Hodge-Aware Convolutional Learning on Simplicial Complexes
Authors: Maosheng Yang
Geert Leus
Elvin Isufi
Venue: Accepted by TMLR ()

Abstract: Neural networks on simplicial complexes (SCs) can learn representations from data residing on simplices such as nodes, edges, triangles, etc. However, existing works often overlook the Hodge theorem that decomposes simplicial data into three orthogonal characteristic subspaces, such as the identifiable gradient, curl and harmonic components of edge flows. This provides a universal tool to understand the machine learning models on SCs, thus, allowing for better principled and effective learning. In this paper, we study the effect of this data inductive bias on learning on SCs via the principle of convolutions. Particularly, we present a general convolutional architecture that respects the three key principles of uncoupling the lower and upper simplicial adjacencies, accounting for the inter-simplicial couplings, and performing higher-order convolutions. To understand these principles, we first use Dirichlet energy minimizations on SCs to interpret their effects on mitigating simplicial oversmoothing. Then, we show the three principles promote the Hodge-aware learning of this architecture, through the lens of spectral simplicial theory, in the sense that the three Hodge subspaces are invariant under its learnable functions and the learning in two nontrivial subspaces is independent and expressive. Third, we investigate the learning ability of this architecture in optic of perturbation theory on simplicial topologies and prove that the convolutional architecture is stable to small perturbations. Finally, we corroborate the three principles by comparing with methods that either violate or do not respect them. Overall, this paper bridges learning on SCs with the Hodge theorem, highlighting its importance for rational and effective learning from simplicial data, and provides theoretical insights to convolutional learning on SCs.

Published in Transactions on Machine Learning Research (08/2025)
Hodge-Aware Convolutional Learning on Simplicial Complexes
Maosheng Yang
m-yong-28tuded
Deportment of Intelligent Systems Delft University of Technology
Geort Leus
g.j.t. B.j.t.ees Jeusstudellf...
Department Microclectronics
Delft University of Technology
Elvin Isufi
e.isufi- @tudelft..d
Department of Intelligent Systems Delft University of Technology
Reviewed on OpenReview: https: // operreview. net/ pet//fornt forum? id- N955
Abstract
Neural networks on simplicial complexes (SCs) can learn representations from data residing on simplices such as nodes, edges, triangles, etc. However, existing works often overlook the Hodge theorem that decomposes simplicial data into three orthogonal characteristic subspaces, such as the identifiable gradient, curl and harmonic components of edge flows. This provides a universal tool understand the machine learning models on SCs, thus, allowing for better principled and effective lcarning. In this paper, study the effect of this data inductive bias on learning on SCs via the principle of convolutions. Particularly, we present a general convolutional architecture that respects the three key principles of uncoupling the lower and upper simplicial adjacencies, accounting for the inter-simplicial couplings, and performing higher-order convolutions. To understand these principles, we first use Dirichlet energy minimizations on SCs to interpret their effects on mitigating simplicial oversmoothing. Then, we show the three principles promote the Hodge-aware learning of this architecture, through the lens of spectral simplicial theory, in the sense that the three Hodge subspaces invariant under its learnable functions and the learning in two nontrivial subspaces is independent and expressive. Third, we investigate the learning ability of this architecture in optic of perturbation theory on simplicial topologies and prove that the convolutional architecture stable to small perturbations. Finally, we corroborate the three principles by comparing with methods that either violate or do not respect them. Overall, this paper bridges lcarning on SCs with the Hodge theorem, highlighting its importance for rational and effective learning from simplicial dala, and provides theoretical insights to convolutional learning on SCs.
1 Introduction
In the line of geometric deep learning (Bronstein et al., 2021), there is growing interest in learning from data defined on simplicial complexes. The motivation behind this comes from two limitations of standard graph neural networks (GNNs). First, graphs are limited to model pairwise interactions between data entites on nodes, yet polyadic (mulli-way) interactions often arise in real-world networks (Battiston et al., 2020; Benson al., 2021; Torres al., 2021), such as friendship networks (Newman et al., 2002), collaboration networks (Benson et al., 2018), gene regulatory networks (Masoomy et al., 2021). Second, graphs are often used to support signals the nodes, and standard graph signal processing and GNN approaches often

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revolve around signals and features nodes. Yet, signals involved with multiple entities are less researched compared to signals on nodes (with one entity). They arise as signal flows on edges, signals on triangles and so on. For example, in physical networks, we may encounter water flows in a water supply network (Money et al., 2022), traffic flows in a road network (Jia et al., 2019), trading flows in financial networks (Lim, 2020) and information flows in brain networks (A (Andnd et al., 2022), as well as in human-generated nerkorks, we have collaboration data, such as triadic collaborations in coauthorship networks (Benson et al., 2018).
Simplicial complexes are a popular higher-order network model and have been shown effective to address both limitations of graph- based models (Bick et al., 2021). They are composed of topological objects, namely, nodes, edges, triangles, etc., which are simplices of different orders. Simplicial complexes naturally describe more topological (higher-order) relationships in networks, thus, having more topological expressive power than graphs. This has been the main motivation of recent neural networks developed on simplicial complexes (Roddenberry & Segarra, 2019; Bunch et al., 2020; Ebli al., 2020; Roddenberry et al., 2021; Bodnar et al., 2021b; Chen et al., 2022b; Giusti et al., 2022). We also refer readers to the recent surveys (Papamarkou et al., 2024; Besta al., 2024). In analogy to standard GNNs relying on the adjacency between nodes, the central idea behind these works is to rely on the relationships between simplices to enable learning. Such relations can be twofold: first, two simplices can be lower and upper adjacent to cach other, c.g., an edge can be (lower) adjacent to another via a shared node, and can also be (upper) adjacent to another by locating in a common triangle; and second, there exist inter-simplicial couplings (or simplicial incidences) between simplices of different orders, as shown in Fig. la. The aforementioned works mainly vary in terms of either bessage-passing or convolutional flavor, or the type of simplicial relationships relying, either on only simplicial adjacencies or on both adjacencies and incidences.
Furthermore, signals can be defined on simplices to model the data related to multiple entites in networks. This has been the main focus of topological signal processing literature Barbarossa & Sardellitti (2020); Schaub et al. (2021); Yang et al. (2022a). The celebrated combinatorial Hodge decomposition arising from discrete calculus (Grady & Polimeni, 2010; Lim, 2020) provides a unique and characteristic decomposition of simplicial signals into three components. This is particularly intuilive for edge flows which allows their decomposition into gradient flows, curl flows and harmonic flows, that are, respectively, curl-free, divergence- These notions from discrete calculus interestingly allow to capture some physical properties of the simplicial signals, such as the conservation laws (Grady & Polimeni, 2010). More importantly, this decomposition offers a tool to better analyze simplicial signals, as reported in statistical ranking problems, financial exchange markets (Jiang et al., 2011), traffic networks (Jia et al., 2019), brain networks (Anand et al., 2022) and game theory (Candogan et al., 2011). We hypothesize it will further promote better principled and effective learning methods on simplicial complexes.
or both.
Given this context, we reckon that the aforementioned works on simplicial neural networks mostly focus on the pure topological aspect of simplicial complexes. It lacks theoretical analyses of their learning capabilities from the Hodge spectral perspective. Also, since SCs are often built from data and are prone to estimation uncertainty, the learning on SCs benefits from a stability analysis to investigate their robustness against perturbations on the simplicial topologies. Thus, in this paper, after reviewing some background on simplicial complexes and simplicial signals in Section 2, we propose a more general and unified framework, namely, simplicial compler convolutional neural network (SCCNN), and we focus on the following three theoretical aspects.
Contributions
In Section we introduce SCCNN and emphasize its three principles, namely, uncoupling the lower and upper simplicial adjacencies, accounting for the inter-simplicial couplings, and performing higher-order convolutions. We then use the Dirichlet energy minimization on SCs to understand how uncoupling the lower and upper adjacencies in Hodge Laplacians, as well as the inter-simplicial couplings can mitigate simplicial oversmoothing,
In Section 4, we characterize the spectral behavior of SCCNN and its expressive power under the help of spectral simplicial theory (Steenbergen, 2013; Barbarossa & Sardellitti, 2020; Yang et al., 2021). We show that an SCCNN performs independent and expressive learning in the three subspaces of the
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Hodge decomposition, which are invariant under its learning operators. This Hodgo-awarenes (or Hodge-aided bias) allows for effective and rational learning on SCs compared to MLPs or simplicial message-passingg networks (Bodnar et al., 2021b).
In Section 5, we obtain theoretical stability bound on the SCCNN outputs against small perturbations on the simplicial connections. This allows to see how the three principles and other network factors can affect the stability, well as the limitations of SCCNNs. This analysis in turn guides the design of convolutional architectures.
In Section 6, validate theoretical findings and highlight the effect of the three principles, the need for the Hodge-aware learning, as well as the stability, based on different simplicial tasks including recovering foreigu currency exchange (forex) rates, predicting triadic and tetradic collaborations, and ocean current trajectories. Finally, we conclude the paper in Section with a discussion on this work and its relations to existing works.
2
Background
We first review simplicial complexes and data supported on simplices, which are natural generalizations of the corresponding notions on graphs. Then, we introduce discrete calculus on simplicial complexes, which is linked to the incidence matrices. Finally, we discuss the Hodge decomposition, which uniquely characterizes the simplicial signals from three subspaces.
2.1 Simplicial complex and simplicial signals
Given a V= (I,..,.. .., no of vertices, a k-simplex & is subset of V with cardinality k + 1. Geometrically, node is a 0-simplex, edge connecting two vertices 1-simplex and triangular face (we shorten as triangle) is 2-simplex. A subset, with cardinality k, of s is face. A coface of s is a (k + 1)-simplex that has s as face. Furthermore, one can collect k-simplices for k = O...., K to form simplicial complex (SC) S of order K with the inclusion restriction that if a simplex is in the SC, so are its subsets. A graph is an SC of order one and by including some triangles, we obtain an SC order two, as shown in Fig. 1a. We denote the set of all k-simplices in S as S = {s jisl.... where nu = |S*|, [, i.e. = Sk
Simplicial adjacency For any two k-simplices, we say they are lower (upper) adjacent if they share a common face (cofaco), which natualy defines the notion of simplicial neighborhoods. For example, two nodes are (upper) adjacent in graph if they are connected by edge. In Fig. 1a, edges e and es are lower neighbors they share node 1; while and are upper neighbors since they are located in the triangle 41
Orientation For computational purposes, we annotate each simplex with an orientation, as an ordering of the labels of its vertices (a node has a trivial orientation). Here we consider the increasing ordering as the reference orientation, that is, a triangle s² = {i, k} is oriented as [i, k] for j < k, and an edge s {i,j} oriented [ for i j.
Algebraic representation We use the incidence matrix Bg R to describe the relationships between (ks - 1)- and k-simplices. Thus, Bi encodes the node-to-odgo incidence and B2 the edge-to-triangle incidence. In an oriented SC S, the entries of B and Bd are given by
L, for
[i,j], [i,
1, for e = [t,
-1, 1, for = [ k]. [i,j,k]
Bike
1, for [
[Blett
(1)
1,
for e = ki, [[.jj,k]
0,
otherwise.
0. otherwise.
We further define the k- Hodge Laplacian
L B BB- B B
(2)
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-2
2.50
-1.34
0.52
10
%
Un5
C
19
04
1/4
027
522 -0.
-1.0
3.1
053
(a) SC
(b) Edge flow
(c) Gradient flow
(d) Curl flow
(e) Harmonic flow
Figure I: (a) A simplicial 2-complex where green shaded triangles denote 2-simplices and the arrows denote the chosen reference orientations. (b) An edge flow where we denote its divergence (div) and curl in purple and orange, respoctively. (o)-(d) The Hodge docomposition of the odge flow in (b). The gradient flow is the gradient of some node signal (in blue) and is curl-free. The curl flow can be obtained from some triangle flow (in red), and is div-free. The harmonic flow has zero div and zero curl, and is circulating around the hole {1,3, 4). Note that in this figure, the flow numbers are rounded up to two decimal places. Thus, at some nodes or triangles with dero-div or zero-curl, the div or curl might not be exactly zero.
with the lower Laplacian LE,d B By and the upper Laplacian Lu BB+1 We have a set of Lg, k = I....., K Topologically, Lı.a and Lg, encode the lower and upper adjacencies of k-simplices, respectively. For example, Lia encodes the edge-to-edge adjacencies through nodes while L1 encodes the adjacencies through triangles.
in an SC of order K with Lu = B BT the graph Laplacian, and Lx = Bx
2.2 Simplicial signals and Hodge decomposition
Simplicial signals A k-simplicial signal (or data) ay E R.* supported on the simplicial set S is defined by an alternating map f& SS + R", which assigns a real value a simplex, with the condition that if the orientation of a simplex is anti-aligned with the reference orientation, then the signal will change the sign (Lim, 2020). For convenience, we call xo node signal. We also refer to 1-simplicial signal œ1 as an edge flow and a 2-simplicial signal æ2 as a triangle flow. A d-dimensional simplicial feature X, E R can be defined for a rich representation learning of simplices. For simplicity, we restrict our analysis to d d=11 1.
Incidence matrices as derivatives on SCs Given a simplicial signal Tks we can measure its variability with respect to the faces and cofaces of k-simplices by computing Bg and 1 (Grady & Polimeni, 2010). Specifically, BJ zo computes the gradient of a node signal vo as the signal difference between the adjacent nodes, i.e, [B] voljs.]] = [mol] [æo]i,, which is often used in the GNN literature. For an edge flow 21, B 11 computes its divergence, which the difference between the total in-flow and out-flow at node j, i.e., [By Ei<sleilssl - Σj<x(zilgaas Moreover, B x1 computes the curl of æ i.e, [B]=)] = [ [an] |( + [ which is the net-flow circulation in triangle = [i. k). As illustrated in Fig. 1b, these two computations provide divergent and rotational variation measures of an edge flow, which are analogous to the notions of divergence and curl for vector fields in continuous domains. In the following, we introduce the Hodge decomposition (Hodge, 1989; Lim, 2020) which unfolds an edge flow into three unique characteristic components.
