Directed Hypergraph Signal Processing

• DHGSP is a tensor-spectral framework that extends graph signal processing to directed hypergraphs by preserving asymmetric higher-order interactions.
• It employs a topologically faithful adjacency tensor via canonical B-hyperarc decomposition to accurately model joint source-target dynamics.
• The framework utilizes t-product and t-SVD for lossless spectral analysis, achieving improved denoising performance on real-world traffic data.

Searching arXiv for the core and contextual papers to ground the article in current records. {"^^^^^^^^2query2 OR \2^^^^^^^^"A Framework for Directed Hypergraph Signal Processing via tensor t-SVD\"","max_results":5} {"^^^^^^^^2query2 "max_results": 5} Directed Hypergraph Signal Processing (DHGSP) is a spectral signal processing framework for data supported on directed hypergraphs—domains in which interactions are simultaneously higher-order and asymmetric. Introduced in "A Framework for Directed Hypergraph Signal Processing via tensor t-SVD" (&&&^^^^2query2^^^^&&&), it extends graph signal processing by defining a directed-hypergraph adjacency tensor, a topologically local shift operator through the tensor t-product, and a lossless Directed Hypergraph Fourier Transform (t-DHGFT) based on tensor singular value decomposition (t-SVD) of a tensorized directed Laplacian after self-adjoint dilation. The framework is motivated by citation systems, metabolic pathways, and traffic propagation, where several sources may jointly affect one or more targets and neither ordinary graphs, directed graphs, nor undirected hypergraphs preserve the full semantics of the interaction (&&&^^^^2query2^^^^&&&).

^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^. Domain, motivation, and signal model

The underlying object is a directed hypergraph

PRESERVED_PLACEHOLDER_^^^^2query2^^^^

with PRESERVED_PLACEHOLDER_^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^, in which each hyperarc is an ordered pair of disjoint subsets

$e=(T(e),H(e)).$

Here $T(e)\subseteq V$ is the tail or source set and $H(e)\subseteq V$ is the head or target set. A directed graph is recovered as the special case $|T(e)|=|H(e)|=1$, while an undirected hypergraph may be viewed as arising by symmetrizing directions (&&&^^^^2query2^^^^&&&).

DHGSP starts from the observation that a matrix-based digraph operator cannot distinguish whether a target depends jointly on a set of sources or independently on pairwise edges from them. In a traffic setting, for example, several upstream roads may jointly affect congestion at a downstream segment, and the relation is not symmetric. This is the canonical situation in which a directed graph loses polyadic structure and an undirected hypergraph loses directionality (&&&^^^^2query2^^^^&&&).

A signal begins as a node signal

$x\in\mathbb{R}^N.$

To represent combinations of nodes that participate jointly in directed polyadic interactions, DHGSP lifts $x$ to a tensor signal by an $(M-1)$-fold outer product, where $M$ is the maximum hyperarc cardinality:

PRESERVED_PLACEHOLDER_^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2query2^^^^

This tensor signal is the object on which the shift and filtering operators act (&&&^^^^2query2^^^^&&&).

Earlier directed-hypergraph operator constructions already modeled a hyperedge as an ordered tail/head pair and built propagation from that structure. The random-walk/PageRank line represents directed hypergraphs through separate tail and head incidence matrices and a Markov operator PRESERVED_PLACEHOLDER_^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^ (Tran et al., 2019), while the directed hypergraph neural network literature uses the same tail/head split to construct a PageRank-symmetrized propagation operator PRESERVED_PLACEHOLDER_^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \22^^^^ for semi-supervised learning and node classification (Tran et al., 2020). DHGSP differs in that it is explicitly a tensor-spectral framework rather than an incidence-matrix Markov or neural propagation framework (&&&^^^^2query2^^^^&&&).

2. Topologically faithful adjacency tensor

The central structural contribution of DHGSP is its directed hypergraph adjacency tensor. The paper identifies two problems in earlier tensor representations of directed hypergraphs: identifiability, meaning that one can lose track of which tensor indices correspond to sources versus targets, and signal cross-talk, meaning that shift operations can incorrectly mix information among different targets. The proposed remedy is a canonical B-hyperarc decomposition in which every directed hyperarc is decomposed into one B-hyperarc per head node, so that each resulting hyperarc has a single head (&&&^^^^2query2^^^^&&&).

