Topological Signal Processing With Unoriented Operators

# Topological Signal Processing with Unoriented Operators
Source: [https://arxiv.org/html/2609.25310](https://arxiv.org/html/2609.25310)
###### Abstract

Topological signal processing \(TSP\) processes signals on simplicial complexes with oriented boundary operators, which is the natural choice for flow signals or when the topological invariants play a role for the task at hand\. However, many higher\-order signals carry no orientation, and applying oriented operators to them is not well\-defined since it introduces an arbitrary choice of simplex orientation\. We study an unoriented TSP \(UTSP\) framework that replaces oriented boundaries with unoriented incidence matrices\. First, we show that unoriented incidence and Laplacian matrices between arbitrary simplicial levels admit graph\-like spectral properties\. Second, since dropping orientation removes the Hodge decomposition, we introduce an unoriented counterpart, termed interaction\-order decomposition, which quantifies how much of a higher\-order signal is explained by aggregating lower\-order signals\. Third, we use this decomposition to derive regularizers for signal reconstruction that penalize each interaction order separately\. Experiments on real\-world data show that the order\-aware regularizers outperform oriented baselines, with the largest gains when the signal energy is unevenly distributed across orders\.

###### Index Terms:

Topological Signal Processing, Simplicial Complexes, Higher\-Order Networks

††address:Delft University of Technology, Delft, Netherlands## 1Introduction

Signal processing on irregular domains has moved beyond graphs to model multi\-way relationships\. While graphs are restricted to pairwise interactions, many real\-world systems, such as co\-authorship networks, biochemical reactions, and social groups, exhibit higher\-order relationships involving three or more entities\[[2](https://arxiv.org/html/2609.25310#bib.bib7),[3](https://arxiv.org/html/2609.25310#bib.bib8)\]\. Simplicial complexes model these systems by generalizing nodes and edges to triangles, tetrahedra, and higher\-dimensional simplices\[[17](https://arxiv.org/html/2609.25310#bib.bib12)\]\.

To process signals defined on these domains, Topological Signal Processing \(TSP\)\[[1](https://arxiv.org/html/2609.25310#bib.bib9),[24](https://arxiv.org/html/2609.25310#bib.bib21),[15](https://arxiv.org/html/2609.25310#bib.bib11)\]builds upon algebraic topology\[[5](https://arxiv.org/html/2609.25310#bib.bib13)\]and equips each simplex with an orientation\. Edges point from a source to a target node, triangles are traversed clockwise or counterclockwise, and the boundary operators record these choices as positive and negative signs\. Thanks to this orientation, TSP operators encode topological invariants of the complex \(e\.g\., connected components or holes\) and simplicial signals can be decomposed into a gradient, a curl and a harmonic part \(the Hodge decomposition\), which is beneficial for filtering, detection and learning\[[29](https://arxiv.org/html/2609.25310#bib.bib22),[15](https://arxiv.org/html/2609.25310#bib.bib11),[16](https://arxiv.org/html/2609.25310#bib.bib30)\]\. Fixing an orientation is natural for flows, i\.e\., signals that flip sign when the orientation of their simplex is reversed, such as electrical currents and traffic or water flows\[[23](https://arxiv.org/html/2609.25310#bib.bib10)\]\.

However, many higher\-order signals carry no orientation at all, e\.g\., the number of papers written by a group of authors, the correlation between brain regions, or the strength of a molecular bond\. Applying oriented operators to such signals is not well\-defined\. Re\-orienting simplices transforms every Hodge Laplacian by a diagonal±1\\pm 1similarity, so the output of any oriented filter on an unoriented signal depends on an arbitrary choice\. This motivatesunorientedoperators, in which the signs of the boundary matrices are dropped and only set inclusion is retained\. Dropping orientation removes also the Hodge decomposition, and with it the structure that TSP uses to organize signals\. To still be able to carry on a similar analysis, we introduce its unoriented counterpart: a signal decomposition by interaction order, which measures how much an unoriented simplicial signal is explained by aggregation of lower\-order signals\.

Related works\.In the graph domain, unoriented operators such as the signless Laplacian are well studied theoretically\[[8](https://arxiv.org/html/2609.25310#bib.bib4),[9](https://arxiv.org/html/2609.25310#bib.bib23)\]and employed empirically\[[25](https://arxiv.org/html/2609.25310#bib.bib18),[22](https://arxiv.org/html/2609.25310#bib.bib3)\]\. On simplicial complexes,\[[7](https://arxiv.org/html/2609.25310#bib.bib1),[11](https://arxiv.org/html/2609.25310#bib.bib2)\]derive spectral bounds for the signless Laplacian;\[[19](https://arxiv.org/html/2609.25310#bib.bib5)\]uses a signless node–triangle Laplacian for triangle\-aware clustering\. In learning, the signless down\-Laplacian serves as a shift for orientation\-invariant edge signals\[[12](https://arxiv.org/html/2609.25310#bib.bib6)\], and unsigned incidence matrices are considered in message\-passing based topological neural networks\[[4](https://arxiv.org/html/2609.25310#bib.bib14),[14](https://arxiv.org/html/2609.25310#bib.bib15)\]\. For hypergraphs, incidence\-based Laplacians\[[31](https://arxiv.org/html/2609.25310#bib.bib24)\], hypergraph signal processing\[[30](https://arxiv.org/html/2609.25310#bib.bib16)\]and their spectra\[[6](https://arxiv.org/html/2609.25310#bib.bib17)\]are well developed\. Our operators between adjacent levels coincide with them, but they differ across non\-adjacent levels\. Finally, decomposing a function of several variables into main effects and interactions of increasing order is classical in statistics\[[10](https://arxiv.org/html/2609.25310#bib.bib25)\], and the same idea is used to quantify high\-order epistasis in fitness landscapes\[[18](https://arxiv.org/html/2609.25310#bib.bib26)\]\. However, these works leave untreated the closure of simplicial complexes under inclusion\. We show that this closure induces a nested structure on the signal spaces of consecutive levels, and develop a signal processing treatment that connects it to signal filtering and reconstruction\.

