_J)\,. (32) ∎ Proof of Thm. VI.1. Since ∈ℬx _K, there exists a unique vector ∈ℝb α ^b_K such that =.x=V_K α. (33) Hence, =.x_S= _SV_K α. (34) The coefficient vector α is uniquely determined if and only if the matrix _SV_K has full column rank, proving rank()=b,rank\! ( _SV_K )=b_K, (35) Since ℬ=Im()B_K=Im(V_K), condition (35) is equivalent to requiring that no nonzero bandlimited signal belongs to ker() ( _S), yielding ℬ∩ker()=.B_K∩ ( _S)=\0\.. Finally, when (35) holds, ^=()†, α= ( _SV_K ) x_S, (36) and substituting this expression into =x=V_K α gives ^=()†. x=V_K ( _SV_K ) x_S. (37) ∎ Proof of Cor. VI.2. Perfect recovery requires rank()=b.rank\! ( _SV_K )=b_K\,. (38) Since _SV_K has |||S| rows, its rank cannot exceed |||S|. Therefore, ||≥b|S|≥ b_K. ∎ Proof of Cor. VI.3. Since F0()=0(J)∈Im(0)L_FD_0(v)=D_0(L_Jv) (D_0 ) for every v, the subspace Im(0)Im (D_0 ) is FL_F-invariant. Since FL_F is 0G_0-self-adjoint, its 0G_0-orthogonal complement is invariant as well. Hence δ _δ commutes with FL_F, with every spectral projector kP_k, and therefore with B_K, giving the stated splitting. For the explicit form, 0D_0 is injective and maps ℰkE_k isometrically into ℱkF_k, so mk′≤mkm _k≤ m_k. The kU_k bases are mutually ℍ0H_0-orthogonal, corresponding to eigenspaces of JL_J for distinct eigenvalues, whence Im(∥)=Im(0)Im(B_K )=Im(D_0U_K). ∎ Supplementary Section 3 Multiplicity of the spectrum Unlike graph Laplacians, whose eigenvalues are generically simple, the spectrum of a sheaf Laplacian often exhibits eigenvalues with multiplicity greater than one. This phenomenon is not accidental, but reflects the geometric structure encoded by the restriction maps. Whenever the latter transport information coherently across the sheaf, the associated eigenspaces acquire additional degrees of freedom, leading to repeated eigenvalues. In this subsection we illustrate two representative mechanisms generating spectral multiplicity. Example 1 (Hierarchical Stiefel embeddings). Suppose that the restriction maps i⊴eJ_i e are Stiefel matrices belonging to St(ce,ci)St(c_e,c_i), acting as isometric embeddings (ci≤cec_i≤ c_e). Each edge can then be naturally oriented from the lower-dimensional stalk towards the higher-dimensional one. Assume furthermore that every node is reachable through an oriented path starting from a node N whose stalk has minimum dimension cN=hc_N=h. Then dim(ker(J))=h, \! ( (L_J) )=h, (39) and every global section is uniquely determined by a single h-dimensional vector assigned to the stalk at node N, which is propagated consistently throughout the sheaf via the Stiefel restriction maps [2]. Consequently, whenever h>1h>1, the zero eigenvalue has multiplicity h. Any orthonormal basis returned by an eigensolver is simply one among infinitely many orthonormal bases spanning the same null space, and no individual basis vector carries an intrinsic meaning. Example 2 (Connection graph). The previous example concerns only the null eigenspace. A much stronger form of multiplicity arises in the homogeneous case ci=ce=c_i=c_e=c, where the restriction maps are special orthogonal matrices i⊴e=ij∈SO(c)J_i e=O_ij (c) defining a connection graph. If the connection is consistent, namely the transport maps compose to the identity around every cycle of G, there exist orthogonal matrices ii∈\O_i\_i such that ij=i⊤j,O_ij=O_i O_j\,, (40) and the sheaf Laplacian admits the factorization J=⊤(⊗c),L_J=O (L_G _c )O\,, (41) where =blkdiag(1,…,N)O=blkdiag(O_1,…,O_N) and L_G denotes the scalar graph Laplacian [3]. In this case, every eigenvalue of the graph Laplacian is repeated exactly c times. The graph topology and the stalk geometry completely decouple, and each graph frequency generates a c-dimensional eigenspace of the sheaf Laplacian. Exs. 1 and 2 illustrate two different mechanisms leading to spectral multiplicity. In Ex. 1, multiplicity originates from the hierarchical embedding structure of the restriction maps and affects only the null space of the sheaf Laplacian. In Ex. 2, it results from a geometric symmetry that propagates throughout the entire spectrum. More generally, these examples suggest that spectral multiplicity is an intrinsic consequence of the geometric structure induced by the restriction maps. SM References [1] G. Strang, Linear Algebra and its Applications, 4th ed. Brooks/Cole Publishing, 2006. [2] G. D’Acunto, P. Di Lorenzo, and S. Barbarossa, “Networks of causal abstractions: A sheaf-theoretic framework,” arXiv preprint arXiv:2509.25236, 2026. [3] F. Chung, W. Zhao, and M. Kempton, “Ranking and sparsifying a connection graph,” Internet Mathematics, vol. 10, no. 1-2, p. 87–115, 2014.