After this decomposition, one tensor mode can be reserved exclusively for the target. Let the directed hypergraph have PRESERVED_PLACEHOLDER_^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^3 nodes and maximum hyperarc cardinality PRESERVED_PLACEHOLDER_^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^4. The adjacency tensor

PRESERVED_PLACEHOLDER_^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^5

is defined so that, for each B-hyperarc PRESERVED_PLACEHOLDER_^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^6,

PRESERVED_PLACEHOLDER_^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^7

where the first index PRESERVED_PLACEHOLDER_^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^8 is reserved for the head node, the remaining indices PRESERVED_PLACEHOLDER_^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^9 range over all ordered length-$e=(T(e),H(e)).$^^^^2query2^^^^ permutations with repetition of the tail nodes such that every tail node appears at least once, and $e=(T(e),H(e)).$^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^ is the number of such permutations (&&&^^^^2query2^^^^&&&). This construction is described as topologically faithful because the target node is always identified by the first index and each head node is treated independently, preventing cross-talk.

From $e=(T(e),H(e)).$2, DHGSP defines the in-degree tensor and in-Laplacian:

$e=(T(e),H(e)).$3

The in-degree of a node is therefore the total incoming mass across all tensor entries whose first index is that node (&&&^^^^2query2^^^^&&&).

This tensorial encoding is the main reason DHGSP is not a straightforward generalization of digraph signal processing. A matrix has one source and one target index; a directed hyperarc with several tail nodes and one head inherently requires multiple source positions. The tensor order $e=(T(e),H(e)).$4 reflects maximum hyperarc cardinality, and every mode beyond the first participates in encoding the source tuple (&&&^^^^2query2^^^^&&&).

3. Shift operators, t-product algebra, and Laplacian structure

The algebraic engine of DHGSP is the t-product and the associated t-SVD for third-order tensors. Although the original adjacency information is $e=(T(e),H(e)).$5-way, the framework recasts the relevant operators into a third-order tensor format compatible with the t-product, with tensors of the form

$e=(T(e),H(e)).$6

In this algebra, the t-product is defined by circular convolution along tubes, or equivalently by multiplication of frontal slices after applying the discrete Fourier transform along the third dimension. Computationally, this reduces tensor factorization to a sequence of ordinary matrix factorizations in the Fourier domain (&&&^^^^2query2^^^^&&&).

A tensor $e=(T(e),H(e)).$7 is t-orthogonal if

$e=(T(e),H(e)).$8

and the t-SVD of the symmetrized in-Laplacian has the form

$e=(T(e),H(e)).$9

where $T(e)\subseteq V$^^^^2query2^^^^ and $T(e)\subseteq V$^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^ are t-orthogonal and $T(e)\subseteq V$2 is f-diagonal. The singular values are arranged in ascending order (&&&^^^^2query2^^^^&&&).

The shift operator is defined by

$T(e)\subseteq V$3

with $T(e)\subseteq V$4 typically chosen as either the adjacency tensor $T(e)\subseteq V$5 or the in-Laplacian tensor $T(e)\subseteq V$6. To make this compatible with the t-product formalism, the framework applies a symmetrization step along the higher dimensions, producing

$T(e)\subseteq V$7

Because only the higher modes are reflected, the frontal slices remain intact, so the directional information encoded in the first two modes is preserved (&&&^^^^2query2^^^^&&&).

The shift is topologically local in a precise sense. Theorem ^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^ in the paper states that if a signal is localized at node $T(e)\subseteq V$8, then the first-order shifted signal is nonzero only at nodes $T(e)\subseteq V$9 for which there exists a hyperarc $H(e)\subseteq V$^^^^2query2^^^^ with $H(e)\subseteq V$^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^ and $H(e)\subseteq V$2. After $H(e)\subseteq V$3 shifts, the signal is nonzero only at nodes reachable from $H(e)\subseteq V$4 in exactly $H(e)\subseteq V$5 directed hyperarc steps. This establishes the operator as a proper notion of local propagation on a directed hypergraph (&&&^^^^2query2^^^^&&&).