Lemma 1 (Lim, 2020)). We have BJ BT 0, ie.,, the curl of the gradient is zero. Theorem 2 (Hodge decomposition). The k-simplicial signal space R.* admits direct sum decomposition
R = im(B.) @ ker( L.) @ im BB-1), thus, I IF.G + DK.H + 02,C.
(3)
where &G.G = BJ xx-1 for some #g-1, and &&.C = BB+1E for some T++1 Moreover, we have ker ( im ( @ ker(Ln, Lx) and ker( BB) = (t) @ (B(1.))).
In the node space, this decomposition is trivial as R = (e) @ im im(Bi) where the kernel of Lo contains constant node data and the image of B, contains nonconstant data. In the edge case, the three subspaces carry more tangible meaning: the gradient space im(B] collects edge flows as the gradient of some node signal, which are fure-free; the curl space (m()) consists of flows cycling around triangles, which are div-frea and flows in the harmonic space ker( are both div- and curl-free. In this paper, we inherit the names
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610
c10
C10
83
(a) SC example
(b) Lower edge conv.
(c) Upper edge conv.
(d) Inter-simplicial locality
Figure 2: (a) An SC where arrows indicate the reference orientations of edges and triangles. 2-simplices are (fille)) triangles shaded in green and open triangle {1,3, 43 is not in the SC. (b) Lower convolutton via H, and H1,d on edge e1. (c) Upper convolution via H1 and H1,, on 61. (d) Node 1 (in black) contains information from its neighbors {2, 3, 4) (nodes in rod), and projected information from edges which contribute to these neighbors (denoted by arrows in red from edges to nodes), and from triangles {tz, t2,tt) which contribute to those edges (denoted by double arrows in red from triangle centers to edges). This interaction is the coupling between the intra- and the extended snter-simplicial locality.
of three edge subspaces to general k-simplices. The above theorem states that any simplicial signal æ can be uniquely expressed as 3& = #G + TRHH + tx.C with the gradient part Fx.G = Bg Øx-1, the curl part for some 1, and the harmonic part following LI = 0. Figs. le to le provide the
FK.C = three Hodge components of the edge flow in Fig. 1b.
3 Simplicial Complex CNNs
We first introduce the general convolutional architecture on SCs, then discuss the propreties of SCCNN and study the effects of the three principles from energy minimization perspective.
In an SC, taking " a and ati as inputs, an SCCNN at layer 1 = 1,..., L. computes the k-simplicial output s via a map
SCCNN fo/ k-1 alt , x + 2h, a = HH
k.d 1-1 + Haa 1 + H Hk (,
(4)
with
Ta H₂ = Σ Wk.d Li
Tu Wk Wkn, t
Ta
To Σ t=0
Ha.d =
Σ w 1=0
H
u Lt
Lk
(5)
Here, H& denotes a simplicial convokutional filter (SCF, (Yang et al., 2022b)) with two sets of learnable coefficients (wk.d,c), (wkna)), while Hy and Ha are the lower and upper SCFs, respectively. Moreover, a/d = B - and a--1 = B 1-1 are the lower and upper projections from (k + 1)-simplices via incidence relations to k-simplices, respectively, and a(.) is an elementwise nonlinearity. The convolutionn operations in this SCCNN can be understood as follows: 1) The previous k-simplicial output a is passed to SCF H of orders Ti, which performs a linear combination of the signals from the lower-adjacent (up !-1 to Ta-hop) and upper-adjacent (up to Tu Tu-hop) simplices. 2) The previous k + 1-simplicial outputs # are first projected to k-simplices, which are then convolved using a lower SCF and an upper SCF, respectively. Eaample 3. In Fig. 2, we provide
example of SCCNN for the edge case k k= 1. We focus edge ei and consider the cases Ta = T. = 2. On edge e1, the SCF H; linearly combines the signals from its direct lower neighbors (edges in blue) and two-hop lower neighbors (edges in purple), as shown in Fig, 2b. It also combines the signals from the direct upper neighbors (edges in red) and two-hop upper neighbors (edges in orange), as shown in Fig. 2c. At the same time, the signals on nodes are projected on the edges, denoted by arrows in blue and purple from nodes to edges in Fig. 2b, which are then combined to edge ei by the lower SCF Hi,d- The signals on triangles are projected on the edges as well, denoted by double arrows in red and orange in Fig. 2c, which are combined to edge ej by the upper SCF Hı.u-
This architecture subsumes the convolutional learning methods on SCs in Bunch et al. (2020): Ebli al. (2020); Roddenberry et al. (2021); Yang et al. (2022a); Chen et al. (2022b); Yang et al. (2022c). We refer
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to Appendix C for detailed discussion. Particularly, we here emphasize the key three principles of an SCCNN layer:
(P1) It uncouples the lower and upper parts in the Hodge Laplacian. This Icads to an independent treatment of the lower and upper adjacencies, achieved by using two sets of learnable weights. We shall see Section 4 that how this relates to the independent and expressive learning in the Hodge subspaces given in Theorem 2.
(P2) It accounts for the inter-simplicial couplings via the incidence relations. The projections 24,d and kknu carry nontrivial information contained in the faces and cofaces of simplices (by Theorem 2).
(P3) It performs higher-order convolutions. We consider Ta. Tu 2 1 in SCFs which leads to a multi-hop receptive field on SCs.
In short, each SCCNN layer propagates information across SCs based on two simplicial adjacencics and two incidences in multi-hop fashion.
3.1 Properties
Simplicial locality The simplicial convolutions admit an intra-simplicial locality where the output Hgæ is localized in Ta-hop lower and Ty-hop upper k-simplicial neighborhoods (Yang et al., 2022b). An SCCNN preserves such property o-) does not alter the information locality. It also admits inter-simplicial locality between k- and (k + 1) siimplisss due to the inter-simplicial couplings. This further extends to simplices of orders k + / if L 2 because (Bo(Br+1) * 0 (Schaub et al., 2021). Moreover, the intra- and inter-simplicial localities are coupled in a multi-hop way through higher-order convolutions such that, for example, a node not only interacts with its incident edges and the triangles including it, but also with those further hops away, as shown in Fig. 2d. We refer to Appendix B.1 for a more formal discussion.
Complexity An SCCNN layer has paramcter complexity of order O(Ta + Tu) and a computational complexity Olk(n= + m441) + ngm4 + Tu)), which are linear in the simplex dimensions. Here, my is the maximum number of neighbors for k-simplices. We refer to Appendix B.2 for more details.
Equivariance SCCNNs permutatioon-oquiiaaannt,,, which allows us to list simplices in any order. They are also orientatiinn-quuivaraaal if the activation function a(.) is odd, which gives us the freedom to choose reference orientations. In Appendix B.3, we provide a more formal discussion on such equivariances and why permutations form a symmetry group of an SC and orientation changes are symmetries of data spaces but not SCs.
3.2 A simplicial Dirichlet energy perspective
Here we analyze the convolution architecture in Eq. (4) from an energy minimization perspective. First, we extend the notion of Dirichlet energy from graphs to SCs.
Definition 4. The Dirichlet energy of k-simplicial signal ae is
D(xx) = Da(mx) + D. = B + B 12
(6)
This Dirichlet energy returns the graph Dirichlet energy when k 0. In this case, D(x)) [| x E,E,loo, 2o, 2 is the fz-norm of the gradient of the node signal 2n- For edge flow æτ, D(x1) consists of two parts, | and |B] æ1 2112,, which measure the total divergence and curl of œτ, respectively, i.e., the odge flow variations w.r.t. nodes and triangles. In the general case, Da( and D,(=;) mcasure the lower and upper k-simplicial signal variations w.r.t. the faces and cofaces, respectively. A harmonic k-simplicial signal 2y has zero Dirichlet energy, e.g., a constant node signal and div- and curl-free edge flow.

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Simplicial shifting as Hodge Laplacian smoothing Bunch et al. (2020); Yang et al. (2022c) considered H3 to be weighted variant of I - Lx: generalixing the graph convolutional network (GCN) (Kipf & Welling, 2017). This is necessarily a Hodge Laplacian smoothing as in Schaub et al. (2021) given an initial signal ag. we consider the Dirichlet energy minimization:
min B 12 7 1 > O,
gradient descont: [+1 k.ggd = (I nLsa - my Lx,u,
with step size 1j >0 O. The simplicial shifting ax{1 = wo (I - L is a gradient descent step with 1) = 7 = and weighted by wi. A minimizer of Eq. (7) with y = is in fact in the harmonic space ker( Thus, a neural network composed of simplicial shifting layers may generate an output with an exponentially decreasing Dirichlet energy as it deepens, formulated by the following proposition. We refer to this simplicial oversmoothing, a notion that generalizes the oversmoothing of a GCN and its variants (No & Maehara, 2019; Cai & Wang, 2020; Rusch et al., 2023).
Proposition 5. If will I Lx 1 < 11 in a nearal network of simplicial shifling layers exponentially converges to zero.
However, when uncoupling the lower and upper parts of Lg in this shifting, associated to the case y F 1, the decrease of D(zx)) can slow down or cease because the objective function in Eq. (7) instead looks for a solution primarily in either ker(Bx) (for y < 1) ker ker(B (B (for > 1), not necessarily in ker Lx)( as we shall corroborate in Section 6.
Inter-simplicial couplings as sources Given some nontrivial 1 and #x+1, we consider the optimization
min Bu xh- 12 + B Th 02+1
(8)
gradient descent: kgdd = -I LL + n7(x, + Ik.u)
with step size > O. It resembles the convolutional layer, at = woll - Lx)ack + Wia + u2Kkku with some learnable weights, in Bunch et al. (2020); Yang al. (2022c)-
Proposition 6. We have the sollowing bounds for the Dirichlet eneryy: || 112 + || < D( will [ LA (((()( + wf Amfxmax(e (Lax(L 2x,22 + wg)max (Lx (Lx,a)|
The signal projections from the lower and upper simplices act as energy sources for æhe and also the objective function in Eq. (8) looks for an I& in the image spaces of Bx+1 and B instead of ker(Lx) This ensues that D(a may not converge to zero, but rather to a nontrivial value || + || Thus, inter-simplicial couplings pay a role in mitigating the oversmoothing.
Here showed that simply generalzing GCNs to simplices will inherit its oversmoothing risks. However, by uncoupling the lower and upper Laplacians and accounting for the inter-simplicial couplings could mitigate this issuc. This can also be explained by mcans of a diffusion process on SCs (Ziegler et al., 2022), which we discuss in Appendix B.4.
4 From convolutional to Hodge-aware
In this section, we first introduce the Hodge-ineariant operator, which is an operator such that the three Hodge subspaces are invariant under it. Then, we show that the SCF is such an operator and SCCNN, guided by the three principles (P1-P3), performs invgge-nnvarion learning, allowing for rational and effective learning on SCs while remaining expressive. Throughout the exposition, we rely on the simplicial spectral theory (Barbarossa & Sardellitti, 2020; Yang et al., 2021; 2022b), which also allows us to characterive the expressive power of SCCNNs. We refer to the detailed derivations and proofs in Appendix E.
Definition 7 (Invariant subspace). Let V be a dimensiimeniion vector space over IR with dim(V) 2 1, and lct T: V + V be an operator in V. A subspace U C V is an invariant subspace under T if Tu E U for all u E U. i.c., the image of every vector in U under T remains within U. We denote this as Tlu U - U where Tlo is the restriction of 7 on U.
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Given the notion of invariant subspace, we then define the Hodge-invariant operators.
Definition 8 (Hodge-invariant operator). Let O E (im(B)) im im((x+1),, ker())) be any Hodge subspace of R A linear transformation F: Th= + "R is Hodge-invariant operator if for all 2 E C it holds that F(2L)= E D. That is, any simplicial signal in certain Hodge subspace remains in that subspace under F. Definition ((Barbarossa & Sardellitti, 2020)). The simplicial Fourier transform (SFT) of z is ët = U w where the cigenbasis Un of L acts as the simplicial Fourier basis and the cigenvalues A₁ = diag (xx) are simplicial frequencies.
Proposition 10 (Yang et al. (2022b)). The SFT basis can be found as Ux = [Uk, Us,G Uk.el where
UK is the eigenvector matrir associated to the zero eigenvalues ARH = (k,,),,,H), named as harmonic frequencies,
UR,G is associaled to the nonzero eigenvalues in Ar.G (R, of LE,d, named gradient frequencies, and
Un.C is associated to the nonzero eigenvalues in As.C diag(Ak.)) named curl frequencies.
Moreover, they spon the Hodge subspaces:
span(Ux.H) = ker(L))) (pan(Ux.))=im(((
span(Un.)) = im(BA+1)
99)
where span(.) denotes all possible linear combinations of columns of Remark 11. The frequency notion in general carries the physical meaning of signal variations. In the simplicial case, gradient frequencies reflect the degree of lower variations Da(us,G) of the associated gradient Fourier basis, and curl frequencies reflect the degree of upper variations Du (u,,c) of the associated curl basis. Harmonic frequencics (zcros) correspond to the basis having zero lower and upper variations. In the odge case, the gradient and curl frequencies, respectively, correspond to the total divergence and total curl, measuring how divergent and rotational the associated basis is (Yang et al., 2022b).
Proposition 12. The SCF H₃ is mndge-inwariant operator. That is, for any 2y E D, have H& E for O E (im(B)) im B&+1), ker(Lu)). Moreover, the SCF operation can be implicitly written
Haw = H& Hilim(my ( CR + Hil (er( + Hxlim( CK,C
(10)
where HAlia(BT) = Iia wxdd La + (wx.d. + wk is the restriction of Hy on the gradient space im (, Halker(Le) = (Wk,d,0 wk. I is the restriction of H3 on the harmonic space, and Halim((a4)) ET= W. (wk,d, I is the restriction on the curl space.
Provided with the Hodge-invariance of H2 and the SFT, perform spectral analysis, which is of interest to further understand the SCCNN since simplicial frequencies reflect the variation characteristics of simplicial signals.
