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arxiv: 2605.22853 · v1 · pith:XUXO5KZ5new · submitted 2026-05-18 · 📡 eess.SP · cs.LG· q-bio.QM

Topological Signal Processing: An Application-Oriented Tutorial

Flavia Petruso , Maria Giulia Preti , Dimitri Van De Ville This is my paper

Pith reviewed 2026-05-25 00:11 UTC · model grok-4.3

classification 📡 eess.SP cs.LGq-bio.QM
keywords topological signal processingsimplicial complexescombinatorial Hodge Laplaciangraph signal processinghigher-order interactionsedge signalsbrain imaginglagged interactions
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The pith

Topological Signal Processing generalizes graph signal processing to signals on edges and higher simplices via the combinatorial Hodge Laplacian.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper provides an accessible tutorial on Topological Signal Processing as a direct extension of Graph Signal Processing. It centers on techniques that use the combinatorial Hodge Laplacian to handle signals defined on nodes, edges, triangles, and higher simplices modeled as simplicial complexes. The authors connect these methods to concrete data examples and include a brain imaging demonstration where an edge-level signal captures lagged interactions. The goal is to make TSP usable for analyzing higher-order relationships that graphs alone cannot represent.

Core claim

TSP generalizes GSP by representing data as simplicial complexes and applying the combinatorial Hodge Laplacian to extend filtering, Fourier transforms, and related operations to the topological level, which enables the study of higher-order interactions; this is illustrated by constructing an edge signal from lagged nodal interactions and applying it to brain imaging data to reveal nontrivial interactions between sets of regions.

What carries the argument

The combinatorial Hodge Laplacian, which extends the graph Laplacian from nodes to signals on edges and higher-dimensional simplices.

Load-bearing premise

The combinatorial Hodge Laplacian supplies the right generalization of the graph Laplacian for signals on higher-order simplices in actual datasets.

What would settle it

Applying the TSP pipeline to the brain imaging dataset yields no additional interactions beyond those already visible from standard node-signal analysis.

Figures

Figures reproduced from arXiv: 2605.22853 by Dimitri Van De Ville, Flavia Petruso, Maria Giulia Preti.

Figure 1
Figure 1. Figure 1: A simplex can indeed be viewed as a geometric object [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Simplices and Simplicial Complexes. a) Simplices of different orders depicted as geometric objects in the plane. b) Corresponding simplicial complexes obtained by including all faces of each simplex. c) Different types of simplicial complexes (top) and an example application for brain imaging modeling (bottom). For the brain mesh, the neighborhood relationships are determined by the cortical surface. In co… view at source ↗
Figure 2
Figure 2. Figure 2: Oriented Simplices, Boundary Matrices, and Operations on Edge Signals. a) Example of oriented simplicial complex of order 2 and its associated boundary matrices. Empty entries indicate zero. b) Edge signal living on the complex. The sign of each entry indicates whether the signal is aligned or anti￾aligned with the reference orientation of the corresponding edge, whereas the thickness of the arrow is propo… view at source ↗
Figure 3
Figure 3. Figure 3: Eigenvectors of the Laplacian of a regular mesh. a) First eigenvectors of L1↑, L1↓, and L1 are shown. The underlying mesh is composed of edges with the same unit weight, and all triangles are included in the simplicial complex. Note that the eigenvectors are not visualized directly at the level of the edges, but rather as vectors at the barycenter of the mesh triangles to facilitate visual interpretation. … view at source ↗
Figure 4
Figure 4. Figure 4: Geometric interpretation of the proposed edge signal. Schematic representation of different configurations of vectors yielding positive (left), negative (center) and zero (right) edge signals. The edge represents the signed area spanned by the vectors xi = (xi[t − 1], xi[t]) and xj = (xj [t − 1], xj [t]). Proof. Using linearity and JWSS, we find that E[Ei,j (t)] = E[Xi(t − 1)Xj (t) − Xj (t − 1)Xi(t)] = E[X… view at source ↗
Figure 5
Figure 5. Figure 5: Schematic of the dataset structure and processing. From left to right: magnetic resonance imaging (MRI) diffusion and functional imaging data are first obtained for every subject. On the domain side, diffusion imaging data are then preprocessed to extract the tractogram, which includes all the white matter fibers connecting different brain voxels. Then, a brain atlas is used to average the fibers connectin… view at source ↗
Figure 6
Figure 6. Figure 6: From GSP to TSP: Different perspectives on brain connectivity eigenvectors. a) Different views of the first nonzero eigenvector from the graph Laplacian L and lower component of the Hodge Laplacian (edge Laplacian) L1↓, which are associated with the lowest frequency of variation over the domain. Arrow thickness is proportional to the magnitude of the eigenvector on the corresponding edge. Note that each ar… view at source ↗
Figure 7
Figure 7. Figure 7: Brain edge signals and projections on the subspace of the Hodge Laplacian. Center: matrix showing the mean edge signal (across time and subjects) for each pair of brain regions, with ordering and grouping based on brain networks. The lower triangular matrix shows X¯ (1) ↓ , i.e., the projection of the average edge signal onto the space of L↓, whereas the upper triangular matrix shows X¯ (1) ↑ , i.e., the p… view at source ↗
Figure 8
Figure 8. Figure 8: Divergence over brain regions. a) Regions surviving the nonparametric sign-flip test on the mean (p-value ≤ 0.05, Bonferroni-corrected), colored by the empirical z-score of the mean with respect to the null distribution zˆ = µ−µperm σperm . Positive average divergence (red) and negative average divergence (blue) indicate that the region is consistently following or anticipating the ones it is connected to … view at source ↗
read the original abstract