The framework therefore provides the directed-hypergraph analogue of the primitive shift in classical GSP, but with locality defined on asymmetric polyadic topology rather than on pairwise adjacency.

4. t-DHGFT, frequency, and spectral interpretation

The paper’s most distinctive spectral construction is the t-DHGFT. The difficulty is that the directed-hypergraph in-Laplacian is asymmetric, so a direct tensor eigendecomposition would not generally yield a real orthonormal basis and could inherit the familiar directed-operator problems of complex spectra, non-orthogonality, and possible lack of diagonalizability. DHGSP resolves this through self-adjoint dilation, extending an SVD-based strategy from directed graph signal processing to the tensor setting (&&&^^^^2query2^^^^&&&).

The dilation places the symmetrized directed in-Laplacian and its Hermitian transpose in off-diagonal blocks,

$H(e)\subseteq V$6

producing a tensor operator that is symmetric under the t-product. Its orthonormal tensor eigenbasis is then built from the left and right t-singular vectors of $H(e)\subseteq V$7, exactly as SVD-based directed graph Fourier transforms use left and right singular vectors of a directed matrix operator (&&&^^^^2query2^^^^&&&).

The original tensor signal is embedded isometrically into the dilated space, and the forward and inverse t-DHGFT are defined from that basis. Because the dilation basis is t-orthogonal, the transform is lossless, perfectly invertible, and Parseval-preserving (&&&^^^^2query2^^^^&&&).

Frequency is defined by directed hypergraph total variation:

$H(e)\subseteq V$8

For a right singular vector $H(e)\subseteq V$9,

$|T(e)|=|H(e)|=1$^^^^2query2^^^^

Hence the singular values

$|T(e)|=|H(e)|=1$^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^

order the modes from smooth to oscillatory: low singular values correspond to signals that vary little under the directed-hypergraph Laplacian, while high singular values correspond to rapidly varying components (&&&^^^^2query2^^^^&&&).

Relative to other direction-aware spectral constructions, this is one of several possible routes to frequency on asymmetric domains. "Graph Signal Processing of Indefinite and Complex Graphs using Directed Variation" defines graph frequencies by ordering orthogonal basis vectors according to direction-aware variation on pairwise directed graphs rather than hypergraphs (Schultz et al., 2020). "Directional Sheaf Hypergraph Networks" instead builds a complex-valued Directed Sheaf Hypergraph Laplacian that is Hermitian, positive semidefinite, and spectrally decomposable on directed hypergraphs, with a Dirichlet energy defined through direction-dependent sheaf incidences (Mule et al., 6 Oct 2025). DHGSP differs from both by keeping a genuinely asymmetric tensor operator and recovering orthonormality through dilation and t-SVD rather than through a directly Hermitian hypergraph Laplacian (&&&^^^^2query2^^^^&&&).

5. Position within the directed-hypergraph literature

DHGSP is presented as a unification of earlier theories. The paper explicitly claims the hierarchy

$|T(e)|=|H(e)|=1$2

If $|T(e)|=|H(e)|=1$3, the adjacency tensor reduces to the ordinary $|T(e)|=|H(e)|=1$4 adjacency matrix of a directed graph. If the operator is additionally symmetric, one recovers ordinary undirected GSP. If directionality is removed by symmetrizing hyperarcs, one obtains undirected HGSP (&&&^^^^2query2^^^^&&&).

This situates DHGSP relative to several other directed-hypergraph formalisms. The PageRank/random-walk line defines the directed-hypergraph transition matrix

$|T(e)|=|H(e)|=1$5

and uses its stationary distribution to build Laplacian-like operators (Tran et al., 2019). The directed hypergraph neural network line uses a PageRank-symmetrized operator

$|T(e)|=|H(e)|=1$6

and a corresponding propagation operator $|T(e)|=|H(e)|=1$7 for semi-supervised learning and node classification (Tran et al., 2020). A different branch develops a nonlinear diffusion operator on directed hypergraphs with stationary vertices, together with a directed-hypergraph energy

$|T(e)|=|H(e)|=1$8

and a Cheeger inequality for directed hyperedge expansion (Chan et al., 2017). These constructions are operator-theoretic and native to directed hypergraphs, but they are not tensor t-SVD frameworks.