4.1 Spectral analysis
Consider the SFT 24 = I k. K.G. G... T of æ; where each component is the intensity of D& at certain simplicial frequency. We can understand how an SCCNN convolutional layer ys = H Ha Tk egulates/learns from the simplicial signals at different frequencies by performing the SFT
Ük.H h ④ T
Ús.G hand O Tk.a h O ŌN.G
(11)
ÜK.C c @ & + h ① Ōt.u:
with @ the elementwise multiplication. The n, dimecnnsioonal vector hg = duag(U, H₂U,) = [hī h hT is the frequency response vector of H₃ with
hg (Wk,d,0 + I,, hu. Ele Wk. dk EGG + wkn,o, hac =hh Wk.u i k.C + wk,dl (12)
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(b)
(c)
Figure 3: (a) (top): Independent gradient and curl learning responses. (bottom): Stability-selcctiiity tradeoff of SCFs where he has better stability but smaller selectivity than IG- (b) Information spillage of nonlincarity. (lop): the SFT of input with only gradient components. (bollom): the SFT of the output shows that after applying a nonlinearity the output also contains information in non- non-gradient frequencies. (c) The distance between the perturbed outputs and true when node adjacencics are perturbed. (top): = 1, triangle output remains clean. (bottom): L = 2, triangle output is perturbed.
where of is the elementwise 1-th power of a vector. Likewise,
0t h = Ei wh kkdd K.G + wkk,, and h = Ein w k K.O.
+ wk,a,
(13)
are the frequency response vectors of Ha.d and Ha The spectral relation in Eq. (11) shows that the gradient SFT & learned by a gradient response hg,G, while the curl SFT øR.C is learned by curl response hy.c- The two learnable responses are independent and they only coincide at the trivial harmonic frequency, as shown by the two individual curves Fig. 3a. Morcover, the lower and upper projections are independently learned by ha and hg,u, respectively.
The elementwise nonlinearity induces an information spillage that one type of spectra could be spread over other types. As illustrated in Fig. 3b, the top figure shows the SFT of an input with only gradient components, and the bottom figure plots the SFT of (y)),, showing that it also contains information in harmonic or curl subspaces. This results from the nonlinearity, since applying a linear SCF leads to an output with only gradient components. In the following, we characterize the expressive power SCCNN.
Proposition 13. An SCCNN layer with inputs Dk,d, Zb: Dk,u is at most expressive as an MLP (G',, + Gaar + Dk..) with G5 Gk.d + Gk where GE and Gea are analytical matrie functions of Ly,a, while GE and Gh are analyticol matriz functions of Lk This expressivity can be achieved by setting Ta Ta nx.G and Tu T n&,C in Eq. (4) with ng,G the number of distinct gradient frequencies and ng.C the namber of distinct curl frequencies.
The proof follows from Cayley- Hamitonn theorem (Horn & Johnson, 2012). This expressive power can be better understood from the spectral perspective - The gradient SFT &GGG can be learned as expressive by an analytical vector-valued function gg.G, which collects the eigenvalues of Gk,d at gradient frequencies. The curl SFT ž can be learned as expressive as by another analytical vector-valued function Gk,c, which collects the eigenvalues of Gk, at curl frequencies. These two functions need only to coincide at the harmonic frequency. In addition, the SFTs of lower and upper projections can be learned expressive as by two independent analytical vector- rector-valued functions as well.
4.2 Hodge-and aware learning
Given the expressive power in Proposition 13 and the spectral relation Eq. (11), we show that SCCNN performs Hodge-aware learning in the following sense, which comes with advantages over the existing approaches.
Theorem 14. An SCCNN Hodge-oware: 1) The SCF H2 a Modge-innariee learning operator. Specif- ically, three Hodge subspaces are invariant under H 2) The lower SCF Had and upper SCF H are,
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odge flow Xy X
lonvrr contribution X = Bj %o
upper contribution Kiu
simplicial frequencies op
gradient freg,
B
curl froq. Ac
Aaz Ac.
harmonic free. Ast
La Kia
25
gradien flow
Hilwjery (1m(B]) [ (m(7)
carl
Hi : 3) im((;))
harmonic flow
Hi ball kor(L.) - kor(L.)
lower SCF His im mn{B] - mB{
upper
Hus?
SCF Hi R
im(B,) - im(Ba)
R
Hilw(s))
X10
[B[]]
in(aa
His Baentt
Y1
kon(L)
LH
X
Her
Hilleay
X,c
In(Bg)
(g
mm/B.)
Aup
Aci
An illustration of the Hodge-aware learning of an SCCNN. We show that how edge flow æ together with the lower and upper projections 2i,d and Li,u, are transformed by an SCCNN in the spectral domain. The implicil operation H1 (in the dashed box on the right) reflects the Hodge-aware learning: 1) H1 is Hodge invariant: each component learned within their own subspace, and H1 does not mic up the three subspaces; 2) The learning in the gradient and curl subspaces are independent where features at shared frequencies λg and Acς1 can be separately learned; and 3) The learning operators are expressive in the sense that the spectral responses as expressive as any analytical functions in the gradient and curl frequencies.
Figure 4:
respectively, gradient- and curl-invariant learning operators; 3) The learnings in the gradient and curl spaces are independent; And 4) the learnings the grudient and curl spaces are expressive as in Proposilion 13.
This theorem shows that an SCCNN performs an expressive and independent learning in the gradient and curl subspaces from the three inputs while preserving the three subspaces to be invariant w.r.t its learnable SCFs. This allows for the rational and effective learning SCs, as illustrated in Fig. 4, from the two aspects. These three-fold properties of an SCCNN, respectively, come from the convolutional architecture choice, the uncoupling of the lower and upper adjacencies, and the higher-order convolutions in the SCCNN.
On the one hand, Proposition 12 shows that the operation of H₃ on the simplicial signal space is equivalent to a summation of its restrictions Hylo on three smaller subspaces D. This Hodgo-invariant nature of the learnable SCFs substantially shrinks the learning space of an SCCNN and allows for an effective learning. On the other hand, simplicial signals often present. implicit or explicit properties that different Hodge subspaces capture. For example, water flows, traffic flows, electrical currents (Grady & Polimeni, 2010; Jia et al., 2019) follow flow conservation, i.c., being div-frec in the gradient space ker( (B)) while exchange rates can be modelled as curl-free edge flows (Jiang et al., 2011). Owing to the learning in the nontrivial subspaces, an SCCNN can capture such characteristics of real-world edge flows effectively. When it comes to regression tasks on SCs, an SCCNN can generate outputs respecting these physicial laws.
of H1 and its independent
Remark 15 (Relation to message passing networks). Message Kesssage-psssigg simplicial networks (MPSNs) (Bodnar et al., 2021b) using MLP to aggregate and update are non-Hodggoaaaar. Their Icarning functions pursue direct mappings between the much larger signal space R"* thus, requiring more training data for accurate learning, as well as a larger computational complexity. Moreover, MPSN does not preserve the Hodge subspaces, i.e., it is not Hodge-invariant. Thus, they might generate outputs with small losses (e.g., mean-squared-orron in regression tasks, yet not respecting the physical laws being either div- or curl-free propertics such as the above simplicial signals. We shall corroborate this in Appendix G.
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Remark 16 (Relation to other convolutional methods). While most convolutional networks SCs use Hodge- invariant learning operators, they are not strictly Hodge-aware, resulting in practical limits. For example, Ebli et al. (2020) considered Hy = =, w; L which preserves the Hodge subspaces yet does not uncouple the lower and upper parts of gg- This makes it strictly less expressive and non-Hodgo-aware. Consider two froquencies A = Àc which share a common value but correspond to the gradient and curl subspaces, respectively. The simplicial signal components at these two frequencics are always learned in the same fashion, which induces contradicting issues when the underlying component in one subspace should be diminished while the one in the other subspace should be preserved. This underlines the importance of uncoupling the two adjacencies because the lower and upper Laplacians operate in different subspaces. Roddenberry et al. (2021) applied H₂ with Ta = Tu = 1. Spatially, this limits the receptive field of each simplex to its direct neighbors. Spectrally, it leads to a linear learning frequency response. A similar trealment was considered in Bunch et al. (2020); Yang et al. (2022c) which simply generalized the GCN without uncoupling the two adjacencies, and gave a limited low-pass lincar spectral response, as shown Fig. 3a and discussed in Scction 3.2.
5 How robust are SCCNNs to domain perturbations?
In practice, an SCCNN is often built on a weighted SC to capture the strengths of simplicial adjacencies and incidences. We defer the explicit formulations Appendix F.1 since it has the same form as Eq. (4) in this case, except for that the Hodge Laplacians are weighted, as well as the incidence matrices. These matrices are often defined following Grady & Polimeni (2010); Horak & Jost (2013); Gugliclmi et al. (2023). For example, Bunch et al. (2020); Yang et al. (2022c) considered particular random walk formulation (Schaub et al., 2020). They can also be learned from data, e.g., via an attention method (Goh et al., 2022; Giusti et al., 2022). For the weighted incidence matrices B, B we use operators Ra,d, Ra. in this section.
To highlight the need for a stability analysis, note that, on the one hand, we may lack the true underlying topologies in SCs as they are often estimated from noisy data; and we may undergo adversarial attacks on the topologies. On the other hand, we want to characterize the stability-selectiiity tradeoff of SCCNN, in analogy to the study for CNNs (Bruna & Mallat, 2013; Qiu et al., 2018; Bietti & Mairal, 2017) and GNNs (Gama et al., 2019b; 2020a; Kenlay et al., 2021; Parada-Mayorge et al., 2022).
This motivates us to investigate the stability of SCCNN: how for are the outputs of an SCCNN before and after perturbations are applied to SC&? We consider the following relative perturbation model, generalizing the graph perturbation model in Gama et al. (2019b)
Definition 17 (Relative perturbation). Consider some perturbation matrix of an appropriate dimension. For the weighted Hodge Laplacian La,,, its relative perturbed version is Lu.d = LL,a + La.d + LE,d Ex.a with perturbation Ex,d: likewise for LR, by E For the weighted incidence matrix Rk,d, its relative perturbed version is Re.d = Rx,d + Jx with perturbation Ja,d; likewise for RR by Jk.u-
This models the domain perturbations on the strengths of adjacent and incident relations, c.g., large weight is applied when two edges are weakly or not adjacent, or data on a node is projected on an edge not incident to it. Moreover, this quantifies the relative perturbations with respect to the local simplicial topology in the sense that weaker connections in an SC are deviated by perturbations proportionally less than stronger connections. We further define the integral Lipschitz property of spectral filters to measure the variability of spectral response functions of Hy-
Definition 18 (Intergral Lipschitz SCF). An SCF H3 is integral Lipschitz with constants Ck.d: Ck. W W O if the derivatives of its spectral response functions ha,o(A) and ha,c(A) follow that [MM,((A) Cr.d and (AH),((A) < ck.u.
This property provides a stalilityyselecciiityy tradeoff of SCFs independently in the gradient and curl froquencies. A spectral response can have both a good selectivity and stability in small frequencies (a large [MG_ for A + O), while it tends to be flat for having better stability at the cost of selectivity (a small variability for large X) in large frequencics, as shown in Fig. 3a. As of the polynomial nature of responses, all SCFs of SCCNN are integral Lipschitz. We also denote the integral Lipschitz constant for the lower SCFs Ha,a by Gr.d and for the upper SCFs Hg by Ck without loss of generality.
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Under the following assumptions, we now characterize the stability bound of SCCNN.
Assumption 19. The perturbations are small such that || én,a, alls, En,d, LEEuulls exEu and || Ek.u. where || = max 2 is the operator norm (spectral radius) of a matrix A. Assumption 20. The SCFs H₂ of an SCCNN have a normalized bounded frequency response (for simplicity, though unnecessary), likewise for HE,d and Ha.u-
Assumption 21. The lower and upper projections are finite such that || S -k,d and || I Tk.u- Assumption 22. The nonlinearity o(),, e.g,, relu, tanh, sigmoid, is co- -Tppscccittz with Co 2 0.
Assumption 23. The initial inpuls atk, for all k, are finile, such that || l2 S [3)x- We collect them in B = -B0 1B.....,,.8 ......,].'
Theorem 24. Let ækk be the k-simplicial signal output of an L-layer SCCNN on weighted SC. Let ak be the outpul of the same SCCNN but relatively perturbed SC. Define òū,a = (l|Vi,a and ŌR, = (l 11 with VE. and Vk, the eigerweclors of Ex, and Ek,u; which measure. the eigenvector misaliguments between the perturbations and Laplacians. Under Assumptions 19 to 23, the Euclidean distance between the two outputs is finite and upper-bounded
| atll2 S [d]& with d = dh Σ 2---Z2
(14)
where for K = 2,
to
to.u
To.u
1
fo.u
T =
l1,d
h
41u
Z =
Fi,d
1l,u
2 =
Aid
f
(15)
t2
t2
F2dd
1
f2.
1
with fR.d = -Tr,al1++ss, and fk.u = Tk (1 +E,,)) Notice that T, Z and Z are tridiagonal and follow a similar structure for general K. The diagonal entries Tf are l = a + CR, a The off-diagonal entries are tg,d = te.. and AK. 2(1 + 5k.,) VTk.
and tk = TR,.Ek, ue..
where D = 2(1 /nz
We refer to Appendix F.2 for a two-step proof. This result bounds the outputs of an SCCNN on all simplicial levels, showing that they are stable to small perturbations the simplicial adjacencies and incidences. Specifically, we make two observations from the complex expression. First, the stability bound depends on i) the degree of perturbations including their magnitudes ck. and εk,, and the eigenspace misalignment 6z,;; ii) the number of simplices rx; ili) the integral Lipschitz properties ck, of SCFs; and, iv) the degree of projections rk... Second, the stability of the k-output depends on the factors related to not only k-simplices, but also simplices of adjacent orders due to inter-simplicial couplings. For example, when L = 1, the node output bound do is affected by factors in the node space, as well the edge space factored by the projection degree. As the layer deepens, this mutual dependence expands further. When I. 2, the factors in the triangle space also affect the stability of the node output do, we observe in Fig. 3c.