Contribution\.We study the Unoriented Topological Signal Processing \(UTSP\) framework, which replaces oriented boundary operators by unoriented incidence matrices and analyzes the structure of an unoriented higher\-order signal\. Our contributions are three\-fold\.

\(C1\)We show that unoriented incidence and Laplacian matrices between arbitrary simplicial levels have analogous spectral properties to the graphs they induce on simplices, thus making graph signal processing directly available for unoriented simplicial signals\.

\(C2\)We prove that signals on any simplicial complex admit a nested orthogonal decomposition by*interaction order*, and we show that unoriented Laplacians can only process signals up to a limited interaction order\.

\(C3\)We derive regularizers for signal reconstruction, including one that shrinks each interaction order separately\. On real datasets, these regularizers outperform oriented and naive baselines, with larger gains where the energy is more unevenly distributed across orders\.

Figure 1:Graphs induced by different unoriented Laplacians𝐋p,qu​n\\mathbf\{L\}\_\{p,q\}^\{un\}\(in blue\)\. Nodes correspond to simplices at levelpp, edges connect them if they share a simplex at levelqq\(shared simplices are orange\)\.
## 2Background and Unoriented Operators

In this section, we first recall simplicial complexes and oriented TSP\. Then, we introduce the unoriented incidence and Laplacian matrices and discuss their spectral properties\.

Simplicial complexes\.LetVVbe a finite vertex set\. App\-simplex is a subset ofVVwithp\+1p\+1elements, and a simplicial complex𝒦\\mathcal\{K\}is a collection of simplices closed under inclusion:σ∈𝒦\\sigma\\in\\mathcal\{K\}andτ⊆σ\\tau\\subseteq\\sigmaimplyτ∈𝒦\\tau\\in\\mathcal\{K\}\. A simplexτ\\tauis afaceofσ\\sigmaifτ⊆σ\\tau\\subseteq\\sigmaand aco\-faceofσ\\sigmaifσ⊆τ\\sigma\\subseteq\\tau\. We write𝒦p\\mathcal\{K\}\_\{p\}for the set ofpp\-simplices andNp=\|𝒦p\|N\_\{p\}=\|\\mathcal\{K\}\_\{p\}\|for their number; a*signal at levelpp*is a vector inℝNp\\mathbb\{R\}^\{N\_\{p\}\}\. Oriented representations, following algebraic topology\[[5](https://arxiv.org/html/2609.25310#bib.bib13)\], assign each simplex an orientation and define boundary matrices𝐁p∈\{0,±1\}Np−1×Np\\mathbf\{B\}\_\{p\}\\in\\\{0,\\pm 1\\\}^\{N\_\{p\-1\}\\times N\_\{p\}\}satisfying𝐁p​𝐁p\+1=𝟎\\mathbf\{B\}\_\{p\}\\mathbf\{B\}\_\{p\+1\}=\\mathbf\{0\}\. Here we drop orientations\.

Unoriented incidence\.For levelsp<qp<q, the unoriented incidence matrix𝐐p,q∈\{0,1\}Np×Nq\\mathbf\{Q\}\_\{p,q\}\\in\\\{0,1\\\}^\{N\_\{p\}\\times N\_\{q\}\}has\(𝐐p,q\)σ,τ=1\(\\mathbf\{Q\}\_\{p,q\}\)\_\{\\sigma,\\tau\}=1iffσ⊂τ\\sigma\\subset\\tau\. We use the conventions𝐐−1,q:=𝟏⊤\\mathbf\{Q\}\_\{\-1,q\}:=\\mathbf\{1\}^\{\\top\}\(the empty face lies in every simplex\) and𝐐q,q:=𝐈\\mathbf\{Q\}\_\{q,q\}:=\\mathbf\{I\}\. Unlike oriented boundaries, products of incidence matrices do not vanish\. Instead, they satisfy the following identity:

𝐐p,r​𝐐r,q=\(q−pr−p\)​𝐐p,q,p<r≤q\.\\mathbf\{Q\}\_\{p,r\}\\mathbf\{Q\}\_\{r,q\}=\\binom\{q\-p\}\{r\-p\}\\mathbf\{Q\}\_\{p,q\},\\quad p<r\\leq q\.\(1\)That is, going from levelppto levelqqthrough an intermediate levelrrequals going directly, up to a combinatorial constant\. This result is the foundation of the interaction\-order decomposition in Theorem[1](https://arxiv.org/html/2609.25310#Thmtheorem1)\.

