t-DHGFT: Directed Hypergraph Fourier Transform

• t-DHGFT is a spectral transform designed to analyze signals on directed hypergraphs by capturing both polyadic interactions and directional flow.
• It employs a tensor-based framework that integrates t-product shift operators, canonical B-hyperarc decomposition, and self-adjoint dilation to preserve source-target semantics.
• The method orders frequency components via directed hypergraph total variation, enabling applications like effective denoising in complex network data.

Searching arXiv for the cited paper and closely related work on directed hypergraph signal processing and tensor t-SVD. The directed hypergraph Fourier transform (t-DHGFT) is the spectral transform introduced within the Directed Hypergraph Signal Processing (DHGSP) framework to analyze signals defined on directed hypergraphs, that is, combinatorial structures with both higher-order and asymmetric relations. In DHGSP, directionality and polyadic interaction are treated simultaneously by combining a directed hypergraph adjacency tensor, a t-product-based shift operator, and a tensor singular value decomposition (t-SVD) construction based on self-adjoint dilation. The resulting transform is described as lossless, admits perfect reconstruction, and orders spectral components through a directed hypergraph total variation, thereby extending graph Fourier analysis beyond pairwise and undirected settings (Mundo-Levano et al., 23 Jun 2026).

1. Scope and motivation

The motivating claim of DHGSP is that standard graph signal processing is limited in two distinct respects. First, many systems of interest are asymmetric, including citations, flows, traffic, causality, influence, and one-way interactions. Directed graph signal processing can model such asymmetry, but its operators often have complex eigenvalues, non-orthogonal eigenvectors, or may be non-diagonalizable, which complicates a clean Fourier theory. Second, pairwise edges are inadequate for higher-order interactions such as multi-author collaboration, coalition influence, or intersection dynamics among multiple vehicles. Hypergraph signal processing addresses higher-order relations, but classical formulations are typically undirected (Mundo-Levano et al., 23 Jun 2026).

DHGSP is positioned as a unified framework that handles higher-order or polyadic structure and asymmetric or directional structure simultaneously. The paper explicitly presents it as a framework that subsumes graph signal processing, directed graph signal processing, and hypergraph signal processing as special cases. In this formulation, t-DHGFT serves as the spectral transform associated with that broader signal-processing model.

A common misconception is to treat directed hypergraphs as merely a straightforward fusion of digraphs and undirected hypergraphs. The framework rejects that simplification. Its central claim is that combining directionality and higher-order structure requires a representation that preserves source-target semantics and avoids spectral pathologies inherited from naive asymmetric constructions.

2. Directed hypergraph model and canonical B-hyperarc decomposition

A directed hypergraph is written as

$DH = (V,E),$

where $V$ is the vertex set, $|V| = N$, and each hyperarc $e \in E$ is an ordered pair

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

with $T(e)$ the tail or source set, $H(e)$ the head or target set, and $T(e)\cap H(e)=\emptyset$. The model reduces to a directed graph when $|T(e)|=|H(e)|=1$, while symmetrizing directions yields an undirected hypergraph.

The paper identifies two deficiencies in prior tensor representations of directed hypergraphs: identifiability and signal cross-talk. Identifiability means that the tensor alone may fail to indicate whether a node is functioning as a source or a target. Signal cross-talk means that shift operations may mix information across distinct targets. To resolve both issues, the framework introduces a canonical decomposition of each directed hyperarc into a collection of B-hyperarcs, one per head node. Under this decomposition, each resulting hyperarc has a single head.

This decomposition is structurally important because it separates target nodes explicitly. The paper’s interpretation is that this preserves semantic faithfulness and prevents interference among distinct targets during shifting. A plausible implication is that the decomposition is not merely a representational convenience but a prerequisite for defining a spectral theory aligned with directed higher-order topology.

3. Adjacency tensor, in-Laplacian, and semantically faithful encoding

Let $N$ denote the number of nodes and $V$0 the maximum hyperarc cardinality. Under the canonical B-hyperarc decomposition, the adjacency tensor is

$V$1

with $V$2 modes in total. For each B-hyperarc

$V$3

the tensor entry is assigned so that the first index is reserved for the head node, while the remaining indices enumerate the tails via all ordered length-$V$4 permutations with repetition of the tail nodes, subject to each tail appearing at least once. The entry is written as

$V$5

where $V$6 is the multinomial normalization factor equal to the total number of such permutations (Mundo-Levano et al., 23 Jun 2026).

Three properties are emphasized. The first index always identifies the head or target. The remaining modes encode the tail or source group. This is the mechanism by which identifiability and cross-talk are resolved. When $V$7, the tensor reduces to the usual adjacency matrix of a directed graph, so the construction is consistent with the matrix case.

The framework also defines the in-degree tensor $V$8, with in-degrees on the superdiagonal, and the in-Laplacian tensor

$V$9

The in-degree of a node $|V| = N$0 is obtained by summing tensor entries corresponding to hyperarcs arriving at $|V| = N$1. This use of an in-Laplacian rather than a symmetrized Laplacian reflects the framework’s commitment to preserving flow orientation instead of erasing it through undirected averaging.