More importantly, this stability bound provides intuitive practical implications for convolutional learning on SCs. While inter-simplicial couplings may be beneficial, SCCNN becomes less stable as the number of layers increases due to the mutual dependence between outputs on different simplicial levels. Thus, to maintain the expressive power, we expect to use higher-order SCFs in exchange for shallow layers. This yet does not harm the stability in the following two aspects:
First, high-frequency components can be spread over the low frequencies due to the information spillage of nonlinearity (cf. Fig. 3b] where the spectral responses are more selective and have better stability. If the signal has large high gradient frequency components, we need the SCCNN to be selective in high gradient frequencies. However, to guarantee the stability, the frequency response should be smooth (less selective) in these frequencies, as illustrated by hG in Fig. 3a (bottom). This selcctivity-stabiiity tradcoff can be mitigated by the nonlincarity the information in high gradient frequencies could spill over in lower frequencies, where the spectral responses are more selective and have better discriminating ability.
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Second, higher-order SCFs are casier to be learned with smaller integral Lipschitz constants than lower-order ones due to the increased degree of freedom, thus, leading to an increased stability. This can be easily seen by comparing one-order and two-order cases. We experimentally investigate this in Section 6.4. Morcover, we introduce the following regularizer to the loss function during training so to promote the integral Lipschitz property.
Ta Σ
Tv twnk u. N..
rIL -1 (AR,G))) + C
twk twk.a
Σ
(16)
for AE, E {AR,G.)T=1 and AR,C E (Ar.C.);=1 RkC. which are the gradient and curl frequencies. To avoid computing the eigendecomposition of the Hodge Laplacian, we can approximate the true frequencies by sampling certain number of points in the frequency band (0, and (0, AR,C.m] where the maximal gradient and curl frequencies can be computed by efficient algorithms, e.g., power iteration.
6 Experiments
The goal of this section is to answer the following four research questions with experiments various simplicial-level regression and classification tasks:
RQ 1
What the effects of the three principles of SCCNN, i.e., uncoupling the lower and upper parts of
Hodge Laplacians (P1), the inter-simplicial couplings (P2), and higher-order convolutions (P3)?
RQ 2 How do the uncoupling of the lower and upper parts of Hodge Laplacians and the inter-simplicial couplings affect the simplicial oversmoothing?
RQ 3
How does the Hodge-awar aware property of SCCNN play role in different tasks on SCs, compared to non-woagee mothods?
RQ 4 How do different factors affect the stability of SCCNN, and how maintain the stability while keeping the expressive power?
For comparison, consider the following learning methods single-level simplices:
simplicial neural network (SNN) (Ebli al,, 2020), which does not respect P1 and P2 and is non-Hodge-awarc;
principled simplicial neural network (PSNN) (Roddenberry et al., 2021), which does not respect P2 and P3 and is non-Hodee nou-Hodge-aware;
simplicial convolutional neural networks (SCNN)' (Yang et al., 2022a), which does not respect P2 but is Hodge-aware;
and the following learning methods on simplicial complexes:
Bunch (Bunch et al., 2020), which does not respect P1 and P2 and is non- Hon-Hodgo-awarc; MPSN (Bodnar et al., 2021b), which based on cessage-passingg and not Hodge-aware.
We also considered the MLP and standard GNN (Defferrard et al., 2016) as baselines to highlight the effect of SC topology on simplicial-level tasks. We refer to Appendix C for the detailed comparisons between these methods and SCCNN, as well as to Appendix for the full experimental details.
6.1 Foreign currency exchange (RQs 1, 3)
In forex problems, to build a fair market, the arbitray-free condition implies that for any currencies i, k, it follows that //ips/h = pi/k where /31 is the exchange rate between and j. That is, the exchange path
Note that the difference between SCCNN and SCNN lies in that the latter does include the lnyer-layer projections, detailed in Appendix C thus, we refer method, simplicial complex CNN.
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Table 1: Forex results (umse]total arbitrage, 4).
Methods
Random Noise
Curl Noise
Interpolation
Input
0.11910.004 [29. 29.199...77
122.4 0.552
15.90
0.717
106.440..m 100902
Bascline (l2 rogularizer)
0.036 005
0.29
0.050 002 11.12:0.537
0.534:0.040)
99.67
SNN
0.110 23.24=1.03
0.446==0.017
(8..9542.20
0.702 033
104.744+1.00
PSNN
0.008+0.001
(0.984
0.000
0.000 0000
0.009 001
[1.13 0..199
MPSN
0.039
77.44...8
0.076=o..02|114992:22
0.117
223. 111.7
SCCNN, id
0.027 00.000
0000=00..000.00010......
0.265
00.000
SCCNN, tanh
0.002=0.000 00335
1.000+...w//00003300...
0.003 0.279
i k provides no profit or loss over direct exchange i - k. Following Jiang et al. (2011), we model exchange rates as edge flows in an SC of order two, specifically, via [x]] = log(r"/3). This conveniently translates the arbitrage-free condition into # being curl-free, i.e., [21][e,]] + [ [ 0 in any triangle [ kj.. We consider a real-world forex market at three timestamps, which contains certain degree of arbitrage (Jia et al., 2019; Yang et al., 2024). We focus on recovering a fair market two scenarios, first, denoising from noisy exchange rates where random noise and noise only in the curl space modelling random arbitrage ("curl noise") are added, and second, interpolation, when only 50% of the total rates are observed. To evaluate the performance, we measure the normalized mean squared error (nmse) and total arbitrage (total curl), both equally important for achieving a fair market.
From Table 1, we make the following observations on the impacts of P1 and P3, as well as the Hodge-awarenes..
1) MPSN performs poorly at this task: although it reduces the mmse, it outputs unfair rates with large arbitrage, against the forex principle, because it is not Hodgo-aware and unable to capture the arbitrage-free property with small amount of data (ef. Remark 15).
2) SNN performs poorly as well: as discussed in Remark 16, it restricts the gradient and curl spaces to be always learned in the same fashion and makes it impossible to perform disjoint learning in two subspaces. However, since there are eigenvalues which share a common value but live in different subspaces in this SC, it requires preserving the gradient component while removing the curl one here. 3) PSNN can reconstruct relatively fair forex rates with small mmse. The reconstruction from curl noise is perfect, while in the other two cases, the mısee and arbitrage are three times larger than the proposed SCCNN due to the limited expressivity of linear learning responses.
4) SCCNN performs the best in both reducing the total error and the total arbitrage, ultimately, corroborating the impact of performing Hodge-aware learning.
We notice that with identity activation function (o = id), the arbitrage-free rule is fully learned by an SCCNN. However, it has relatively large errors in the random noise and interpolation cases due to its limited linear expressive power. With a nonlinearity a = tanh, an SCCNN can tackle these more challenging cases, finding a good compromise between overall errors and data characteristics.
6.2 Simplicial oversmoothing analysis (RQ 2)
1n
We use simplicial shifting layers (i.e., Eq. (7) composed with tanh) to illustrate the evolution of Dirichlet energies of the outputs on nodes, edges and triangles in an SC of order two with respect to the number of layers. The corresponding inputs are randomly sampled from a uniform distribution (([-5,5), 5)). Fig. 5 (the dashed lines) shows that simply generalizing the GCN on SCs as in Bunch method could lead to oversmoothing on simplices of all orders. This aligns with our theoretical results in Section 3.2. However, uncoupling the lower and upper parts of Lj (e.g., by setting- 7 2 in Eq. (7)) could mitigate
10-1-
10-7
shilLrode
10-11
with proj made
10-13
shift.dge
with proj.cdge
10-13
uncouple lover pwprr/uppr.uuu adge
shift.tri.
10-21
with proj .ri.
Layer
Figure Simplicial oversmoothing.
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Table 2: Simplex prediction (AUC, 1).
MLP
GNN
SNN
PSNN
SCNN
Bunch
MPSN
SCCNN
2-simplex
68.5+1..
93.9
92.0+1..8
95.611.3
98.3 16.5=1.5
98.7 98.1+0.5 99.2+0..3
3-simplex 69.0+2222
96.6+00.5
95.1+1.2
98.1+0.5
98.3+0.4
98.5+0055
0..
Table 3: Ablation study on different components in an SCCNN: the best results with the corresponding hyperparameters, and the results under the same hyperparameters.
Missing component
2-simplex
Hyper Params.
2-simplex
Hyper Params.
98.7+0.5
== 2, = 2
98.7
L= ==2,T = 2 L=2,T 2 L=2,T =2
Edge-to-Node
93.9+1.0
== 2
89.8+1.9
Node-to-Node
98.7+004
== 2
96.1
Fddge-t--ddge
98.5++100
==3,7 2
97.4+0.8
1=2,7'
Node-to- Edde
98.8+0.3
I= L=4,7 2
98.5+
1=2,7'=
Node input
08.2+0.55
L= 2, = 4
98.1+0.77
L. 2,=,7'== =2
Edge input
98.1+0.4
L = 2,T = 3
97.2+0.9
= 2,T = =2
the oversmoothing on edges, as shown by the dotted line. Lastly, when we account for the inter-simplicial coupling, as shown by the solid lines (where we applied Eq. (8)), it could almost prevent the oversmoothing, since it provides energy sources. We refer to Appendix G.I for more results.
6.3 Simplex prediction (RQs 1, 3-4)
We consider the prediction task of and 3-simplices which extends the link (1-simplex) prediction in graphs Our approach is to first learn the ropresentations of lower-order simplices and then use an MLP with their concatenation as inputs to identify if a simplex is closed or open, which generalizes the link prediction method of Zhang & Chen (2018). Considering a coauthorship dataset (Ammar et al., 2018), we built an SC following Ebli et al. (2020) where nodes represent authors and (k- - })-immplicss thus represent the collaborations of k-authors. The input simplicial signals are the numbers of citations, e.g., In and B2 are those of dyadic and triadic collaborations. Thus, 2-simplex (3-simplex) prediction amounts to predicting triadic (tetradic) collaborations. We evaluate the AUC (area under the curve) performance.
From Table 2, make three observations on the effect of the three key principles. 1) SCCNN, MPSN and Bunch methods outperform the ones without inter-simplicial couplings. This highlights that accounting for contributions from faces and cofaces increases the representation power of the network. 2) SCNN performs better than an SNN, which shows that uncoupling the lower and upper parts in LE improves the represent ation learning. 3) SCCNN performs better than Bunch (similarly, SCNN better than PSNN), showing that higher- order convolutionn further improves predictions. 4) While MPSN performs similar to SCCNN, it has throe times more parameters than an SCCNN (Appendix G.3.6) under the settings of the best results.
Ablation study We then perform an ablation study to investigate the roles of different. components an SCCNN. As reported in Table 3, we remove certain simplicial relations in the SCCNN and evaluate the prediction performance. Without the edge-to-node incidence, when inputting the node features to the MLP predictor, it is equivalent to GNN, which has poor performance. When removing other adjacencies or incidences, the best performance remains similar but with an increased model complexity (more layers required). This however is not preferred, because the stability decreases as the architecture deepens and the model gets influenced by factors in other simplicial spaces, as discussed in Section 5 and shown in Fig. 3c. When keeping the hyperparametors fixod, the performance docreases more severely. We also consider the case with limited input where the input on nodes or on edges missing. The best performance of an SCCNN only slightly drops with an increase of the convoluttonn order.
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11"1
10-11
15711
18:1
20'1
18
35
18
10
12"
1
(a) Node pert. =1 (b) Node pert.I.
(c) Edge pert.
(d) Tri. pert
(e) Tri. pert. T.
Figure 7: The relative difference of SCCNN outputs on simplices of different orders when different levels of perturbations are applied to only nodes, edges and triangles and the number of layers varies.
6.3.1 Stability analysis (RQ 4)
Stability bounds To investigate the stability bound Eq. (14), we add perturbations to relatively shift the cigenvalues of the Hodge Laplacians and the singular values of the projection matrices by e [0, (ef. Assumption 19). We compare the bound in Eq. (14) to the experimental & distance on each simplex level. As shown in Fig. 6 where the dashed lines are the theoretical stability bounds whereas the solid ones are the experimental stability bounds, we see the bounds become tighter as perturbation increases.
10P
Nods
Node thm.
Edge
Edge thm
thm.
109-
10-11
10-2
10-1
10
10-2
10
10
Stability dependence across simplices For 2-simplex prediction of K = 2, we measure the distance between the simplicial outputs of SCCNN with and without perturbations on nodes, edges, and triangles, i.e., || - l ||, for k: = 0, 1,2. Fig. 7 shows that overall the stabilities of different simplicial outputs are dependent each other. Specifically, we see that the triangle output is not influenced by the perturbation on node weights until I, = 2; likewise, the node output is not influenced by the perturbations on triangle weights when L = 1. Also, perturbations on the edge weights will perturbe the outputs on nodes, edges, triangles when = 1. This corroborates our discussions Section 5.
Figure 6: Stability bounds.
Effect of number of simplices. We see that the same level of perturbations added to different simplices leads to different. degrees of instability, owing to the effect of ng on stability in Eq. (14). Since no mn < f12; the perturbations on node weights cause less instability than those edge and triangle weights.
Effect of number of layers. As the number of layers increases, Fig. 7 also shows that the stability of SCCNN degrades, which corresponds to our analysis of using shallow layers.
6.4 Trajectory prediction (RQ 1, 4)
We consider the task of predicting trajectories in a synthetic SC and ocean drifters from Schaub et al. (2020), following Roddenberry et al. (2021). From Table 4, we first observe that the SCCNN and Bunch methods do not always perform better than those without inter-simplicial couplings. This is because zero inputs are applied on nodes and triangles following Roddenberry et al. (2021), which makes inter-simplicaal couplings inconsequential. Secondly, an SCCNN performs better than Bunch on average, and SCNN better than PSNN, showing the advantages of higher-order convolutions. Note that the prediction here aims to find the best candidate from the neighborhood of the end node, which depends the node degree. Since the average node degree of the synthetic SC is 5.24 and that in the ocean drifter data is 4.81, random guess has around 20% accuracy. The high standard derivations may result from the limited ocean drifter data size.