Unoriented Laplacians\.For levelsp≠qp\\neq qwe define the unoriented Laplacian acting on levelppthrough levelqqas

𝐋p,qu​n=\{𝐐p,q​𝐐p,q⊤p<q,𝐐q,p⊤​𝐐q,pp\>q\.\\mathbf\{L\}^\{un\}\_\{p,q\}=\\begin\{cases\}\\mathbf\{Q\}\_\{p,q\}\\mathbf\{Q\}\_\{p,q\}^\{\\top\}&p<q,\\\\ \\mathbf\{Q\}\_\{q,p\}^\{\\top\}\\mathbf\{Q\}\_\{q,p\}&p\>q\.\\end\{cases\}\(2\)Its diagonal counts theqq\-simplices incident to app\-simplexσ\\sigmaand its off\-diagonal\(σ,τ\)\(\\sigma,\\tau\)entry counts theqq\-simplices incident to bothσ,τ\\sigma,\\tau, i\.e\., it encodes connectivity at levelppvia shared simplices at levelqq\(see Figure[1](https://arxiv.org/html/2609.25310#S1.F1)for examples\)\. Therefore,𝐋p,qu​n\\mathbf\{L\}^\{un\}\_\{p,q\}is the weighted adjacency matrix of the graph where simplices at levelppare nodes and edges connect two simplices if they share a simplex at levelqq, plus a diagonal term\. Consequently, the spectral properties of any single𝐋p,qu​n\\mathbf\{L\}^\{un\}\_\{p,q\}follow from adjacency\-based graph signal processing\[[21](https://arxiv.org/html/2609.25310#bib.bib31),[20](https://arxiv.org/html/2609.25310#bib.bib20)\]\. First, it is symmetric positive semidefinite\. Second, being entrywise non\-negative, on a connected graph, its top eigenvector is single\-signed and concentrates on the densest structures of that graph, i\.e\., cliques and hubs\[[8](https://arxiv.org/html/2609.25310#bib.bib4)\]\. Third, its zero eigenvalues correspond to signals whose values cancel when summed over everyqq\-simplex \(for𝐋1,0u​n\\mathbf\{L\}^\{un\}\_\{1,0\}, alternating signs along even cycles or pairs of odd cycles\)\. Figure[2](https://arxiv.org/html/2609.25310#S2.F2)illustrates these facts on a small complex\. Operators between non\-adjacent levels, such as the triangle\-through\-node Laplacian𝐋2,0u​n\\mathbf\{L\}^\{un\}\_\{2,0\}, have no oriented counterpart, since𝐁1​𝐁2=𝟎\\mathbf\{B\}\_\{1\}\\mathbf\{B\}\_\{2\}=\\mathbf\{0\}\. What single\-graph theory does not describe is the relation*between*the operators of the family\{𝐋p,qu​n\}q\\\{\\mathbf\{L\}^\{un\}\_\{p,q\}\\\}\_\{q\}, which we study next\.

![Refer to caption](https://arxiv.org/html/2609.25310v1/fig1_edge_operators.png)Figure 2:Top and bottom eigenvectors of the unoriented edge Laplacians𝐋1,0u​n,𝐋1,2u​n\\mathbf\{L\}^\{un\}\_\{1,0\},\\mathbf\{L\}^\{un\}\_\{1,2\}\. For both, the top eigenvectors localize on cliques and hubs\. The bottom eigenvector of𝐋1,0u​n\\mathbf\{L\}^\{un\}\_\{1,0\}is highly alternating on a 4\-cycle, while that of𝐋1,2u​n\\mathbf\{L\}^\{un\}\_\{1,2\}centers on a disconnected edge\. All 3\-cliques are filled by a triangle\.
## 3Interaction\-Order Decomposition

We now consider signals𝐬p∈ℝNp\\mathbf\{s\}\_\{p\}\\in\\mathbb\{R\}^\{N\_\{p\}\}, which assign a value to each simplex at levelpp\. The incidence operators allow lifting signals from one level to another\. For example, the operation𝐐p,q⊤​𝐬p\\mathbf\{Q\}\_\{p,q\}^\{\\top\}\\mathbf\{s\}\_\{p\}assigns to each simplexσ∈𝒦q\\sigma\\in\\mathcal\{K\}\_\{q\}the sum of the signals of its facesρ\\rhoat levelpp, whereρ∈𝒦p,ρ⊂σ\\rho\\in\\mathcal\{K\}\_\{p\},\\rho\\subset\\sigma\. In this way, one can generate edge signals as aggregations \(orlifts\) of node signals, triangle signals as lifts of node and edge signals, and so on\. Using this approach, one can decompose a signal𝐬p\\mathbf\{s\}\_\{p\}into contributions of lower\-level simplices, as we formalize next\.

We begin by defining the space𝒱k:=range⁡\(𝐐k,p⊤\)⊆ℝNp\\mathcal\{V\}\_\{k\}:=\\operatorname\{range\}\(\\mathbf\{Q\}\_\{k,p\}^\{\\top\}\)\\subseteq\\mathbb\{R\}^\{N\_\{p\}\}fork=−1,0,…,pk=\-1,0,\\dots,p, which identifies the space of level\-ppsignals that are lifts of signals onkk\-simplices, i\.e\., sums of contributions of\(k\+1\)\(k\{\+\}1\)\-entities\. We say that a signal in𝒱k\\mathcal\{V\}\_\{k\}hasinteraction orderat mostkk, i\.e\., it can be written by contributions of smaller simplices up to levelkk\. By the conventions in Section[2](https://arxiv.org/html/2609.25310#S2), we get𝒱−1=span⁡\(𝟏\)\\mathcal\{V\}\_\{\-1\}=\\mathrm\{span\}\(\\mathbf\{1\}\)and𝒱p=ℝNp\\mathcal\{V\}\_\{p\}=\\mathbb\{R\}^\{N\_\{p\}\}\. With this in place, we state the decomposition in the following theorem\.