Many modern datasets are large and carry complex structural relationships. Graph-based methods have traditionally been used to represent networked data, modeling individual elements as nodes and pairwise interactions as edges. Furthermore, Graph Signal Processing (GSP) has been developed to analyze signals on graph nodes, such as temperature measurements (node signals) across different regions of a country represented as a graph. Topological Signal Processing (TSP) is an emerging field that generalizes GSP, enabling the analysis of signals defined not only on nodes but also on edges, triangles, and higher-dimensional network elements, modeled as simplicial complexes and related topological structures. This makes TSP naturally well-suited for studying higher-order interactions in complex systems by extending classical signal processing concepts, such as filtering and Fourier transforms, to the topological level. Despite its versatility, TSP remains challenging for many practitioners. Therefore, we present an accessible overview of TSP foundations while drawing connections with application-oriented settings. We focus on processing techniques based on the combinatorial Hodge Laplacian, which generalizes the graph Laplacian to simplicial complexes. In particular, we review key TSP concepts, relate them to real-world examples, and discuss how higher-order structures and signals can be derived from datasets. For instance, we introduce an edge-level signal capturing lagged interactions between nodal signals, and demonstrate its use in a case study on TSP-based analysis of brain imaging data, revealing nontrivial interactions between sets of brain regions. Overall, we aim to promote a broader adoption of TSP by bridging methodological developments with applications, fostering its use among a wide community of theoretical and applied researchers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript is an application-oriented tutorial on Topological Signal Processing (TSP). It presents TSP as a generalization of Graph Signal Processing (GSP) that extends signal analysis from nodes to edges, triangles, and higher simplices via the combinatorial Hodge Laplacian. The paper reviews core TSP concepts (Hodge decomposition, topological Fourier transforms, filtering), connects them to datasets, shows how to construct higher-order signals (including an edge-level lagged interaction signal derived from nodal data), and illustrates the framework with a brain-imaging case study that claims to reveal nontrivial higher-order interactions among brain regions.

