Hodgelets: Spectral Wavelets for Edge-Flows

• Hodgelets are wavelet atoms constructed from the combinatorial Hodge Laplacian, encoding oriented edge-flows along curl-free, divergence-free, and harmonic components.
• They utilize spectral filtering and polynomial approximations to achieve localized, multiscale representations, enabling efficient sparse coding of flow data.
• Empirical evaluations demonstrate that separate Hodgelets enhance sparsity and interpretability in clustering and analyzing flow dynamics on networks.

A Hodgelet is a wavelet atom for representing oriented edge-flows (1-cochains) on a simplicial complex, constructed from the spectrum of the combinatorial Hodge Laplacian. Analogous to spectral graph wavelets for node-signals, Hodgelets constitute a localized, multiscale dictionary for edge-flows. By leveraging the Hodge decomposition, Hodgelets offer representations that distinctly respect the curl-free (gradient), divergence-free (circulation), and harmonic components of flows, enabling interpretable and highly sparse representations for physical or information flows on networks (Roddenberry et al., 2021).

1. Mathematical Structures and Preliminaries

A simplicial complex $X$ comprises a finite collection of simplices closed under nonempty subset inclusion, where $k$-simplices have cardinality $k+1$. The sets $X_0$, $X_1$, and $X_2$ denote nodes, edges, and triangles, respectively. Each $k$-simplex $\sigma$ adopts a reference orientation, and the space of real-valued functions on oriented $k$-simplices is $C^k(X) \cong \mathbb{R}^{N_k}$, with $k$0 specifically modeling oriented edge-flows.

Key operators include the boundary maps $k$1 (node–edge incidence, $k$2) and $k$3 (triangle–edge incidence, $k$4), as well as coboundaries given by the adjoints $k$5 and $k$6.

The combinatorial Hodge 1-Laplacian is defined as:

$k$7

where $k$8 and $k$9 are the lower and upper Laplacians, acting on $k+1$0. The eigenstructure $k+1$1 of $k+1$2 forms the spectral backbone for Hodgelet construction.

2. Spectral Construction of Hodgelets

Given the eigendecomposition $k+1$3 of $k+1$4 over $k+1$5 and a family of real-valued wavelet kernels $k+1$6 (e.g., log-scaled Hann windows), a Hodgelet at scale $k+1$7 and edge basis vector $k+1$8 is defined via functional calculus as:

$k+1$9

with the explicit spectral representation:

$X_0$0

The choice of $X_0$1 localizes each atom around $X_0$2-bands, while spectral polynomial approximation by $X_0$3 ensures spatial localization. This construction enables atoms that efficiently interpolate between bandlimitedness and spatial focus for edge-flows.

3. Hodge–Helmholtz Decomposition Alignment

Hodgelets can be aligned with the canonical Hodge decomposition:

$X_0$4

where:

• $X_0$5: divergence-free (curl) flows
• $X_0$6: curl-free (gradient) flows
• $X_0$7: harmonic flows

Two construction paradigms emerge:

• Joint Hodgelets: $X_0$8, mixing both curl and gradient modes.
• Separate Hodgelets: Employ kernel families $X_0$9 and $X_1$0 (each vanishing at zero) to define:

$X_1$1

Harmonic modes may be captured by zero-frequency scaling functions as required. This separation enables Hodgelets to respect and isolate the physically relevant subspaces associated with vector calculus analogues.

4. Frame Properties of Hodgelet Dictionaries

The dictionaries formed by Hodgelets are overcomplete and can be analyzed via frame theory:

• Joint Frame Bounds: For $X_1$2, the joint Hodgelet dictionary $X_1$3 is an $X_1$4-frame in $X_1$5 with

$X_1$6

When $X_1$7 is constant, the frame is tight.

• Separate Frame Bounds: For

$X_1$8

the union of upper and lower Hodgelets forms an $X_1$9-frame with:

$X_2$0

$X_2$1

Frame properties ensure stable, redundant representations suitable for sparse coding and analysis.

5. Algorithmic Procedure

The construction of Hodgelets for a given simplicial complex $X_2$2 proceeds as follows:

1. Boundary Matrices: Assemble $X_2$3 (node–edge incidence) and $X_2$4 (edge–triangle incidence).
2. Laplacians: Compute $X_2$5, $X_2$6, and $X_2$7.
3. Spectral Operations:
   • (A) Obtain eigenpairs $X_2$8 of $X_2$9 via full diagonalization ($k$0), or
   • (B) Approximate $k$1 via $k$2-term Chebyshev or Lanczos polynomial filters in $k$3 time per application.
4. Wavelet Atoms: For scales $k$4 and edges $k$5, compute $k$6 (or, for separate Hodgelets, $k$7, $k$8).
5. Dictionary Construction: Compile the resulting $k$9 atoms as a wavelet dictionary for downstream analysis (e.g., sparse coding, clustering).

Polynomial filtering reduces computational costs to $\sigma$0 for practical $\sigma$1.

6. Empirical Evaluations and Comparative Performance

Empirical studies highlight the practical advantages of Hodgelets, especially in settings where distinguishing flow subspaces is significant.

Synthetic Hexagonal Grid (N₀=225, N₁=629, N₂=405):

• A 2-ball flow on a planar hexagonal complex is discretized as an edge-flow $\sigma$2.
• Sparse representations are compared using orthogonal matching pursuit (OMP) across node-linegraph wavelets, pure Fourier modes of $\sigma$3, joint Hodgelets, and separate Hodgelets.
• Separate Hodgelets attain equivalent $\sigma$4-error tolerance with far fewer atoms, indicating enhanced sparsity.

Ocean Drifter Trajectories near Madagascar (N₀=133, N₁=320, N₂=186, $\sigma$5 flows):

• Each buoy trajectory is modeled as an oriented edge-path. A Hodgelet dictionary is constructed and sparse $\sigma$6-means (with feature selection) is used to cluster trajectories into $\sigma$7 flows.
• Performance is evaluated on held-out paths by measuring the average maximum normalized inner product with centroids.
• Separate Hodgelets produce clusters whose centroids align more closely with new trajectories than those recovered by the edge, Fourier, or joint Hodgelet dictionaries.

These results demonstrate that, particularly where curl and gradient components are separated, Hodgelets yield representations that are sparse, physically meaningful, and computationally tractable (Roddenberry et al., 2021).

7. Interpretability, Applications, and Significance

Hodgelets extend spectral graph wavelet methodologies to higher-order network flows, providing a cohesive framework for localizing, analyzing, and sparsely encoding edge-flows subject to the intrinsic geometries of simplicial complexes. By explicitly respecting the Hodge–Helmholtz decomposition, they facilitate the recovery, clustering, and analysis of complex flows in both synthetic models and real-world data, such as oceanographic or transportation networks.

A plausible implication is that Hodgelets, especially in their separate form, offer an effective bridge between algebraic-topological signal processing and the physically interpretable analysis of network dynamics, enhancing both theoretical understanding and practical algorithmic tools for flow data on simplicial structures (Roddenberry et al., 2021).