###### Theorem 1\(Interaction\-order decomposition\)

For any simplicial complex and any levelpp:

1. \(a\)𝒱−1⊆𝒱0⊆⋯⊆𝒱p−1⊆𝒱p=ℝNp\\mathcal\{V\}\_\{\-1\}\\subseteq\\mathcal\{V\}\_\{0\}\\subseteq\\cdots\\subseteq\\mathcal\{V\}\_\{p\-1\}\\subseteq\\mathcal\{V\}\_\{p\}=\\mathbb\{R\}^\{N\_\{p\}\};
2. \(b\)for everyq<pq<p,range⁡\(𝐋p,qu​n\)=𝒱q\\ \\operatorname\{range\}\(\\mathbf\{L\}^\{un\}\_\{p,q\}\)=\\mathcal\{V\}\_\{q\}andker⁡\(𝐋p,qu​n\)=𝒱q⟂\\ \\ker\(\\mathbf\{L\}^\{un\}\_\{p,q\}\)=\\mathcal\{V\}\_\{q\}^\{\\perp\}\.

###### Proof\.

\(a\) \([1](https://arxiv.org/html/2609.25310#S2.E1)\) with levels\(k,k\+1,p\)\(k,k\+1,p\)and0≤k≤p−10\\leq k\\leq p\-1gives𝐐k,k\+1​𝐐k\+1,p=\(p−k\)​𝐐k,p\\mathbf\{Q\}\_\{k,k\+1\}\\mathbf\{Q\}\_\{k\+1,p\}=\(p\-k\)\\,\\mathbf\{Q\}\_\{k,p\}\. Transposing,𝐐k,p⊤=1p−k​𝐐k\+1,p⊤​𝐐k,k\+1⊤\\mathbf\{Q\}\_\{k,p\}^\{\\top\}=\\tfrac\{1\}\{p\-k\}\\mathbf\{Q\}\_\{k\+1,p\}^\{\\top\}\\mathbf\{Q\}\_\{k,k\+1\}^\{\\top\}, so every vector in𝒱k\\mathcal\{V\}\_\{k\}is the lift of a vector on\(k\+1\)\(k\{\+\}1\)\-simplices, i\.e\.,𝒱k⊆𝒱k\+1\\mathcal\{V\}\_\{k\}\\subseteq\\mathcal\{V\}\_\{k\+1\}\. Fork=−1k=\-1use𝐐0,p⊤​𝟏=\(p\+1\)​𝟏\\mathbf\{Q\}\_\{0,p\}^\{\\top\}\\mathbf\{1\}=\(p\+1\)\\mathbf\{1\}\. \(b\) With𝐀=𝐐q,p\\mathbf\{A\}=\\mathbf\{Q\}\_\{q,p\},𝐋p,qu​n=𝐀⊤​𝐀\\mathbf\{L\}^\{un\}\_\{p,q\}=\\mathbf\{A\}^\{\\top\}\\mathbf\{A\}hasker⁡\(𝐀⊤​𝐀\)=ker⁡𝐀=range⁡\(𝐀⊤\)⟂=𝒱q⟂\\ker\(\\mathbf\{A\}^\{\\top\}\\mathbf\{A\}\)=\\ker\\mathbf\{A\}=\\operatorname\{range\}\(\\mathbf\{A\}^\{\\top\}\)^\{\\perp\}=\\mathcal\{V\}\_\{q\}^\{\\perp\}, hencerange⁡\(𝐋p,qu​n\)=ker⁡\(𝐋p,qu​n\)⟂=𝒱q\\operatorname\{range\}\(\\mathbf\{L\}^\{un\}\_\{p,q\}\)=\\ker\(\\mathbf\{L\}^\{un\}\_\{p,q\}\)^\{\\perp\}=\\mathcal\{V\}\_\{q\}\. ∎

Theorem[1](https://arxiv.org/html/2609.25310#Thmtheorem1)proves that, for a fixed levelpp, the space of signals with interaction order at mostp−2p\-2is a subspace of that of signals with interaction order at mostp−1p\-1, and so on\. Since the spaces are nested,ℝNp\\mathbb\{R\}^\{N\_\{p\}\}splits orthogonally into the*bands*𝒲k:=𝒱k∩𝒱k−1⟂\\mathcal\{W\}\_\{k\}:=\\mathcal\{V\}\_\{k\}\\cap\\mathcal\{V\}\_\{k\-1\}^\{\\perp\},k=−1,…,pk=\-1,\\dots,pwith𝒱−2=\{0\}\\mathcal\{V\}\_\{\-2\}=\\\{0\\\}\. Writing𝐏k\\mathbf\{P\}\_\{k\}for the orthogonal projector onto𝒲k\\mathcal\{W\}\_\{k\}, every level\-ppsignal decomposes uniquely as

𝐱=∑k=−1p𝐏k​𝐱,πk​\(𝐱\):=‖𝐏k​𝐱‖22‖𝐱‖22,\\mathbf\{x\}=\\sum\_\{k=\-1\}^\{p\}\\mathbf\{P\}\_\{k\}\\mathbf\{x\},\\qquad\\pi\_\{k\}\(\\mathbf\{x\}\):=\\frac\{\\\|\\mathbf\{P\}\_\{k\}\\mathbf\{x\}\\\|\_\{2\}^\{2\}\}\{\\\|\\mathbf\{x\}\\\|\_\{2\}^\{2\}\},\(3\)where𝐏k​𝐱\\mathbf\{P\}\_\{k\}\\mathbf\{x\}is the*pure order\-kk*component andπk\\pi\_\{k\}the fraction of energy at orderkk\. The projectors are computed by least squares on the incidence matrices \(𝐏𝒱k=𝐐⊤​\(𝐐𝐐⊤\)\+​𝐐\\mathbf\{P\}\_\{\\mathcal\{V\}\_\{k\}\}=\\mathbf\{Q\}^\{\\top\}\(\\mathbf\{Q\}\\mathbf\{Q\}^\{\\top\}\)^\{\+\}\\mathbf\{Q\}with𝐐=𝐐k,p\\mathbf\{Q\}=\\mathbf\{Q\}\_\{k,p\}, then𝐏k=𝐏𝒱k−𝐏𝒱k−1\\mathbf\{P\}\_\{k\}=\\mathbf\{P\}\_\{\\mathcal\{V\}\_\{k\}\}\-\\mathbf\{P\}\_\{\\mathcal\{V\}\_\{k\-1\}\}\)\.