Application-driven work also intersects with DHGSP without becoming a full spectral theory. "Towards Multi-agent Policy-based Directed Hypergraph Learning for Traffic Signal Control" models traffic states as signals on a dynamic directed hypergraph, builds spatial and temporal directed hyperedges, computes hyperedge embeddings through a role-aware incidence-style operator, and performs directed higher-order aggregation with attention before PPO control (Wang et al., 2024). The paper is best understood as a directed-hypergraph neural processing architecture rather than as formal DHGSP, but it demonstrates that time-varying vertex signals on dynamically constructed directed hypergraphs are practically useful in traffic systems (Wang et al., 2024).

At the continuum end, "Vlasov Equations on Directed Hypergraph Measures" replaces finite directed hypergraphs by measure-valued fibers

$|T(e)|=|H(e)|=1$9

and studies higher-order directed interaction dynamics through a Vlasov equation on directed hypergraph measures (&&&3^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^&&&). This is not a spectral DHGSP framework, but it supplies a large-scale dynamical substrate for signal evolution on directed hypergraph domains (&&&3^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^&&&).

6. Empirical validation, computational profile, and open problems

The empirical validation in the DHGSP paper is a denoising experiment on real traffic data from Macheng, China, with ^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^53 nodes, taken from the GSP-traffic dataset. The directed hypergraph is built by grouping directed edges into hyperarcs following a prior directed hypergraph method. Node-level traffic signals are corrupted by additive white Gaussian noise, and reconstruction is performed by low-pass hard-thresholding in the t-DHGFT domain, retaining only the $x\in\mathbb{R}^N.$^^^^2query2^^^^ lowest-frequency components while varying $x\in\mathbb{R}^N.$^^^^2(Mundo-Levano et al., 23 Jun 2026) OR \2^^^^ across the spectrum (&&&^^^^2query2^^^^&&&).

The baselines are Undirected GSP using a symmetric Laplacian, DGSP using the SVD of a directed Laplacian, and Undirected HGSP via t-product. The performance metric is mean absolute error as a function of retained frequency fraction. The paper reports that DHGSP consistently achieves lower MAE than all baselines across the entire frequency range, with especially pronounced gains around 5^^^^2query2^^^^–6^^^^2query2 of the spectrum (&&&^^^^2query2^^^^&&&). This suggests that modeling both higher-order and directional structure improves spectral separation between signal and noise in traffic networks.

The computational profile follows directly from the construction. Building the adjacency tensor requires enumerating, for each B-hyperarc, all length-$x\in\mathbb{R}^N.$2 tail permutations with repetition in which every tail appears at least once, which becomes combinatorially heavier as hyperarc cardinalities grow. Once the operator has been built, the main linear-algebraic cost is FFT-accelerated t-product and t-SVD computation over frontal slices. The higher-mode symmetrization enlarges the third dimension to $x\in\mathbb{R}^N.$3, trading memory and computation for a real-valued spectral structure (&&&^^^^2query2^^^^&&&).

The paper also delineates several limitations. The framework depends on the maximum hyperarc cardinality $x\in\mathbb{R}^N.$4 and on canonical B-hyperarc decomposition; the permutation-based tensor filling can become expensive as tail sizes increase; and the current empirical study focuses on denoising rather than on sampling, prediction, compression, clustering, or learning (&&&^^^^2query2^^^^&&&). The authors identify future directions including sampling theory for directed hypergraphs, compression bounds, spectral clustering, and directed hypergraph neural networks based on learned t-product filter banks (&&&^^^^2query2^^^^&&&).

Taken together, these elements define DHGSP as a tensor-spectral framework whose essential claims are semantic faithfulness of the adjacency tensor, topological localization of the shift, and orthogonal, lossless spectral analysis after dilation. Its main conceptual contribution is to preserve the semantics of directed hypergraphs while still providing a mathematically clean Fourier analysis for signals on asymmetric higher-order domains (&&&^^^^2query2^^^^&&&).