6.4.1 Stability analysis (RQ 4)
Different from Scction 6.3.1, we further investigate the stability in terms of the integral Lipschitz properties and convolutional orders. We consider SCNNs (Yang et al., 2022a) with orders Ta = T, E (1, 5) and train them with regularizations the integral Lipschitz constants. As shown in Fig. 8, the higher-order case has
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accuracy
0.6
Table 4: Trajectory prediction (aceuracy, f).
3
0.4
T= b
Methods
Synthetic trajectories
Ocean drifters
2
SNN
65.5+22
52.5=6.0
T-
T
3
PSNN
63.1#3.1
49.0=8.0
5
SCNN
67.7+1.7
53.0+7.8
Bunch
62.3+4.0
46.0=6.22
0.00
0.25
0.50
0.75
1.00
SCCNN
65.2+4.1
54.5
Figure 8: Stability and accuracy of SCCNN versus convolutional orders.
better stability (smaller l distance between the outputs without and with perturbations) and consistent better accuracy, compared to the lower-order case. This is because the additional flexibility in the higher-order case allows the filters to have better integral Lipschitz propertics, thus, better stability, while maintaining the accuracy. Meanwhile, we also study the effect of the regularizer in Eq. (16) on improving the stability while not losing the performance. We refer to Appendix G.4.4 for the detailed experimental analysis where we numerically mcasure the improved integral Lipschitz propertics with this regularizer.
7 Related Works, Discussions and Conclusion
Related works Our work is mainly related to learning methods on SCs, Roddenberry & Segarra (2019) first usod L, to build neural networks on edges in a graph setting without the upper edge adjacency. Ebli et al. (2020) then generalized convolutional GNNs (Kipf & Welling, 2017; Defferrard al., 2016) to simplices by using the Hodge Laplacian. Roddenberry et al. (2021); Yang et al. (2022a) instead uncoupled the lower and upper Laplacians to perform one- and multi-order convolutions, to which Goh et al. (2022); Giusti et al. (2022); Lee et al. (2022) added attention schemes. Keros et al. (2022) considered a variant of Roddenberry el al. (2021) to identify topological "holes" and Chen et al. (2022b) combined shifling on nodes and edges for link prediction. These works learned within a simplicial level and did not consider the incidence relations (inter-simplicaal couplings) in SCs, which was included by Bunch et al. (2020); Yang et al. (2022c). These works considered convolutional-tyy type methods, which can be subsumed by SCCNNs. Meanwhile, Bodnar et al. (2021b); Hajij et al. (2021) generalized the message passing on graphs (Xu et al., 2018a) to SCs, relying both adjacencies and incidences. Most of these works focused extending GNNs to SCs by varying the information propagation on SCs with limited theoretical insights into their components. Among them, Roddenberry et al. (2021) discussed the equivariance of PSNN to permutation and orientation, which SCCNNs admit as well. Bodnar et al. (2021b) studied the mossgac-passingg on SCs in terms of WL test of SCs built by completing cliques in a graph. The more closely related work is Yang el al. (2022a), which gave only a spectral formulation based on SCFs but not SCCNNs. We refer to Papamarkou et al. (2024); Besta et al. (2024) for an overview of the current progress on learning on SCs.
Discussions In our opinion, the advantage of SCs is not only about them being able to model higher- order network structures, but also support simplicial data, which can be both human-generated data like coauthorship, and physical data like flow-type data. This is why we approached the analysis from the perspectives of both simplicial structures and the simplicial data, i.e., the Hodge theory and spectral simplicial theory (Hodge, 1989; Lim, 2020; Yang et al,, 2021; Barbarossa & Sardellitti, 2020; Steenbergen, 2013; Yang et al., 2022b; Govek et al., 2018). We provided insights into why the three principles (P1-P3) are needed and how they can guide the effective and rational learning from simplicial data. As we have practically found, SCCNNs perform well in applications where data exhibits properties characterized by the Hodge decomposition due to the Hodgo-awareness, while non-Hodgo-aware learners fail at giving rational results. In cases where data does not possess such properties, SCCNNs have better or comparable performance than the ones which violate or do not respect the three principles.
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Concurrently, there are works more general cell complexes, e.g., (Hajij et al., 2020; 2022; Sardellitti et al., 2021; Roddenberry et al., 2022; Bodnar et al., 2021a), where 2-cells inloude not only triangles, but also general polygon faces. We focus on SCs because a regular cell complex can be subdivided into an SC (Lundell et al., 1969; Grady & Polimeni, 2010) to which the analysis in this paper applics, or we can generalize our analysis by allowing Bg to include 2-cells. This is however informal and does not exploit the power of cell complexes, which relies on cellular sheaves, as studied in (Hansen & Ghrist, 2019; Bodnar al., 2022).
Conclusion We proposed three principles (P1-P3) for convolutional learning on SCs, summarized in a general architecture, SCCNNs. Our analysis showed this architecture, guided by the three principles, demonstrates an awareness of the Hodge decomposition and performs rational, effective and expressive learning from simplicial data. Furthermore, our study reveals that SCCNNs exhibit stability and robustness against perturbations in the strengths of simplicial connections. Experimental results validate the benefits of respecting the three principles and the Hodge-awareness, as well as the stability results. Overall, our work establishes a solid theoretical fundation for convolutional learning on SCs, highlighting the importance of the Hodge theorem.
Reproducibility Statement
Bhaming_on__Stt We refer to Learning_ on on_SCs for the reproducibility our experiments. We also note that the proposed architecture is implemented in the TopoModelX framework (Hajij et al., 2024).
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Contents
1
Introduction
1
2 Background
3
2.1 Simplicial complex and simplicial signals
3
2.2 Simplicial signals and Hodge decomposition
4
3 Simplicial Complex CNNs
5
3.1 Properties
6
3.2 A simplicial Dirichlet energy perspective
6
From convolutional to Hodge-aware
7
4.1 Spectral analysis
8
4.2 Hodge-aware learning
9
5
How robust are SCCNNs to domain perturbations?
11
Experiments
13
Foreign currency exchange (RQs 3)
6.2
Simplicial oversmoothing analysis (RQ 2)
6.3
Simplex prediction (RQs 1, 3-4)
15
6.3.1 Stability analysis (RQ 4)
16
6.4
Trajectory prediction (RQ 1, 4)
16
6.4.1 Stability analysis (RQ 4)
16
7 Related Works, Discussions and Conclusion
17
A Illustration for Background
25
B Simplicial 2-Complex CNNs and Details on Properties
26
C Related works
28
D Proofs for Section 3
30
E Proofs for Section 4
31
F Proofs for Section 5
32
G Experiment details
38
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A Illustration for Background
Ths paper relies on the Hodge decomposition and the spectral simplicial theory. To case the exposition, we illustrate them for the edge flow space. We refer to Barbarossa & Sardellitti (2020); Yang et al. (2021; 2022b) for more details.
2
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-0
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0042
0.38
0.77
0:55
5
0.08
0.5
0.44
20
-038
011
021
000
0.55
0.59
-0.59
0.46
0.55
(a) MG.1, A (0.80)
(b) uG (1.
(c) M..3 AG.3 (2.43)
(d) UG.4 (3.9(3
(e) UG,3, (5.1()
55
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-043
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-024
1000
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-044
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-0.42
(f) uG.G (6.08)
(g) UC,1,
(h) UC22 UC22,A,,2 (3.00))0))
(i) Uc,3 (4.4(4.41) (i) UH, An
Figure 9: (a)-(1) Six gradient frequencies and the corresponding Fourier basis. We also annotate their divergences, and we see that these eigenvectors with a small eigenvalue have a small magnitude of total divergence, i.e., the edge flow variation in terms of the nodes. Gradient frequencies reflect the nodal variations. (i) Three curl frequencies and the corresponding Fourier basis. We annotate their curls and we see that these eigenvectors with a small eigenvalue have a small magnitude of total curl, ie.., the edge flow variation in terms of the triangles. Curl frequencies reflect the rotational variations. (j) Harmonic basis with a zero froquency, which has zero nodal and zero rotational variation.
A.1 Spectral simplicial theory
Here we show how the eigenvalues of La carry the notion of simplicial frequeney Yang et al. (2022h). Specifically, we show for k = 1 an eigenvalue measures the total divergence or curl of the eigenvector.
Gradient Frequency: the nonzero eigenvalues associated with the eigenvectors Ui. of Li,d, which span the gradient space im im(B)), admit L, = XI, for any eigenpair u1,G and A Thus, we have * = uI.GLiau = "f,G B B = [] ui,cll3, which is an Euclidean norm of the divergence, i.c., the total nodal variation of u1.G. If an eigenvector has a larger eigenvalue, it has larger total divergence. For the SFT of an edge flow, if the gradient embedding T has a large weight on such an eigenvector, it contains components with large divergence, and we say it has a large gradient frequency. Thus, we call such cigenvalues associated with UG, gradient frequencics.
Curl Frequency: the nonzero eigenvalues associated with the eigenvectors U1, of Li,B, which span the curl space (n(2), admit U1, = cu, for any eigenpair u1cc and Aı,-- Thus, we have AL, = uI, = B,B,BBBB u1,C = [[B] u1,ell, which is an Euclidean norm of the curl, ie., the total rotational variation of u1,c- If an eigenvector has a larger eigenvalue, it has a larger total curl. For the SFT of an edge flow, if the curl embedding G has a large weight on such an cigenvector, it contains components with large curl, and say it has large curl frequency. Thus, we call such eigenvalues associated with Ui. curl frequencies.
Harmonic Frequency: the zero cigenvalues associated with the cigenvectors UL., which span the harmonic space kir(Li), admit L = 0 for any eigenpair uiH and ÅI = 0. From the definition of L1, we have B = B, U1HH = O. That is, the eigenvector 41H is divergence- and curl-free. We also say such an cigenvector has zero signal variation in terms of the nodes and triangles. This resembles the constant graph signal in the node space. We call such zero eigenvalues as harmonic frequencies.
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Fig. 9 shows the simplicial Fourier basis and the corresponding simplicial frequencies of the SC, from which we see how the eigenvalues of L1 can be interpreted as the simplicial frequencies.
For == 0, the eigenvalues of Lo carry the notion of graph frequency, which measures the graph (node) signal smoothness w.r.t. the upper adjacent simplices, i.., edges. Thus, the curl frequency of coincides with the graph frequency and a constant graph signal has only harmonic frequency component. For more general k, there exist these three types of simplicial frequencies, which measure the k-simplicial signal total variations in terms of faces and cofaces.
B Simplicial 2-Complex CNNs and Details on Properties
We give two examples where the first is SCCNN SC of order two, and the second is the form of SCCNN with multi-fcatures. Example 25. For = 2, a SCCNN layer reads as
z oH 1-1 H B,a
á = oH B v + Hi 1 + H Bra 11
(17)
222 o B
B2 a Hzab-
Recursively, we see that a SCCNN layer takes as inputs farp-1 To xx-2 x/2221 to compute zq-- One may find this familar as some type of skip connections in GNNs Xu et al. (2018b). Example 26 (Multi-Feature SCCNN). A multi-fcature SCCNN at layer takes {x x each of which has Fi-1 features, and generates an output X with Fj features as
as inputs,
Ta Σ
Ta Σ 0=
Tu
To B
X
k.d B
x k- k.d +
W.
x
W
k,d, +
Σ L t=0
x
W k.u, + >
L
x k+l
W k,u,
t=0
t=0
(18)
where L' indicates the matrix (-power of L, while superscript indicates the layer index.
B.1 Simplicial locality in details
The construction of SCFs has an cuttra-iimplicill locality. Haæk which consists of basic operations L2 and Lk They are given, on simplex sf by
[x = EjENt, [Laalo)ax))
[I = E EjENY, upy [n [mx];
(19)
where of aggregates signals from its lower and upper neighbors, Wit and Art We can compute the t-step shifting recursively as T = Lx,a k,d #x), one-step shifting of the (t - 1)-shift result; likewise for L Ts A SCF linearly combines such multi-step simplicial shiftings based on lower and upper adjacencies. Thus, the output H is localized in Ta-hop lower and T,-hop upper k-simplicial neighborhoods (Yang et al., 2022b). SCCNNs preserve such intra-simen intru-simplicial locality the elementwise nonlinearity does not alter the information locality, shown in Figs. 2b and 2c.
A SCCNN takes the data on k- and (A: + 1)-simplices at layer - 1 to compute z/ causing interactions between k-simplices and their (co)saces when all SCFs are identity. In turn, s/-1 contains information on (k sissimlleess from layer (- 2. Likewise for a/-1, thus, æy also contains information up to (A + 2 simmmiicee if L 2.2 2, because B (Bx+1) ) # O. Accordingly, this inter-simplicial locality extends to the whole SC if L 2 K, unlike linear filters in a SC where the locality happens up to the adjacent simplices (Schaub et al., 2021; Isufi & Yang, 2022), which limits its expressive power. This locality is further coupled with the intra-locality through three SCFs such that a node not only interacts with the odges incident to it and direct triangles including it, but also edges and triangles further hops away which contribute to the neighboring nodes, shown in Fig. 2d.
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B.2 Complexity
In SCCNN layer for computing æ there are 2 + Ta + Tn filter coefficients for the SCF Hi: and 1 1 Ta and 1+ Tu for His and H respectively, which gives the parameter complexity of order OTTa + Tu). This complexity will increase by F:Fi-1 fold for the multi-fcature case, and likowise for the computational complexity. Given the inputs { 2 *+1h,, we discuss the computation complexity of æ in Eq. (4).
First, consider the SCF operation H As discussed in the localities, it is a composition of Ta-step lower and Tu-step upper simplicial shiftings. Each simplicial shifting has computational complexity of order ((ngm)) dependent on the number of neighbors my where ny is the number of k-simplices. Thus, this operation has a complexity of order O(ngm (Ta + T.)).