Significance. If the tutorial content is accurate and the case study convincingly isolates higher-order effects, the work could lower the barrier for practitioners in neuroscience and other complex-systems fields to adopt TSP. The explicit construction of an edge signal from nodal time series and the use of the Hodge Laplacian are concrete bridges between theory and data; however, the significance hinges on whether the brain-imaging demonstration adds information beyond standard node-level GSP.

major comments (2)
  1. [brain-imaging case study] Case-study section (brain-imaging demonstration): the claim that the edge-level lagged signal and Hodge decomposition reveal 'nontrivial interactions between sets of brain regions' is not supported by any reported baseline comparison against node-level GSP on the same data. No quantitative metric (unique variance explained by the curl/harmonic components, ablation of the 2-simplex terms, or statistical test against a graph-Laplacian null model) is provided, leaving open the possibility that the observed effects are recoverable from the underlying graph Laplacian alone.
  2. [edge-level signal construction] Section on construction of the edge-level signal: the lagged-interaction edge signal is introduced as a motivating example, yet the manuscript does not specify the precise lag-selection procedure, the normalization used to obtain a well-defined edge flow, or any sensitivity analysis showing that the subsequent Hodge decomposition is robust to these choices. This detail is load-bearing for reproducibility of the claimed higher-order findings.
minor comments (2)
  1. [foundations] Notation for the combinatorial Hodge Laplacian is introduced without an explicit comparison table to the graph Laplacian; adding such a side-by-side definition would improve accessibility for GSP readers.
  2. [application examples] Several real-world examples are mentioned in passing (temperature, brain imaging) but lack pointers to the exact public datasets or preprocessing pipelines used; supplying these references would aid readers who wish to replicate the constructions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our tutorial manuscript. We agree that the two major points identify areas where additional clarity and supporting analysis would strengthen the presentation, particularly for reproducibility and for substantiating the illustrative claims in the case study. We outline our responses and planned revisions below.

read point-by-point responses

  1. Referee: [brain-imaging case study] Case-study section (brain-imaging demonstration): the claim that the edge-level lagged signal and Hodge decomposition reveal 'nontrivial interactions between sets of brain regions' is not supported by any reported baseline comparison against node-level GSP on the same data. No quantitative metric (unique variance explained by the curl/harmonic components, ablation of the 2-simplex terms, or statistical test against a graph-Laplacian null model) is provided, leaving open the possibility that the observed effects are recoverable from the underlying graph Laplacian alone.

    Authors: We acknowledge that the brain-imaging demonstration is primarily illustrative and that the current text does not include a quantitative baseline comparison against node-level GSP. To address this, the revised manuscript will add a dedicated subsection that reports (i) the fraction of variance in the edge signal uniquely captured by the curl and harmonic components after regressing out the gradient component, (ii) an ablation removing contributions from 2-simplices, and (iii) a direct comparison of reconstruction error or prediction performance against a standard graph-Laplacian model on the same nodal time series. These additions will clarify whether the higher-order effects provide information beyond what is recoverable from the underlying graph structure. revision: yes

  2. Referee: [edge-level signal construction] Section on construction of the edge-level signal: the lagged-interaction edge signal is introduced as a motivating example, yet the manuscript does not specify the precise lag-selection procedure, the normalization used to obtain a well-defined edge flow, or any sensitivity analysis showing that the subsequent Hodge decomposition is robust to these choices. This detail is load-bearing for reproducibility of the claimed higher-order findings.

    Authors: We agree that the construction details are essential for reproducibility. In the revised manuscript we will expand the relevant section to state: (a) the lag-selection rule (maximum lagged cross-correlation within a physiologically plausible window), (b) the normalization step that converts pairwise lagged products into an antisymmetric edge flow consistent with the oriented incidence matrix, and (c) a sensitivity analysis that recomputes the Hodge decomposition for a range of lags and reports stability of the resulting curl/harmonic energy ratios. These additions will be placed immediately before the brain-imaging case study. revision: yes

Circularity Check

0 steps flagged

Expository tutorial presents no derivations or predictions.

full rationale

The paper is an application-oriented tutorial reviewing TSP foundations via the combinatorial Hodge Laplacian, relating concepts to examples, and illustrating an edge-level lagged signal in a brain imaging case study. No equations, first-principles derivations, parameter fits, or predictions are claimed that could reduce to inputs by construction. All content is expository and self-contained against external literature; no self-citation chains or ansatzes are load-bearing for any result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard domain assumptions from algebraic topology and signal processing without introducing new free parameters or invented entities.

axioms (1)
  • domain assumption The combinatorial Hodge Laplacian is the natural operator for extending signal processing concepts to simplicial complexes.
    Invoked as the foundation for TSP processing techniques in the abstract.

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