Interpretation\.Consider a triangle signal\. Part of it may be explained by each node having a strength and the triangle value being their sum \(order00\); a further part by each pair contributing \(order11\); what is left is genuinely three\-way \(order22\)\. Theorem[1](https://arxiv.org/html/2609.25310#Thmtheorem1)\(a\) states that these explanations are nested, so the split into “main effects, pairwise effects, genuinely higher\-order effects” is well defined on any complex\. The reason is \([1](https://arxiv.org/html/2609.25310#S2.E1)\): a node\-additive triangle signal is also pair\-additive, since assigning each edge half the sum of its endpoint strengths reproduces the same result\. Nesting requires closure under faces, so this structure is specific to simplicial complexes\. Figure[3](https://arxiv.org/html/2609.25310#S3.F3)shows a visualization of this signal decomposition\. Theorem[1](https://arxiv.org/html/2609.25310#Thmtheorem1)\(b\) then identifies exactly what each unoriented Laplacian sees:𝐋p,qu​n\\mathbf\{L\}^\{un\}\_\{p,q\}acts on the components of order at mostqqand is identically zero on all higher orders\. For example, the triangles\-through\-nodes Laplacian𝐋2,0u​n\\mathbf\{L\}^\{un\}\_\{2,0\}sees the node\-additive part only; the triangles\-through\-edges Laplacian𝐋2,1u​n\\mathbf\{L\}^\{un\}\_\{2,1\}sees orders≤1\\leq 1; neither touches the pure three\-way band\. For oriented operators there is no such statement since their ranges are mutually orthogonal \(the Hodge decomposition\) rather than nested, and they mix the bands\.

Comparison with oriented TSP\.Two main differences follow from the definitions\. \(i\) The oriented framework organizes signals by flow structure \(gradient, curl, harmonic\); the unoriented one organizes them by interaction order\. \(ii\) The kernels of the Hodge Laplacians identify topological invariants \(e\.g\., number of connected components or topological holes\), while the kernel of the unoriented Laplacian corresponds to different structures:dimker⁡𝐋0,1u​n\\dim\\ker\\mathbf\{L\}^\{un\}\_\{0,1\}counts bipartite components;ker⁡𝐋1,0u​n\\ker\\mathbf\{L\}^\{un\}\_\{1,0\}is spanned by alternating signals on even cycles and pairs of odd cycles joined by a path\[[13](https://arxiv.org/html/2609.25310#bib.bib29)\]\(cf\. Figure[2](https://arxiv.org/html/2609.25310#S2.F2)\)\.

![Refer to caption](https://arxiv.org/html/2609.25310v1/fig_bands_illustration.png)Figure 3:Interaction\-order decomposition of a triangle signal𝐱\\mathbf\{x\}\(Theorem[1](https://arxiv.org/html/2609.25310#Thmtheorem1)\)\.𝐱\\mathbf\{x\}is decomposed into four orthogonal components: one constant, one node\-additive𝐏0​𝐱\\mathbf\{P\}\_\{0\}\\mathbf\{x\}, one pair\-additive𝐏1​𝐱\\mathbf\{P\}\_\{1\}\\mathbf\{x\}, and the remainder three\-way effect\. The triangle below the complex is shown around it\.
## 4Filters and Signal Reconstruction

In this section, we present practical applications of this framework\. Throughout this section the signal𝐬\\mathbf\{s\}lives at levelppand is processed through a lower levelq<pq<p; by Theorem[1](https://arxiv.org/html/2609.25310#Thmtheorem1)\(b\) the Laplacian𝐋p,qu​n\\mathbf\{L\}^\{un\}\_\{p,q\}then acts on the components of𝐬\\mathbf\{s\}of order at mostqqand is zero on the components of orderq\+1,…,pq\+1,\\dots,p\. For a triangle signal \(p=2p=2\) related through nodes \(q=0q=0\), the operator acts on the node\-additive part, a subspace of dimension at mostN0N\_\{0\}insideℝN2\\mathbb\{R\}^\{N\_\{2\}\}, and ignores the pair\-explained and genuinely three\-way content\.

Filters\.Let𝐋p,qu​n=𝐕​𝚲​𝐕⊤\\mathbf\{L\}^\{un\}\_\{p,q\}=\\mathbf\{V\}\\mathbf\{\\Lambda\}\\mathbf\{V\}^\{\\top\}with eigenvaluesλ1≥λ2⋯≥λNp≥0\\lambda\_\{1\}\\geq\\lambda\_\{2\}\\cdots\\geq\\lambda\_\{N\_\{p\}\}\\geq 0\. A linear shift\-invariant filter at levelppis𝐬^=𝐕​g​\(𝚲\)​𝐕⊤​𝐬\\mathbf\{\\hat\{s\}\}=\\mathbf\{V\}g\(\\mathbf\{\\Lambda\}\)\\mathbf\{V\}^\{\\top\}\\mathbf\{s\}for a spectral responsegg\. Since𝐋p,qu​n\\mathbf\{L\}^\{un\}\_\{p,q\}is a non\-negative shift, we define frequency as the distance from the top eigenvalueω=λ1−λ\\omega=\\lambda\_\{1\}\-\\lambda\[[21](https://arxiv.org/html/2609.25310#bib.bib31)\], so that the top eigenvector \(cohesive, single\-signed\) has frequency00and highly alternating eigenvectors have high frequency\. We next show that filters of this form are the solution to signal denoising problems\.