Second, consider the lower SCF operation H B a/ As incidence matrix By sparse, it has ng + 1) nonzero entries each k-simplex has + 1 faces. This leads to a complexity of order O(nyk) for operation BR -. Followed by a lower SCF operation, i..., Ta-step lower simplicial shifting, thus, complexity of order O(kmx + Ta) is needed.
Third, consider the upper SCF operation Bx+1
1-1 Likewise, incidence matrix B&+1 has ni++ (k: + 2)
nonzero entries. This leads to complexity of order ((1411k) for the projection operation B4+1 Followed by an upper SCF operation, i.e., a Ti-step upper simplicial shifting, thus, a complexity of order Okn + ngm.TT) is needed.
Finally, we have computational complexity of order Olk(m= + n441) + M& + Tw)) in total. Remark 27. The lower SCF operation H.,.B.. a can be further reduced n < ng. Note that we have
Tc Lg
Ta
H B k
w k.d.
B
a = B
W Wkdd
L
1, 2
(20)
(=0
1=0
where the second equality comes from that La = B.B.B = B. L BR ( (B, BT B (Br B Lx-1,u and likewise for general t. Using the RHS of Eq. (20) where the simplicial shifting is performed in the (k: - 1-)impiplliaaal space, we have a complexity of order Oknn + Tu - Similarly, we have
To Σ wh Li BRT1 = B
Te L
H B BeHa H-1
k+ a
1-1
(21)
t=0
where the simplicial shifting is performed in the (k + simiplllia space. If it follows that nx+1 < A2g we have smaller complexity of O(kn441 7441m4+17aa T.)) by using the RHS of Eq. (21).
B.3 Symmetries of SCs and simplicial data, Equivariance of SCCNNs
Permutation symmetry of SCs. There exists permutation group Pre for each set S in a SC of order K. For K 0, this gives the graph permutation group. We can combine these groups for different simplex orders by group product to form larger permutation group p= X PRy, which is symmetry group of SCs and simplicial data, assuming vertices in each simplex are consistently ordered. That is, we have, for PK) E P, Ip- Lajes = [ ()) (p- Ba]e =
(0) G) and [p æx); [#u)) This permutation symmetry of SCs gives us the freedom to list simplices in any order.
p=
Orientation symmetry of simplicial data. The orientation of simplex is an equivalence class that two permutation Lim (2020); Munkres (2018). Thus, for a
orientations are equivalent if they differ by simplex o = .i.....,h i&} with k > 0, we have an orientation symmetry group Oz,: = for, o by a group homomorphism which maps all the even permutations of (fo...., ik] to the identity element of and all the odd permutations to the reverse operation og,-'
We can further combine the orientation groups of all simplices in a SC as O X OK, by using a group product. This however is not symmetry group of oriented SC because oK Lg changes the signs of
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LE elements in ith column and row, and OK,s B& changes the ith row, resulting in different SC topology. Instead, it is symmetry group of the data space, due to its alternating nature w.r.t. simplices. For o E O we have [o- æxl; = ok.i fu(sf) = fu(o s{), i.e, [æu] remains unchanged w.r.t. the changed orientation of s This gives the freedom to choose reference orientations of simplices when working with simplicial data. Theorem 28 (Permutation Equivariance). A SCCNN in Eq. (4) is P-equivariant. For all p E P, we have p. SCCNN, {Pk-1 : Pk-1,Pk Pk+1 [x+1] + Pk Theorem 29 (Orientation Equivariance). SCCNN in Eq. (4) is O-equivariant if (.) is odd. For all o E O, we have o SCCNN& (ok-1 x 1k 0k+1 [ + 0g ak.
Proof. (informal) Both the permutation group and orientation group have linear matrix representations. By following the same procedure in (Bodnar et al., 2021b, Appendix D) or Roddenberry et al. (2021), we can prove the oquivariance.
B.4 Diffusion process SCs
Diffusion process graphs can be generalized to SCs to characterize the evolution of simplicial data over the SC, in analogy to data diffusion on nodes Anand et al. (2022); Ziegler et al. (2022); Grady & Polimeni (2010).
Here we provide an informal treatment of how discretizing diffusion equations on SCs can give resemblances of simplicial shifting layers. Consider diffusion equation and its Euler discretization with a unit time step
(t) = - Lt)x((t) Euler step: (x + 1) = (t) La = -I - Lx)xxttx())
(22)
with an initial condition (xx(t) xQ. The solution of this diffusion is xxx exp ( (-Lxt)æ As the time increases, the simplicial data reaches to steady state 2 (1) = 0, which lies in the harmonic space ker(La). The simplicial shifting layer resembles this Euler step with weight and nonlinearity when viewing the time step as layer index. Thus, a NN composed of simplicial shifting layers can suffer from oversmoothing on SCs, giving outputs with decreasing Dirichlet energies as the number of layers increases.
Now let us consider the case where the two Laplacians have different coefficients
dx(()= = -La (f) - (t)), Euler step: (x(()= = (1 Lk,d 7 (z).)
(23)
The steady state of this diffusion equation follows (Lu.d + 7 L = O, where (1) would be in the kernal space of Lg still. However, before reaching this state, when the time increases, (t) would primarily approach to the kernel of B i >> 1, in which the lower part of the Dirichlet energy remains, i.e., the decrease of (tx(t)) slows down.
When accounting for inter-simplicial couplings, consider there are nontrivial 0g-1 and a++1 and the diffusion equation becomes
zt (t) - (t) B xx-1 B
(24)
which has source terms B zt-1 B B+411 Consider steady state æts 0. We have Lg.x(()= Dkkd + Øk,us where ay is not in the kernel space of Lg. The Euler discretization gives
a (t + 1) (I (t) Dk,d 0k,u-
(25)
The layer in Bunch al. (2020) = wo([ Lx)ak + wiFk.d + w2mk,a a weighted variant of above stop when viewing time steps layers.
C Related works
We first compare SCCNN with other architectures on if they respect the three principles in Section 3 in Table 5. We then describe how the SCCNN in Eq. (18) generalize other NNs on graphs and SCs in Table 6. For simplicity, we use Y and X to denote the output and input, respectively, without the index l. Note that for GNNs, Lo,d is not defined.
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Table 5: Comparisons between SCCNN and other architectures on they respect the three principles.
Methods
Scheme
P1 P2
P3
MPSN Bodnar et al. (2021b)
cossago-passing
yes yes
no, only direct neighborhoods
Eq. (11) of MPSN, or Bunch et al. (2020)
convolutional convolutional
no yes
no, only direct neighborhoods
Eq. (27) of MPSN
yes yes
no, only direct neighborhoods yes
SNN Ebli et al. (2020)
convolutional
no
PSNN Roddenberry et al. (2021)
convolutional
yes
no
no, only direct neighborhoods yes
SCNN Yang et al. (2022a)
convolutional
yes no
SCCNN
convolutional
yes yes
yes
Table 6: SCCNNs generalize other convolutional architectures on SCs.
Methods
Parameters (n.d. denotes "not defined")
Ebli et al. (2020)
w d, w/ H d. H n.d.
Roddenberry et al. (2021)
Ta = Tu = 1, H R.d Hy.
n.d.
Yang et al. (2022a)
H H n.d.
Bunch et al. (2020)
Ta Tu 1, H H.
=I
H R.u
Bodnar et al. (2021b)
Ta = Tu 1, H
R,u
Gama et al. (2020a) proposed to build a GNN layer with the form
Yo
Σ 1=
Lo.oo Wo.u,
(26)
where the convolution step is performed via a graph filter (Sandryhaila & Moura, 2013; 2014; Gama et al., 2019a; 2020b). This GNN can be easily built a special SCCNN without contributions from edges. Furthermore, Defferrard et al. (2016) considered a fast implementation of this GNN via a Chebyshev polynomial, while Wu et al. (2019) simplified this by setting Wo,t,u as zeros for t < T.. Kipf & Welling (2017) further simplified this by setting Tu = 1, namely, GCN.
Yang et al. (2022a) proposed a simplicial convolutional neural network (SCNN) to learn from k-simplicial signals
T. X k,u
YA
Σ L 1=0
We.de k.d.
Σ
L k
(27)
L=
where the lincar operation is also defined simplicial convolution filter in Yang et al. (2022b). This is special SCCNN with focus one simplex level without taking into the lower and upper contributions consideration. The simplicial neural network (SNN) of Ebli et al. (2020) did not differentiate the lower and the upper convolutions with form of Yv = (Σio D & Wr,), which leads to a joint processing in the gradient and curl subspaces analyzed in Section 4.
While Roddenberry et al. (2021) proposed an architecture (referred to as PSNN)of a particular form of Eq. (27) with Ta = Tú = 1, performing only one-step simplicial shifting Eq. (19). Keros et al. (2022) also performs a one-step simplicial shifting but with an inverted Hodge Laplacian to localize the homology group in an SC. An attention mechanism was addod to both SCNNs and PSNNs by Giusti et al. (2022) and Goh et al. (2022), respectively. Battiloro et al. (2023) added the attention mechanism to SCCNNs.
To account for the information from adjacent simplices, Bunch et al. (2020) proposed a simplicial 2-complex CNN (S2CCNN)
Yo = (Lo Wo.,, + B,X,
Y a (B. X,W i + L, X, K,X, B2X,Wi,, (..)) Y, = (B, X, W2a00 + L2, W2,,,)
(28)
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which is limited to SCs of order two. Note that instead of Hodge Laplacians, simplicial adjacency matrices with self-loops are used in Bunch et al. (2020), which encode equivalent information as setting all filter orders in SCCNNs as one. It is a particular form of the SCCNN where the SCF is a one-step simplicial shifting operation without differentiating the lower and upper shifting, and the lower and upper contributions are simply added, not convolved or shifted by lower and upper SCFs. That is, Bunch et al. (2020) can be obtained from Eq. (4) by setting lower and upper SCFs as identity, Ha,d = K = I, and setting Wk,d,t = Wk,,, and Ta Tu 1 for the SCF H3. The convolution in Yang et al. (2022c, Eq. 3) the same as Bunch al. (2020) though it was performed in block matrix fashion.
The combination of graph shifting and edge shifting in Chen et al. (2022b) can be again seen as special S2CCNN, where the implementation was performed in a block matrix fashion. Bodnar et al. (2021b) proposed a message passing scheme which collects information from one-hop simplicial neighbors and direct faces and cofaces as Bunch et al. (2020) and Yang et al. (2022c), but replacing the one-step shifting and projections from fcocs by some learnable functions. The same message passing was applied for simplicial represcntation learning by Hajij et al. (2021).
Lastly, there are works on signal processing and NNs cell complexes. For example, Sardellitti et al. (2021); Roddenberry et al. (2022) generalized the signal processing techniques from SCs to cell complexes, Bodnar et al. (2021a); Hajij et al. (2020) performed message passing on cell complexes as in SCs and Hajij et al. (2022) added the attention mechanism. Cell complexes are a more general model compared to SOs, where k-cells compared to k-simplices contain any shapes homeomorphic to a k-dimensional closed balls in Euclidean space, e.g., filled polygon 2-cell while only triangles are 2-simplices. We refer to Hansen & Ghrist (2019) for more formal definition of cell complexes. Despite cell complexes weal-world higher-order structures, SCCNNs can be easily generalized to cell complexes by considering any k-cells instead of only k-simplices in the algebraic representations, and the theoretical analysis in this paper can be adapted to cell complexes as well.
more powerful to model
D Proofs for Section 3
D.1 Dirichlet energy minimization perspective
Hodge Laplacian smoothing. We can find the gradient of problem Eq. (7)
as
0D Seg
= B Bx +
BBx42BB4120 thus, gradient descent step follows as Eq. (7) with step size n7.
Proof of Proposition 5. Consider 1= =1 1.
1 = = Ba( - Li - 7 Ln )ark 12 + w/ll B B (I La,d - -xLx,)=
=wil(I -,- Br / 1/2 + will(T I 7 Lk+11,) B 1/ 2/111
(29)
- -I Lx B + woll(1 - LL 1d)
B k 2/.11
which follows from triangle inequality. By definition, we have || - = || Lx 2 and I|
Lk,u 1/- L lB. Also, we have || Lall3 max Lg 11 || LR,, Thus, we have
D( will La (())((a)) when y = 1. When willl Lxll < 1, Dirichlet energy D(x will exponentially decrease as / increases.
When 7 # from Eq. (29), we have D( Da + D,( which follows
Da(a < will(T - LR,a) 1| (a(a))) and Du Du(a < wi - -xk,,))
(30)
When = 1, the oversmoothing condition is | L 2 = max Lx I12. |I L 113) || 211 || L under the oversmoothing condition, by not restricting y to be 1, w L L can be larger than 1 depending on the choice, which means D=(zl) does not necessarily decrease, so does not (axl))
< 1 If
w
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Hodge Laplacian smoothing with sources. The gradient of the objective in Eq. (8) is given by Lank By 0k-1 B which gives the gradient descent update in Eq. (8) with step size
Consider the layer in Bunch et al. (2020) x ++1 = wo ( - Lx) + w 2ndd + w2Kkk. with some weights. By triangle inequality, we have D(ak I w I L ((a)) + wf>max
l1 w Ama (m, u If the weight Wo is small enough following the condition in Proposition 5, the contribution from the projections, controled by weights W and W2; can compromise the decrease by wo, maintaining the Dirichlet energy.
E Proofs for Section 4
E.1 The SCF is
in Proposition 12
Proof. We first give the following lemma.
Lemma 30. Any finite set of eigenfunctions of linear operator spons an invariant subspace.
Then, the proof follows from Lemma 30 and Proposition 10.
E.2 derivation of the spectral frequency response Eq. (11)
SFT of Ty. First, the SFT of æx is given by [ "K, [ccl T with the harmonic embedding Hx.H U 2 = U in the zero frequencies, the gradient embedding Ōg,G = Ug = G in the gradient frequencies, and the curl embedding 0&.c U2 U K,C in the curl frequencies.