4. Shift operator, symmetrization, and topological localization

Given a node signal $|V| = N$2, the tensor signal is constructed as the $|V| = N$3-fold outer product

$|V| = N$4

where $|V| = N$5 denotes the outer product. The shift is then defined by

$|V| = N$6

where $|V| = N$7 is typically chosen as either $|V| = N$8 or $|V| = N$9, and $e \in E$0 denotes the t-product.

To make the t-product machinery applicable while preserving directionality, the tensor is symmetrized along higher modes, specifically modes $e \in E$1, by appending a zero slice and reflecting entries into a palindrome. The resulting symmetrized tensor is denoted $e \in E$2, with

$e \in E$3

The paper stresses four features of this construction: only the higher modes are symmetrized, the frontal slices are left intact, directional information is preserved, and the palindromic structure makes the DFT slices real, which leads to real tubal singular values (Mundo-Levano et al., 23 Jun 2026).

Theorem 1 establishes topological localization. If $e \in E$4 is localized at node $e \in E$5, meaning $e \in E$6 and $e \in E$7 for $e \in E$8, then the first shifted signal

$e \in E$9

is non-zero only at nodes $e = (T(e), H(e)),$0 such that there exists a hyperarc $e = (T(e), H(e)),$1 with

$e = (T(e), H(e)),$2

More generally, after $e = (T(e), H(e)),$3 shifts, the signal is non-zero only at nodes reachable from $e = (T(e), H(e)),$4 in exactly $e = (T(e), H(e)),$5 directed hyperarc steps. This is the framework’s notion of topological faithfulness: the shift respects the directed hypergraph’s causal or flow structure.

An important clarification follows from this theorem. The shift is not merely algebraically defined; it is constrained by the directed hypergraph topology. That distinction is central to the paper’s claim that DHGSP supplies a genuine signal-processing framework rather than only a tensor representation.

5. Self-adjoint dilation, t-SVD, and the definition of t-DHGFT

The paper argues that ordinary t-eigendecomposition is insufficient because the in-Laplacian tensor $e = (T(e), H(e)),$6 is asymmetric, and a direct t-eigendecomposition is not available in the usual way when symmetry is required for a real orthonormal basis. The proposed remedy is a tensor version of self-adjoint dilation.

For the symmetrized in-Laplacian $e = (T(e), H(e)),$7, the symmetric dilation is

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

where $e = (T(e), H(e)),$9 is the zero tensor block and $T(e)$0 denotes the tensor conjugate transpose under the t-product framework. This dilated tensor is symmetric under the t-product and therefore guarantees a complete orthonormal eigenbasis with real tubal eigenvalues (Mundo-Levano et al., 23 Jun 2026).

The t-SVD of the symmetrized in-Laplacian is

$T(e)$1

where $T(e)$2 and $T(e)$3 are t-orthogonal, $T(e)$4 is f-diagonal, and the singular values are ordered in ascending order. The eigenbasis of the dilation is then constructed from $T(e)$5 and $T(e)$6, yielding a block eigenvector tensor $T(e)$7.

The transform itself is defined by first embedding the signal isometrically into the dilated space and then projecting onto that orthonormal basis. The embedded signal is written using the symmetrized identity tensor $T(e)$8, and the forward transform is

$T(e)$9

This is the t-DHGFT. The inverse transform is

$H(e)$0

Because $H(e)$1 is orthogonal, the inversion is exact.

6. Losslessness, frequency ordering, and empirical denoising

The transform is described as lossless for four stated reasons: the signal is embedded isometrically into the dilated space, the dilation is symmetric and yields an orthonormal basis, the forward and inverse transforms are exact inverses, and Parseval’s identity holds. In consequence, the transform preserves energy and does not discard information. This is contrasted with asymmetric spectral methods that may lose invertibility or rely on non-orthogonal eigenvectors.

Frequency is ordered through the directed hypergraph total variation

$H(e)$2

For a right singular vector $H(e)$3,

$H(e)$4

so singular values ordered as

$H(e)$5

rank the modes from smooth to rapidly oscillatory (Mundo-Levano et al., 23 Jun 2026). Low-frequency components correspond to signals that vary slowly across directed hyperarc flow patterns, whereas high-frequency components fluctuate sharply relative to the directed hypergraph topology. This extends the familiar graph-Fourier smoothness interpretation to directed higher-order structure.

The experimental evidence reported in the paper concerns denoising on a real traffic dataset from Macheng, China, with 153 nodes. Directed hypergraphs are constructed by grouping directed edges into hyperarcs. The signal is corrupted by additive white Gaussian noise, and denoising is performed by low-pass hard-thresholding in the t-DHGFT domain, retaining only the $H(e)$6 lowest-frequency components. The method is compared against undirected GSP, DGSP, and undirected HGSP. DHGSP is reported to achieve lower MAE than all baselines across the spectrum, especially around intermediate bandwidths. The interpretation given in the paper is twofold: the improvement over HGSP indicates that directionality matters, and the improvement over DGSP indicates that higher-order interactions matter.

These results do not by themselves establish universal superiority across all tasks, but they support the narrower claim advanced in the paper: t-DHGFT provides a faithful frequency domain for directed hypergraph signals and is useful for spectral denoising in settings where both flow asymmetry and polyadic interactions are present.