Reconstruction\.Let𝐬∈ℝNp\\mathbf\{s\}\\in\\mathbb\{R\}^\{N\_\{p\}\}and𝐲=𝐌⁡\(𝐬\+𝐧\)\\mathbf\{y\}=\\mathbf\{M\}\(\\mathbf\{s\}\+\\mathbf\{n\}\), where𝐧\\mathbf\{n\}is zero\-mean white noise and𝐌\\mathbf\{M\}is a diagonal0/10/1sampling mask \(𝐌=𝐈\\mathbf\{M\}=\\mathbf\{I\}for denoising\)\. We estimate

𝐬^=arg⁡min𝐱⁡‖𝐌⁡\(𝐲−𝐱\)‖22\+α​𝐱⊤​𝐑​𝐱\+γ​𝐱⊤​𝚪​𝐱,\\hat\{\\mathbf\{s\}\}=\\arg\\min\_\{\\mathbf\{x\}\}\\ \\\|\\mathbf\{M\}\(\\mathbf\{y\}\-\\mathbf\{x\}\)\\\|\_\{2\}^\{2\}\+\\alpha\\,\\mathbf\{x\}^\{\\top\}\\mathbf\{R\}\\,\\mathbf\{x\}\+\\gamma\\mathbf\{x\}^\{\\top\}\\mathbf\{\\Gamma\}\\mathbf\{x\},\(4\)with𝐑⪰0\\mathbf\{R\}\\succeq 0a regularizer encoding the structural prior on𝐬\\mathbf\{s\}, and𝚪⪰0\\mathbf\{\\Gamma\}\\succeq 0a secondary regularizer:𝚪=𝐈\\mathbf\{\\Gamma\}=\\mathbf\{I\}recovers the standard norm\-2 penalty\. The closed form solution is𝐬^=\(𝐌\+α​𝐑\+γ​𝚪\)−1​𝐲\\hat\{\\mathbf\{s\}\}=\(\\mathbf\{M\}\+\\alpha\\mathbf\{R\}\+\\gamma\\mathbf\{\\Gamma\}\)^\{\-1\}\\mathbf\{y\}\. For𝐌=𝚪=𝐈\\mathbf\{M\}=\\mathbf\{\\Gamma\}=\\mathbf\{I\}and𝐑=h⁡\(𝐋p,qu​n\)\\mathbf\{R\}=h\(\\mathbf\{L\}^\{un\}\_\{p,q\}\)the solution is the filterg⁡\(λ\)=1/\(1\+γ\+α​h​\(λ\)\)g\(\\lambda\)=1/\(1\+\\gamma\+\\alpha h\(\\lambda\)\)\. Different choices for𝐑,𝚪\\mathbf\{R\},\\mathbf\{\\Gamma\}follow from theory\.

\(1\) Cohesion regularizer\.If𝐬\\mathbf\{s\}is expected to concentrate on cliques or hubs, set𝐑=λ1​𝐈−𝐋p,qu​n\\mathbf\{R\}=\\lambda\_\{1\}\\mathbf\{I\}\-\\mathbf\{L\}^\{un\}\_\{p,q\}, i\.e\.g⁡\(λ\)=1/\(1\+γ\+α⁡\(λ1−λ\)\)g\(\\lambda\)=1/\(1\+\\gamma\+\\alpha\(\\lambda\_\{1\}\-\\lambda\)\)\. This is a soft low\-pass inω\\omega: the top eigenvector is shrunk by1/\(1\+γ\)1/\(1\+\\gamma\), and every other eigenvector is shrunk further in proportion to its distanceλ1−λ\\lambda\_\{1\}\-\\lambdafrom the top\.

\(2\) Interaction\-order regularizer\.The cohesion regularizer cannot tell the interaction bands apart\. The components of𝐬\\mathbf\{s\}of order aboveqqlie inker⁡𝐋p,qu​n\\ker\\mathbf\{L\}^\{un\}\_\{p,q\}, where the penalty reduces toλ1​𝐈\\lambda\_\{1\}\\mathbf\{I\}: in denoising they are all shrunk by the same scalar whatever they contain, and in imputation, since𝐋p,qu​n\\mathbf\{L\}^\{un\}\_\{p,q\}is the only coupling between simplices, no observed value propagates into them\. The bands of order at mostqqare treated through the spectrum of𝐋p,qu​n\\mathbf\{L\}^\{un\}\_\{p,q\}, which mixes them\. Genuinely high\-order content, e\.g\., the pure three\-way part of a triangle signal, is therefore neither denoised nor inferred, and no choice ofqqallows shrinking one interaction band independently of another\. To act on every band, we use the projectors of \([3](https://arxiv.org/html/2609.25310#S3.E3)\) and define theinteraction\-order regularizeras

𝐑ord=∑k=−1pβk​𝐏k,βk≥0,\\mathbf\{R\}\_\{\\mathrm\{ord\}\}=\\sum\_\{k=\-1\}^\{p\}\\beta\_\{k\}\\,\\mathbf\{P\}\_\{k\},\\qquad\\beta\_\{k\}\\geq 0,\(5\)with band weightsβk\\beta\_\{k\}, whose Tikhonov solution for𝐌=𝚪=𝐈\\mathbf\{M\}=\\mathbf\{\\Gamma\}=\\mathbf\{I\}is𝐬^=∑k\(1\+γ\+α​βk\)−1​𝐏k​𝐲\\hat\{\\mathbf\{s\}\}=\\sum\_\{k\}\(1\+\\gamma\+\\alpha\\beta\_\{k\}\)^\{\-1\}\\mathbf\{P\}\_\{k\}\\mathbf\{y\}: each interaction band of the level\-ppsignal, including the top one, is shrunk by its own factor\. We consider two profiles forβk\\beta\_\{k\}: a smooth profile such asβk=\(k\+1\)ν\\beta\_\{k\}=\(k\+1\)^\{\\nu\}, where a largerν\\nushrinks higher interaction orders more, or a hard cutcutm\\mathrm\{cut\}\_\{m\}whereβk=0\\beta\_\{k\}=0fork<mk<mandβk=1\\beta\_\{k\}=1fork≥mk\\geq m, which leaves the bands below ordermmshrunk only byγ\\gammaand attenuates the rest by1/\(1\+γ\+α\)1/\(1\+\\gamma\+\\alpha\)\.