SFT of HyR By diagonalizing an SCF HІ with UR, we have
HET = H Ts = Un(hx @ kx)
(31)
where H₂ = (haag(hs)) Here, hu = 1h 1 h is the frequency response, given by
harmonic response he, = (Wk,d,0 + wh,n,) gradient response: hRrG = ETm wk,d St wk,ool curl response: h = Et" W. Oz K.C. wk,dol
with ()) the elementwise 1-th power of a vector. Thus, we can express hg @ ãg as
[haR @ æx, (hr. @ āR,G) (hi, @ "x
(32)
SFT of projections. Second, the lower projection aE,d im (B²)) has only nonzero gradient embedding
ãx.d = Lsd Likewise, the upper projection 2x,u E im in( (B+1)) contains only a nonzero curl embedding
kR_ = U The lower SCF Haa has h = Ihh Aea. G.G as the frequency response that modulates the gradient embedding of æ and the upper SCF HA, has h Σn= wh oot the frequency response that modulates the curl embedding of æk,u-
SFT of yk. For the output y Hg HA + U have
Üv. h ① Øg.H:
ÚK, h Tkd heg Ō..G: Uk.C h @ T + hu @ Tk..
(33)
E.3 Expressive power Proposition 13
Proof. From the Cayley-Hamilton theorem Horn & Johnson (2012), we know that an analytical function f(A) of matrix can be expressed as matrix polynomial of degree at most its minimal polynomial degree, which equals to the number of distinct eigenvalues if A is positive semi-definite.
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Consider analytical function Gk,d of Lx,d, defined the spectrum of L,,d via analytical function Jk,G (A) where À is in the set of zero and the gradient frequencies. Then, Gk,d can be implemented by matrix polynomial of L of order up to RA,G where na,G is the number of nonzero eigenvalues of Lk,d, ie., the number of distinct gradient frequencies. Likewise, any analytical function Gk.u of Lx can be implemented by a matrix polynomial of L of order up to nk,c, which the number of nonzero eigenvalues of L ie., the number of distinct curl frequencies.
Thus, as of the matrix polynomial definition of SCFs in SCCNN, the expressive power of H Ha Hk,u is at most & + (G kk Lk when the matrix polynomial orders (convolution orders) follow Tk,d k.d = n and Tk T k.u n.cc-
E.4 Hodge-awand of SCCNN in Theorem 14
Proof. Consider linear mapping T V - V. An invariant subspace W of T has the property that all vectors v E W are transformed by T into vectors also contained in W, i.e, v E W = T(v) E W. For an input x E im(B)); ); the output Hyæ is in im(Br) too, because of
H El E En E im(Ba)
(34)
where the second equality comes from the orthogonality between im(B,) and im(B)+1). Similarly, we can show that for E im (Bx+1), the output Haæ E im(Bs+1): for x ker ker(Lx); the output Hęæ ker (L.). This essentially says the three subspaces of the Hodge decomposition are invariant with respect to the SCF Ha. Likewiso, the gradient space is invariant with respect to the lower SCF Ha,d, which says any lower projection remains in the gradient space after passed by Ha,d; and the curl space is invariant with respect to the upper SCF Ha.u-
Lastly, through the spectral relation in Eq. (11), the learning operator H2 the gradient space controlled by the learnable weights {wx,d.x}. which is independent of the learnable weights (wk,we), associated to the learning of H2 in the curl space. Likewise, the lower SCF learns in the gradient space well but with another set of learnable weights {ukka,),, and the upper SCF learns in the curl space with learnable weights (w/, From the spectral expressive power, we see that above four independent learning in the two subspaces can be as expressive as any analytical functions of the corresponding frequencies (spectrum). This concludes the independent and expressive learning in the gradient and curl spaces.
F Proofs for Section 5
We first give the formulation of SCCNNs weighted SCs, then we proceed the stability proof.
F.1 SCCNN on weighted SCs
A weighted SC be defined through specifying the weights of simplices. We give the definition of commonly used weighted SC with weighted Hodge Laplacians in Grady & Polimeni (2010); Horak & Jost (2013).
Definition 31 (Weighted SC and Hodge Laplacians). In an oriented and weighted SC, we have diagonal
weighting matrices M2 with [M] measuring the weight of ith k-simplex. A weighted kth Hodge Laplacian is given by
L - L.,. = M,B. M MM,BB BB B&+1 M B B, M.
(35)
where Lu.d and L are the weighted lower and upper Laplacians.
A symmetric version fol-
lows L = M 1/2 L 1/2 and likewise, we have Li = M B M B and Li =
M 1/2 B [T, M 1/2 with the weighted incidence matrix is M M M. Guglielmi et al., 2023; Schaub et al., 2020).
1/2
1/2 (Horak & Jost, 2013;
SCCNNs in weighted SC. The SCCNN layer defined a weighted SC is of form
o = oHHH RR + Hoak + H (11
(36)
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where the three SCFs are defined based on the weighted Laplacians Eq. (35), and the lower and upper contributions akaa and dk, are obtained via. projection matrices Ri.d E [R and Ra, E IR instead of B and B For example, Bunch al. (2020) considered Ri,a M, B, and R1.a B2M2.
F.2 Proof of Stability of SCCNNs in Theorem 24
For a SCCNN Eq. (36) weighted SC S, we consider its perturbed version a perturbed SC S at layer 1, given by
$ o R H. H Re
(37)
which is defined based on perturbed Laplacians with the same set of filter coefficients, and the perturbed projection operators following relativ perturbation model.
Given the initial imput æ for Ao 0, 1....... K, goal is to upper bound the Euclidean distance between the outputs 2 and 2 for I = 1, I,
|| - - |o H & H R
(38)
Haa H 1-1 Re
H Rr R
We proceed the proof in two steps: first, we analyze the operator norm [ Halla of a SCF H& and its perturbed version Hg: then we look for the bound of the output distance for a general L-layer SCCNN. To ease notations, we omit the subscript such that || = max||]][==|||ar Aa/// is the operator norm (spectral radius) of a matrix A, and ||>||| is the Euclidean norm vector æ.
In the first step we omit the indices l and I for simplicity since they hold for general and We first give a useful lemma.
Lemma 32. Given the ith eigenvector of L = -UAU, for lower and upper perturbations Ea and Eu, we have
Eau: = U + E ui E, us = quith; + Ezu,
(39)
with eigendecompositioon Ea VaQaVa and Eu = V.QuV, where Va. V, collect the eigenvectors and Qa, Qu the eigenvalues. It holds that || Eill cada and | cnōu, with 6a = () Va - U1|+1)2 - and
6, = (lV -VV-UU++ +1)
measuring the eigenvector misalignments.
Proof. We first prove that Equ; = qarus + Eiu, The perturbation matrix on the lower Laplacian can be written as Ea- = Fd + En with E-1 = UQaUT and Eσ = (Va UVa U) UQa(Va -U) (Va U)Q.UT. For the ith eigenvector u,, we have that
Eau: Esu + E u. gartk + Eru; u;
(40)
where the second equality follows from Ey gart. Since || Eall S ca, it follows that ||aal ca. Then, applying the triangle inequality, we have that
|[ UVa - Qallll I| eall - + Va Ull
U)
- 11 | - U
(41)
=Ea(() -UI+))= = =edda
which completes the proof for the lower perturbation matrix. Likewise, we can prove for E.
7
F.2.1 Step I: Stability of the SCF
Proof. 1. Low-order approximation of H H. Given a SCF H = Ita wa + IT= Wa we
denote its perturbed version by H Ita Wa L Ite Di where the filter cocfficients are the same. The difference between H and H can be expressed as
To Σ
T
H-H
wa (L)
L +
Wn ( -bi),
(42)
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in which can compute the first-order Taylor expansion of L'
L = (Ld + EaLd + La LA + Dae+ Ca
(43)
with Datt ( -- + E parameterized by and Ca following |Cal| Di EaLa + La Eall La 8-7 Likewise, we can expand D as
E = (Ln La + Lau)*= = L Dua + Cu
(44)
with Dus == E r + ". parameterized by and C. following ||Cull Ei=2 Eu + Eull Lu 1_1 Then, by substituting Eq. (43) and Eq. (44) into Eq. (42), we have
Ta Σ t=
Te Σ 1=
H
wa Da +
Wn Du. + Fa F.
(45)
with negligible terms |Fal== Eal and |F Eul because perturbations are small and the cocfficients of higher-order power terms are the derivatives of analytic functions ha(A) and hc(A), which are bounded (ef. Definition 18).
2. Spectrum of (H H'.. Consider simplicial signal a with an SFT t U 1....... Tn] thus, æ = > 1= T,u;, Then, we study the effect of the difference of the SCFs on a simplicial signal from the spectral perspective via
Ta
H H
wa.t
D d,f
+
Wu,t D Du + Faw + F.
(46)
t=0
1=1 1=0
where we have
= - + L EaL and D D
ELEEL + L E -.-r-1 "),,
(47)
r=0
Since the lower and upper Laplacians admit the eigendecompositions for an eigenvector u;
Lau, dasus,
Luu; Aui
(48)
we can express the terms in Eq. (46) as
E = L u, = X Lilaaa + Eiw,) = qai ui + E
(49)
where the second cquality holds from Lemma 32. Thus, we have
EaL u, qai N L E u.
(50)
With the results in Eq. (49) and Eq. (50), we can write the first term in Eq. (46)
Wa D'
Wd.c
2qaiXhiu
t=0
term
(51)
72
t-1
wa.
(N L E1
E u.)
0=0
term 2
Note that they be jointly diagonalized.
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Term 1 can be further expanded as
term 1= 2
Σ Figas
taa c Na = Bagashasic (Aai)u;
(52)
where we used the fact that Etmo twd e dashia (Nai). Using La UA we can write term 2 in Eq. (51) as
term 2 =
Σ T. ddiagg(()))) E
(53)
where gai E Rn has the jth entry
Ta
for (54)
[gall
Σ
Watt
Nai [Aaly + Nai (Aal)t)
=
da - (ha (Aas)) Nac
for + i.
t=
r=0
Now, substituting Eq. (52) and Eq. (53) into Eq. (51), we have
Wa D Di
Σ x C +
Σ ay diaggga)) E u.
(55)
By following the same steps Eq. (51)-Eq. (54), we can express also the second term in Eq. (46) as
Ta
Wu,
D Dr u. = 2 Σ
2 Aue h'uhhe i diiggggu)) Ezu
(56)
t=0
1=1
where Jui E R. is defined as
Ta
22uthl Auit thi Ana- Aay
for j = i, for j (57)
gguil
Wu,t
N + " [Aul]tt)]
(be ho()us))
1=0
r=u
3. Bound of |1 - H) Now we are ready to bound || H Ha based on triangle inequality. First, given the small perturbations Eall s and |EElll I Eu, we have for the last two terms in Eq. (46)
F I ( | and Fa < (1:))
(58)
Second, for the first term |> To Eia wa. Diulll in Eq. (46), we can bound its two terms in Eq. (52) and Eq. (53) as
Ta Wa
x gas Aa (Nas))a +
D
d
- 2
Σ T. Udiag(ga)) Ejs
(59)
1=0
For the first term the RHS of Eq. (59), write
2
Eeqaidar (Aai) (Mai)
< 4
Tr 2 Plaail Parn 4c222110013
(60)
which results from, first, Igail S <a Eal since qai is cigenvalue of Ea; sccond, the integral Lipschitz
property of the SCF (W)) < Ga; and lastly, the fact that En |a.|2 = || = | and ||u:1|2 = 1. We then have
2 t Da
s 2caca||2:11
(61)
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For the second term in RHS of Eq. (59), we have
k (diaggg)))) U Eiu:
S
[ (digg(a)) U
E u.|l
(62)
which stems from the triangle inequality. We further have (gai UT [|diag(gas)] S 2Ca resulting
from |U|l 1 and the ca-integral Lipschitz of (A) [ef. Definition 18). Moreover, it follows that || || S eada
from Lemma 32, which results in
n
Σ
5. Uddiagga:)U Equs] S 2Caaadayynllr)
(63)
where we use that Ei=1 Øyl-
| 1 S vn||al| = Vn|løl| By combining Eq. (60) and Eq. (63), we have
Wd D uu
2eaccl|<>11 + 2Caaeada/
(64)
Analogously, show that
Ta D ui 2c]cu||=||| V
Wn
(65)
Now by combining Eq. (58), Eq. (64) and Eq. (65), we can bound || x as
|| H)æl - || + &a (z]))=
(66)
2,, | 2C, /n | |=
By defining Aa =((1 + bayn) and A 2( + n) we can obtain that
H H S ca + c. Aucu + o(+)) 0(2)).
(67)
Thus, we have || H.l S + where we ignore the second and higher order terms on éj,d and ck Likewise, we have H.e H s CR.d dR for the lower SCF and [ H CR, for the upper SCF.
with Ax.d = 2(1 /nx) and AR. 2(1 + 6x
/nx)
F.2.2 Step II: Stability of SCCNNS
Proof. Given the initial input æg, the Euclidean distance between a and & at layer can be bounded by using triangle inequality and the co Lipschiit property of a(-) [ef. Assumption 22] as
| - æ/ll2 I co (Phhd + t Sk,u),
(68)
with
Rh. = He 12 k-1 - H Re
o = H k H1a -11
(69)
k H k R $ H - Hll-
We now focus on upper bounding each of the terms.
1. Term oy. By subtracting and adding H. 1-1 within the norm, and using the triangle inequality, we obtain
< ( a )|| + [1
H H 110 a
H k
H. 2 1-11
(70)
S || x (Ck. + Ck c
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where used the SCF stability in Eq. (67) and that all SCFs have normalized bounded frequency response in Assumption 20. Note that H. is also characterized by (a()) with the same set of filter coefficients H.
