Topological Signal Processing With Unoriented Operators
Andrea Cavallo, Andrea Cavallo, Varun Sarathchandran, Geert Leus, Elvin Isufi, Elvin Isufi
cs.LG
Quick verdict
Proposes an unoriented TSP framework with a proven interaction-order decomposition; theory is sound and regularizers win on four small real datasets.
Summary
This paper develops an unoriented framework for topological signal processing (UTSP) on simplicial complexes, motivated by higher-order signals that carry no natural orientation (e.g., co-authorship counts, correlations), for which oriented boundary operators introduce an arbitrary sign convention. The authors replace oriented boundaries with unoriented incidence matrices Q^{p,q}, show these induce Laplacians whose spectra follow from adjacency-based graph signal processing (C1), and prove an interaction-order decomposition: level-p signal spaces split into nested orthogonal bands W_k = V_k ∩ V_{k-1}^⊥, where V_k = range(Q^{k,p⊤}), quantifying how much of a signal is explained by aggregating lower-order simplices (Theorem 1). Unlike the Hodge decomposition's orthogonal gradient/curl/harmonic structure, bands here are nested, with kernels tied to signless-graph structures (e.g., bipartiteness) rather than topological invariants. The decomposition yields closed-form Tikhonov regularizers — a cohesion regularizer, an interaction-order regularizer R_ord = Σ β_k P_k, and a node-level penalty G — evaluated for denoising (σ=0.5) and 50% imputation on four small complexes (fp-landscape, acm-coauth, tags-math, NDC-substances; N2 between 204 and 1093), 20 trials each. The order regularizer wins every Table 1 cell, with the largest gains on fp-landscape (0.279 vs 0.728 NRMSE against the strongest competing regularizer for imputation at p=2) and smallest where energy matches the dimension profile. Limitations include small datasets, best-variant-per-cell selection, and no significance testing.
Review complete
Assessment at a glance
4
risk flags identified
● high 0
● medium 2
● low 2
5 strengths · 4 weaknesses · 5 author questions
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Key claims under review
- Unoriented incidence and Laplacian matrices between arbitrary simplicial levels are symmetric PSD, entrywise-nonnegative operators with graph-like spectral properties, making graph signal processing directly applicable to unoriented simplicial signals.
- Signals on any simplicial complex admit a unique nested orthogonal decomposition by interaction order (Theorem 1), an unoriented counterpart to the Hodge decomposition.
- Each unoriented Laplacian L^{un}_{p,q} acts exactly on the interaction bands of order at most q and is identically zero on higher-order bands (Theorem 1b).
- The interaction-order regularizer outperforms oriented and naive baselines in every cell of Table 1, with gains largest when signal energy is unevenly distributed across orders (fp-landscape: 74% of energy in 4% of dimensions).
Strengths
Provable, self-contained core theory
Identity (1), Q^{p,r}Q^{r,q} = C(q−p, r−p)Q^{p,q}, drives Theorem 1(a); Theorem 1(b) follows from ker(A⊤A)=ker A, giving range(L^{un}_{p,q})=V_q exactly — one-line proofs that fully determine which interaction bands each Laplacian can act on.
Sharp problem motivation
Sec. 1 shows that reorienting simplices transforms every Hodge Laplacian by a diagonal ±1 similarity, so any oriented filter output on an unoriented signal is arbitrary — a precise, checkable argument for count/correlation-type data.
Theory-to-algorithm pipeline
Consistent empirical wins with mechanism analysis
Reproducibility and transparency
Weaknesses
Small, low-level evaluation scope
Baseline and selection concerns
No statistical significance testing
Denoising gains are modest
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Frequently asked questions
What is unoriented topological signal processing (UTSP)?
A framework that processes signals on simplicial complexes using unoriented incidence matrices instead of oriented boundary operators. It targets higher-order data with no natural direction — counts, correlations, group-level properties — where choosing simplex orientations would be arbitrary and would change the output of any oriented filter.
How does the interaction-order decomposition differ from the Hodge decomposition?
The Hodge decomposition needs orientations and splits signals into orthogonal flow components (gradient, curl, harmonic). The interaction-order decomposition is orientation-free: it splits a signal into nested orthogonal bands measuring how much of it is explained by sums of contributions from lower-order simplices (nodes, pairs, ...), with the remainder being genuinely higher-order effects.
Is code available for this paper?
Yes. The authors state code is available at github.com/andrea-cavallo-98/Unoriented_TSP. All experiments are closed-form regularized least-squares solves on four public datasets, so results are CPU-reproducible without GPUs.
What datasets were used and how large were the gains?
Four complexes: a fluorescent-protein fitness landscape, ACM co-authorship, math.stackexchange tags, and NDC substances. The order-aware regularizer won every tested condition in Table 1; the largest gain was imputation on the fitness landscape (NRMSE 0.279 vs 0.728 for the strongest competing regularizer at p=2), while denoising gains were modest (e.g., 0.298 vs 0.427-0.446).
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
- Figures 1-4 are available only as extracted captions; the eigenvector plots (Fig. 2), the decomposition illustration (Fig. 3), and the band-energy profiles (Fig. 4a) could not be visually verified.
- Table 1 values were transcribed from linearized text; each mean ± s.d. was cross-checked against the Results narrative, but minor formatting artifacts are possible.
- No supplementary material was included in the extraction; extended proofs or additional experiments may exist beyond the 4-page main text.
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