\(3\) Band\-aware secondary regularizer\.We propose the alternative secondary regularizer𝚪=𝐆\+ϵ/γ​𝐈\\mathbf\{\\Gamma\}=\\mathbf\{G\}\+\\epsilon/\\gamma\\mathbf\{I\}, whereϵ=10−10\\epsilon=10^\{\-10\}ensures a solution to \([4](https://arxiv.org/html/2609.25310#S4.E4)\) and𝐆:=𝐐0,p⊤​\(𝐐0,p​𝐐0,p⊤\)\+2​𝐐0,p=\(\(𝐐0,p⊤\)\+\)⊤​\(𝐐0,p⊤\)\+\\mathbf\{G\}:=\\mathbf\{Q\}\_\{0,p\}^\{\\top\}\(\\mathbf\{Q\}\_\{0,p\}\\mathbf\{Q\}\_\{0,p\}^\{\\top\}\)^\{\+2\}\\mathbf\{Q\}\_\{0,p\}=\\big\(\(\\mathbf\{Q\}\_\{0,p\}^\{\\top\}\)^\{\+\}\\big\)^\{\\\!\\top\}\(\\mathbf\{Q\}\_\{0,p\}^\{\\top\}\)^\{\+\}, so that𝐱⊤​𝐆𝐱=‖𝐜⋆‖2\\mathbf\{x\}^\{\\top\}\\mathbf\{G\}\\mathbf\{x\}=\\\|\\mathbf\{c\}^\{\\star\}\\\|^\{2\}with𝐜⋆\\mathbf\{c\}^\{\\star\}the smallest node signal satisfying𝐐0,p⊤​𝐜⋆=𝐏𝒱0​𝐱\\mathbf\{Q\}\_\{0,p\}^\{\\top\}\\mathbf\{c\}^\{\\star\}=\\mathbf\{P\}\_\{\\mathcal\{V\}\_\{0\}\}\\mathbf\{x\}\. That is,𝐆\\mathbf\{G\}penalizes the node\-level explanation of𝐱\\mathbf\{x\}and vanishes iff𝐱⟂𝒱0\\mathbf\{x\}\\perp\\mathcal\{V\}\_\{0\}\. The plain ridge instead gives, for𝐱=𝐐0,p⊤​𝐜∈𝒱0\\mathbf\{x\}=\\mathbf\{Q\}\_\{0,p\}^\{\\top\}\\mathbf\{c\}\\in\\mathcal\{V\}\_\{0\},‖𝐱‖2=𝐜⊤​𝐋0,pu​n​𝐜\\\|\\mathbf\{x\}\\\|^\{2\}=\\mathbf\{c\}^\{\\top\}\\mathbf\{L\}^\{un\}\_\{0,p\}\\mathbf\{c\}, which weights each node strength by the number ofpp\-simplices containing it\. This matters in imputation when the observed simplices are fewer than the nodes: the node\-additive fit is then underdetermined, and‖𝐱‖2\\\|\\mathbf\{x\}\\\|^\{2\}resolves the ambiguity by favouring nodes that appear in fewpp\-simplices, whereas𝐆\\mathbf\{G\}acts as a ridge penalty on the node coefficients themselves and picks the most parsimonious set of node strengths consistent with the data\.

Figure 4:\(a\) Energy per band \(πk\\pi\_\{k\}, left\) against its share of the dimensions \(right\)\. \(b\) Error in the top band after denoising \(∥𝐏p​\(𝐬^−𝐬\)∥22/∥𝐏p​𝐬∥22\\lVert\\mathbf\{P\}\_\{p\}\(\\hat\{\\mathbf\{s\}\}\-\\mathbf\{s\}\)\\rVert\_\{2\}^\{2\}/\\lVert\\mathbf\{P\}\_\{p\}\\mathbf\{s\}\\rVert\_\{2\}^\{2\}\); values above1\.01\.0mean the estimator injects error into it\. We omitacmsince the top band holds≤0\.05%\\leq 0\.05\\%of the energy\.Table 1:NRMSE atσ=0\.5\\sigma=0\.5/50%50\\%missing, mean±\\pms\.d\. over2020trials; best per block inbold\.†\\daggermarks the two proposed regularizers; the rest are baselines\. cohesion and oriented are at their best variant per cell \(overqq, and over the Hodge Laplacians\)\.
## 5Experiments

