A Framework for Directed Hypergraph Signal Processing via tensor t-SVD

Abstract: We introduce Directed Hypergraph Signal Processing (DHGSP), a unified framework that extends graph signal processing to accommodate both higher-order (polyadic) and asymmetric (directional) relationships simultaneously. Using the tensor singular value decomposition (t-SVD) within the t-product algebra, we define a novel adjacency tensor for directed hypergraphs, a topologically faithful shift operator, and a lossless Directed Hypergraph Fourier Transform (t-DHGFT). Experiments on real traffic networks demonstrate that DHGSP outperforms matrix-based (graph and digraph) and undirected tensor-based (hypergraph) baselines in denoising tasks.

• The paper introduces a unified DHGSP framework that decomposes hyperarcs for lossless, directional Fourier analysis via t-SVD.
• It constructs an identifiable adjacency tensor using B-hyperarc decomposition to prevent signal cross-talk during spectral shifts.
• Empirical results on traffic networks show lower MAE than traditional methods, particularly preserving mid-frequency components.

A Unified Framework for Directed Hypergraph Signal Processing via Tensor t-SVD

Introduction and Context

The framework introduced in "A Framework for Directed Hypergraph Signal Processing via tensor t-SVD" (2606.25112) directly addresses key deficiencies in modern Graph Signal Processing (GSP) by formalizing spectral tools for higher-order, asymmetric relational data. While GSP and its extensions to directed graphs and undirected hypergraphs have enabled rigorous frequency analysis of complex networked signals, they remain fundamentally limited: traditional directed GSP cannot consistently support lossless Fourier analysis due to the nonnormality and complex spectra of core operators, and hypergraph approaches lack any intrinsic directionality. A plethora of real-world applications—such as group-wise influence in social systems, multifactorial signaling in biological networks, and intersection dynamics in traffic—demand a model that simultaneously encodes polyadic and directional interactions. This work establishes such a model, termed Directed Hypergraph Signal Processing (DHGSP), leveraging third-order tensor algebra and t-SVD.

Directed Hypergraph Algebra and Adjacency Tensor Construction

Directed hypergraphs generalize directed graphs by allowing hyperarcs—ordered pairs of vertex subsets—to encode many-to-many, asymmetric relations. Existing matrix or tensor encodings for such hypergraphs have suffered from severe identifiability and signal cross-talk issues, fundamentally restricting their analytical tractability. The paper introduces a canonical construction: every hyperarc is decomposed into its constituent B-hyperarcs, each of which is a many-to-one relation, isolating every target vertex and its incoming sources. The adjacency tensor is then defined using this decomposition, reserving its primary mode for the head vertex and encoding all permissible tail combinations while normalizing for permutation redundancy.

This construction accomplishes two critical properties:

• Identifiability: Every tensor index position has an unambiguous topological interpretation (head versus tail roles).
• Prevention of signal cross-talk: The signal at each target is a function of only its direct sources.

At the $M=2$ cardinality case, the construction reduces precisely to the classical directed adjacency matrix, showing that the proposed framework subsumes established GSP, DGSP, and HGSP as special cases.

Topological Shift Operator and Signal Localization

Spectral analysis on hypergraphs requires well-defined shift operators that reflect both directionality and higher-order relations. DHGSP utilizes the t-product tensor algebra to define a hypergraph shift: the signal is lifted to higher mode order via iterative outer products, symmetrized along auxiliary modes (leaving directionality intact), and shifted under the adjacency or in-Laplacian tensor. A key result (Theorem 1) demonstrates that after $l$ shifts, the nonzero support of the signal lies precisely at nodes reachable from the source via $l$ directed hyperarc steps, preserving the causal, topological semantics of the network.

Directed Hypergraph Fourier Transform via t-SVD

Classical spectral analysis requires orthogonal, real-valued eigenbases; directed Laplacians are inherently non-symmetric and admit complex or even non-diagonalizable spectra. The framework overcomes this by applying self-adjoint dilation—an extension from DGSP to tensor algebra—yielding a symmetric operator with a complete, real-valued t-SVD.

The Directed Hypergraph Fourier Transform (t-DHGFT) is constructed by t-SVD of the (dilated, symmetrized) in-Laplacian tensor, using t-orthogonal bases. The transform guarantees:

• Lossless, invertible analysis: Parseval’s identity holds exactly in the transform domain.
• Explicit frequency ordering: Modes are naturally sorted by total-variation, from globally smooth to highly oscillatory, supporting traditional spectral interpretation and filtering strategies.

Empirical Analysis: Spectral Denoising on Traffic Networks

The practical advantage of DHGSP is validated on real-world traffic data (Macheng, China, 153 nodes). Directed hypergraphs are empirically constructed from road intersection dynamics, and DHGSP is benchmarked against spectral denoising using GSP (symmetric Laplacian), DGSP (SVD of directed Laplacian), and undirected HGSP (t-product).

Key experimental findings include:

• DHGSP achieves lower mean absolute error (MAE) across the entire spectrum for the denoising task, compared to all baselines.
• The improvement is most pronounced at intermediate bandwidths (50-60% conserved frequencies), suggesting DHGSP more effectively preserves informative mid-frequency components sensitive to directionality and higher-order structure.
• The results confirm that directionality encodes critical structural dependencies not present in undirected hypergraphs, while higher-order topology captures dependencies missed by pairwise digraph edges.

Implications and Future Directions

By enabling rigorous frequency analysis for networks with polyadic, asymmetric dependencies, DHGSP opens pathways for new theoretical investigations (e.g., spectral clustering of directed hypergraphs, generalization of uncertainty and sampling principles, spectral theory of causality on networked systems) and novel practical algorithms (e.g., neural architectures parameterized by t-product, spectral feature design for higher-order directed data).

Immediate extensions, as identified by the authors, include:

• Sampling and compressive recovery theory for directed hypergraph-valued signals
• Spectral clustering exploiting DHGSP eigenmodes
• Construction of directed hypergraph neural networks leveraging learned t-product filter banks

Other anticipated developments involve transfer-learning architectures for directed hypergraph-structured domains (e.g., protein interaction networks, citation graphs), advances in spectral estimation on irregular data manifolds, and the generalization of isomorphism testing and motif discovery to higher-order, directional contexts.

Conclusion

The DHGSP framework provides a mathematically principled, computationally tractable, and semantically faithful signal processing architecture for the simultaneous modeling of directionality and higher-order topology in complex networks. Its core innovations—canonical adjacency tensorization via B-hyperarc decomposition, topologically localized t-product shifts, and lossless spectral transforms via t-SVD—establish a unified foundation for future research and applications in directed hypergraph signal processing (2606.25112).