Data and preprocessing\.We use four complexes with signals at levels22and33\(i\.e\., triangles and tetrahedra\)\. Infp\-landscapenodes are the1313mutated sites of a fluorescent protein, akk\-simplex is the genotype carrying that set of mutations, and the signal is its measured brightness\[[18](https://arxiv.org/html/2609.25310#bib.bib26)\]; every subset was measured, so the complex is the full simplex \(N0​…​3=13,78,286,715N\_\{0\\ldots 3\}=13,78,286,715\)\. Two are hypergraph benchmarks\[[3](https://arxiv.org/html/2609.25310#bib.bib8)\]: intags\-math, nodes are tags and simplices are the sets of tags applied to questions on math\.stackexchange\.com, while inNDC\-substanceseach simplex is a drug and the nodes are the substances making it up\. For both, the complex is the downward closure of the observed hyperedges and the level\-kksignal is the count of events whose participant set is*exactly*thatkk\-simplex, transformed aslog⁡\(1\+count\)\\log\(1\+\\text\{count\}\)\. Inacm\-coauththe nodes are authors and akk\-simplex is a set ofk\+1k\+1co\-authors of a paper\[[28](https://arxiv.org/html/2609.25310#bib.bib19)\]; the signal is thekk\-way co\-moment of the TF\-IDF\[[26](https://arxiv.org/html/2609.25310#bib.bib28)\]of their term vectors, i\.e\., the vocabulary the whole group shares\. We select the maximum hyperedge size to ensure that thep=3p=3top band remains non\-empty\. This results in the sizes6,6,86,6,8fortags\-math,NDC\-substancesandacm\-coauthrespectively\. Then, we subsample nodes to maximiseN2/N0N\_\{2\}/N\_\{0\}subject toN2∈\[200,1200\]N\_\{2\}\\in\[200,1200\], which results in the valuesN2=1093,204,689N\_\{2\}=1093,204,689\. Signals are standardised\. Our code is available online111[https://github\.com/andrea\-cavallo\-98/Unoriented\_TSP](https://github.com/andrea-cavallo-98/Unoriented_TSP)\.

Setup\.We solve \([4](https://arxiv.org/html/2609.25310#S4.E4)\) for denoising \(𝐌=𝐈\\mathbf\{M\}=\\mathbf\{I\}\) and imputation \(error on the removed entries\), scoringRMSE/std⁡\(𝐬\)\\mathrm\{RMSE\}/\\mathrm\{std\}\(\\mathbf\{s\}\)over2020trials at noise levelσ=0\.5\\sigma=0\.5and50%50\\%missing\. Every𝐑\\mathbf\{R\}has unit spectral radius;α\\alphaspans\[10−3,103\]\[10^\{\-3\},10^\{3\}\]andγ∈\{10−3,10−1,10\}\\gamma\\in\\\{10^\{\-3\},10^\{\-1\},10\\\}, chosen by SURE\[[27](https://arxiv.org/html/2609.25310#bib.bib27)\]for denoising and a25%25\\%validation split for imputation\. We compare two naive baselines \(ridgefor denoising,neighborhood meanfor imputation\); theorientedTikhonov regularizer \(every Hodge Laplacian, low\- and high\-pass\);cohesionfor everyq<pq<p; and theorderregularizer in \([5](https://arxiv.org/html/2609.25310#S4.E5)\) withβ∈\{\(k\+1\)4,cut1,cut2\}\\beta\\in\\\{\(k\+1\)^\{4\},\\mathrm\{cut\}\_\{1\},\\mathrm\{cut\}\_\{2\}\\\}\. In imputation we also select𝚪\\mathbf\{\\Gamma\}between𝐈,𝐆\\mathbf\{I\},\\mathbf\{G\}, while for denoising we fix𝚪=𝐈\\mathbf\{\\Gamma\}=\\mathbf\{I\}\. For each method, we report the results of the best Laplacian\.

Results\.The order regularizer is best in all cells of Table[1](https://arxiv.org/html/2609.25310#S4.T1)\. The gains w\.r\.t\. the oriented estimator show that the correct assumption on signal orientedness plays a relevant role in reconstruction quality\. The energy per band differs sharply across the four complexes \(Fig\.[4](https://arxiv.org/html/2609.25310#S4.F4)a\), and the advantage of the order regularizer over the naive baseline tracks how far the energy profile departs from the dimension profile\. In particular, it is largest onfp\-landscape, where 74% of the energy occupies 4% of the dimensions, and smallest ontags\-mathp=3p\{=\}3andNDC\-substances, where the two nearly coincide\. Considering the top band energy \(Fig\.[4](https://arxiv.org/html/2609.25310#S4.F4)b\), the order regularizer is the only one to recover it where it has energy and to avoid injecting error where it is noise\-dominated\. Finally, the secondary regularizer𝐆\\mathbf\{G\}is selected in66%66\\%ofacm\-coauthimputations, wheredim𝒱0\\dim\\mathcal\{V\}\_\{0\}exceeds the observed count, against7%7\\%onfp\-landscape\(dim𝒱0=13\\dim\\mathcal\{V\}\_\{0\}=13against143143observations\)\. That is,𝐆\\mathbf\{G\}helps when the node\-additive subspace is not identified by the observations\.

## 6Conclusion

We studied the Unoriented Topological Signal Processing \(UTSP\) framework, which replaces oriented boundary operators with unoriented incidence matrices and Laplacians\. We characterized their spectral properties and established an orthogonal nested decomposition by interaction order, from which we built order\-specific regularizers for signal denoising and imputation\. Empirical evaluations on real\-world datasets confirm that the unoriented framework largely outperforms oriented competitors and interaction order\-specific regularization helps for signals distributed unevenly across orders\.

## 7Acknowledgments

This work was supported in part by the TU Delft AI Labs programme, NWO OTP GraSPA proposal \#19497, NWO VENI proposal 222\.032, and by the SURE\-AI Centre grant \#357482, Research Council of Norway\. Claude Opus 5 \(Anthropic\) was used to assist in writing the manuscript and coding the simulations\. The authors take full responsibility for the results of this paper\.

## 8Compliance with Ethical Standards

This is a numerical simulation study for which no ethical approval was required\.

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