2. Term dgd and o By subtracting and adding a term H Ra within the norm, we have
L - H Re ( k-1 x (H k.d Red H R
(71)
< Re || - + H k.d Re.d - H Re. Ralllla
where we used again triangle inequality and Rall S from Assumption 20. For the term | we have | S | + ||Ja,alll R & Tx,a(1 + ck.a) where we used || S Tk,d in Assumption 21 and I S 6x.d- For the second term of RHS in Eq. (71), by adding and subtracting H Riaa we have
H H. RR a Rs.all = [ Rx.d Hk,d R + H R H. Rk
I H R Rk |H Haaall S Tk.dkkdd + Ch,dd k
(72)
where we use the stability result of the lower SCF H in Eq. (67). By substituting Eq. (72) into Eq. (71), we have
oka S fr.all الاااl + (TR,aER, + A araa By following the same procedure [cf. Eq. (71) and Eq. (72)), we obtain
(73)
Φk S fk,ull x'Ti1l + (MR,uEk, + C uk ul
(74)
3. Bound of | alll- Using the notations 4x 4.a and le in Theorem 24, we then have a set of recursions, for == 0,
| aill Sca (F (| at-ill |l + a | + |
at -01-111 11 + le ||ar [2n
(75)
+ fk.u ||0 - 4| | + tu |
Define vector D' as [b'] = | a with 6 O. Let B collect the energy of all outputs at layer with [8'] = | 1o/-11 We can express the Euclidean distances of all k-simplicial signal outputs for k 0, 1....... K,
b' 3 c. 2b-12114 c T3
(76)
where 2 indicates elementwise smaller than equal, and have
to
lo.u
1 fu,u
4,
h
11.
fidd
1
fi,u
T
and 2
(77)
IK-Ld
1K-1
I--1,u
fk-1d
fK-1,u 1.u
(K,d
tk
We are now interested in building a recursion for Eq. (76) for all layers I. We start with term æq. Based its expression in Eq. (36), we bound it as
|| | < ca H k.
Ra,d ||| [ 1 1 +|+ + ||
[ R |I
(78)
S co (rk, -
+ rk +|
which holds for I 1,....., K. Thus, it can be expressed in the vector form B c, 2 with
1 To.u
Fi.d
1
Fia
Z =
(79)
TK-1d
т- TK-1,u
TKdd
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Similarly, we have 33 -', leading to 8 1 c Z'z with Bo B [ef. Assumption 23]. We can then express the bound Eq. (76) as
b' c, + + T. B.
(80)
Thus, we have
6 0 3 b² (2T3 + b3 3 +TT3+2TZZ+TZ'3 6a 3....
(81)
which, inductively, leads to
b' 1 d 2 7--1T 1T
(82)
i=1
Bt setting / L, we obtain the bound b" d= c/ Ei=1 2 2''TTZ"
" in Theorem 24.
G Experiment details
G.1 Synthetic experiments on Dirichlet energy evolution
We created a synthetic SC with 100 nodes, 241 edges and 135 triangles with the GUDHI toolbox Rouvreau (2015), and we set the initial inputs on three levels of simplices to be random sampled from 4([-5, 5]). We then built a SCCNN composed of simplicial shifting layers with weight. wp and nonlinearities including id, tanh and relu. When the weight follows the condition in Proposition 5, from Fig. 10 (the dashed lines labled "shift"), see that the Dirichlet energies of all three outputs exponentially decrease as the number of layers increases. We then uncoupled the lower and upper parts of the Laplacians in the edge space in the shifting layers by setting 7 # 1. As shown in Fig. 10 (the dotted lines), the Dirichlet energies of the edge outputs decrease at slower rate than before. Lastly, we added the inter-simplicial couplings, which overcome the oversmoothing problems, as shown by the solid lines.
10'
10-2
10-3
11
10-7
nhif,node
shit.node
10
shift.odde
D(x)
10-11
with pmj mode
prjj. mode
D(o))
shift,edge 10
with proj, node
sngee
shift,dgs
10-15
with proj.
10
proj, odge
with proj..ddae
uncouple lower /uppar shif.bri.
10
uncouple loons iupperaddge
10-47
uncouple lowar/ copar//ppaaddgy shift.rf.
10
with proj. tr..
10-30-
with proj. tr..
10-96
with proj. tri.
Layer
Layer
Layer
(a) identity, 7 2
(b) tanh, 7=
(c) relu, 7
Figure 10: Oversmoothing effects of simplicial shifting and the mitigation effects of uncoupling lower and upper adjacencies and accounting for inter-simplicial couplings.
G.2 Additional details on Forex experiments
In the forcx dataset, there are 25 currencies which can be exchanged pairwise at three timestamps. We first
represented their exchange rates on the edges and took the logrithm, i.e., [x1]].s] = log r4/ = - -= Then, the total arbitrage can be computed as the total curl Bg m1
We considered to recover the exchange rates under three lypes of settings: 1) random noise following normal distribution such that the signal-to-noies ration is -3dB, which is spread over the whole simplicial spectrum; 2) 'curl noise" projected from triangle noise following normal distribution such that the signal-to-noise ration is 3dB, which is distributed only in the curl space; and 3) 50% of the total forex rates are recorded and the other half is not available, set as zero values.
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Table 7: Forex results (umse, arbitrage) and the corresponding hyperparameters.
Methods
Random noise
"Curl noise"
Interpolation
Input
0.119 + 0.004, 25.19 0.874
0.552 + 0.027, 122.36 + 5.90
0.717 + 0.030, 106.40 + 0.902
f2-norm
0.036 + 0.005, 2.29 + 0.079
0.050 + 0.002, 11.12 0.537
0.531 +: 0.043 9.67 + 0.082
SNN
0.11 0.005, 23 24 = 1.03
0,446 + 0.017, 86 96.977 = 2.197
0.702 - 0.033, 104.738 + 1.042
L F 64, 4, tanh
== 6, F 61. 3, tanh
L= 2, 61, 1, tanh
PSNN
0.008 + 0.001, 0.984 - 0.17 L= 6, = 64, tanh
0.000 0.000 0.000 0.000
0.009 0.001, 1.128 - 0.329 L= 6, 64, tanh
L= 5
Bunch
0.981 0.0 22.912 = 1.228
0.981 + 0.0, 22.912 = 1.228
0.983 19.887 6.341
MPSN
0.039 + 0.004, 7.748 + 0.943 L = 64, id, sum
0.076 + 0.012, 14.922 + 2.493 L= 4, 64, tanh, mean
0.117 147 11.674
L == 2, 64, tanh, sum
SCCNN, id
0.027 + 0.005, 0.000 0.000
0.000 + 0.000, 0.000 + 0.000 L 5, 1.Th
0.265 + 0.036 0.000 0.000 L 2, 16, O.
L= 2, 16, Ta Tu 3
SCCNN, tanh
0.002 0.000, 0.325 0.082
0.000 1.000000000000000000000000000000 0.003 = 0.003
0.003 0.002, 0.279 0.151
61, 5. Tu = 2
L= 61, = 2, Tu 2
L= 6 5, Tu =
First, as a baseline method, we chose lo norm of the curl Bg regularizer to reduce the total arbitrage, i.c., 21 = (I+ w.I,,) x1 with regularization weight w E j0, 10). For the learning methods, we consider the following hyperparameter ranges: the number of layers to be L E ( 6), the number of intermediate features to be F E {1, 16, 32, 64). For the convolutional methods including SNN Ebli et al. (2020), PSNN Roddenberry et al. (2021), Bunch Bunch et al. (2020) and SCCNN, we considered the intermediate layers with nonlinearities including id and tanh. The convolution orders of SNN and SCCNN are set to be (1. For the message-passingg method, MPSN Bodnar et al. (2021b), we considered the setting from (Bodnar et al., 2021b, Eq. 35) where the sum and mean aggregations are used and each message update function is a two-layer MLP. With these noisy or masked rates as inputs and the clean arbitrage-free rates as outputs, we trained different learning methods at the first timestamp, and validated the hyperparameters at the second timestamp, and tested their performance at the thrid onc. During the training of 1000 epochs, a normalized MSE loss function and adam optimizer with a fixed learning rate of 0.001 are used. We run the same experiments for 10 times. Table reports the best results (umse) and the total arbitrage, together with the hyperparameters.
G.3
Additional details on Simplex prediction
G.3.1 Method in Detail
The method for simplex prediction is generalized from link prediction based on GNNs by Zhang & Chen (2018): For k-simplex prediction, we use an SCCNN in an SC of order up to k to first learn the features of lower-order simplices up to order k- - 1. Then, we concatenate these embedded lower-order simplicial features and input them to a two-layer ML. which predicts if a k-simplex is positive (closed, shall be included in the SC) or negative (open, not included in the SC).
For example, in 2-simplex prediction, consider an SC of order two, which is built based on nodes, edges and (existing positive) triangles. Given the initial inputs nodes wo and edges æ; and zero inputs triangles 22 = 0 since we assume no prior knowledge on triangles, for an open triangle t = ji, k), an SCCNN is used to learn features on nodes and edges (denoted by y). Then, we input the concalenation of the features on three nodes or three edges to an MLP, i.e., MLLPnode[[p (tgo))/llloo) |l ([oo))) or MLPedge [[v] [ر [ل ((ل to predict if triangle is positive or negative. A MLP taking both node and edge features is possible, but we keep it on one simplex level for complexity purposes. Similarly, we consider an SCCNN in an SC of order three for 3-simplex prediction, which is followed by ML. operating on either nodes, edges triangles.
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G.3.2 Data Preprocessing
We consider the data from the Semantic Scholar Open Research Corpus Ammar et al. (2018) to construct a coauthorship complex where nodes are authors and collaborations between k-author are represented by (k- 1 1i-mmmiimpliess. Following the preprocessing in Ebli et al. (2020), we obtain 352 nodes, 1472 edges, 3285 triangles, 5019 tetrahedrons (3-simplices) and number of other higher-order simplices. The node signal Zu. edge flow 21 and Iriangle flow #2 are the numbers of citations of single author papers and the collaborations of two and three authors, respectively.
For the 2-simplex prediction, we use the collaboration impact (the number of citations) to split the total set of triangles into the positive set Tp- = [x2]c > 7] containing 1482 closed triangles and the negalive set Tx= = [[[[=>]] I 7 containing 1803 open triangles such that we have balanced positive and negative samples. We further split the 80% of the positive triangle set for training, 10% for validation and 10% for testing; likewise for the negative triangle set. Note that in the construction of the SC, ie., the incidence matrix B2, Hodge Laplacians Lı and L2,d, we ought to remove negative triangles in the training set and all triangles in the test set. That is, for 2-simplex prediction, we only make use of the training set of the positive triangles since the negative ones are not in the SC.
Similarly, we prepare the dataset for 3-simplex (tetrahedron) prediction, amounting to the tetradic collabora- tion prediction. We obtain balancod positive and negative tetrahedron sets based on the citation signal æ3. In the construction of B3, L2, and Lz,d, we again only use the tetrahedrons in the positive training set.
G.3.3 Models
For comparison, we first use heuristic methods proposed in Benson et al. (2018) as baselines to determine if a triangle 1 = [ k] is closed, namely, 1) Harmonic mean: 8t = 3/(33/ 3[(=))]]] 1-1 + [
[=)] 2) Geometric mean: limp-o [[)]] + [2]][j. + [<) [=)((,,))))/,,, and 3) Arithmetic mean: 5, = ([x)) + [2]]]]] + (slls.))33 which compute the triangle weight based on its three faces. Similarly, we generalized these mean methods to compute the weight of 3-simplex [i. k m] based on the four triangle faces in 3-simplex prediction.
We then consider different learning methods. Specifically, 1) "Bunch" " by Bunch et al. (2020) (we also generalized this model to 3-dimension for 3-simplex prediction); 2) Message passing simplicial network ("MPSN") by Bodnar et al. (2021b) which provides a baseline of message passing scheme in comparison to the convolution scheme; 3) Principled SNN ("PSNN") by Roddenberry et al. (2021); 4) SNN by Ebli et al. (2020); 5) SCNN by Yang et al. (2021); 6) GNN by Defferrard et al. (2016); 7) MLP: providing as a bascline for the effect of using inductive models.
For MLP, Bunch, MPSN and our SCCNN, we consider the outputs in the node and edge spaces, respectively, for 2-simplex prediction, which are denoted by suffix "-Nodc" or " "-Edge". For 3-simplex prediction, the output in the triangle space can be used as well, denoted by a suffix "-Tri." where we also build SCNNs in both edge and triangle spaces.
G.3.4 Experimental Setup and Hyperparameters
We consider the normalized Hodge Laplacians and incidence matrices, a particular version of the weighted ones Horak & Jost (2013); Grady & Polimeni (2010). Specifically, we use the symmetric version of the normalized random walk Hodge Laplacians in the edge space, proposed by Schaub et al. (2020), which were used in Bunch al. (2020); Chen et al. (2022a) as well. We generalizod the definitions for triangle predictions.
Hyperparameters 1) the number of layers: L 4,, 2) the number of intermediate and output fcatures to be the same as F E {16, 32}; 3) the convolution orders for SCCNNs are set to be the same, i.c., T-= = Ta = Tn = T = T E {1, 2, 3, 4, 5). We do so to avoid the exponential growth of the parameler search space. For GNNs (Defferrard et al., 2016) and SNNs (Ebli et al., 2020), we set the convolution orders to be Te 21, 3, 4,,455 5,5) while for SCNNs (Yang et al., 2022a), we allow the lower and upper convolutions to have different orders with Ta, Tu E (1,2,3, 4, 5); 4) the nonlincarity in the feature learning phase: LeakyReLU with negative slope 0.01; 5) MPSN is set Bodnar et al. (2022); 6) the MLP in the prediction phase: two
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layers with sigmoid nonlinearity. For 2-simplex prediction, the number of the imput features for the node features is 3F, and for the edge features is 3F. For 3-simplex prediction, the number of the input features for the node features is 4F, for the edge features is 6/" and for the triangle features is 4F since a 3-simplex has four nodes, six edges and four triangles. The number of the intermodiate features is the same as the input Teatures, and that of the output leatures is one; and, 7) the binary cross entropy loss and the adam optimizer with a learning rate of 0.001 are used; the number of the epochs is 1000 where an early stopping is used. We compute the AUC compare the performance and run the same experiments for times with random data splitting.
G.3.5 Results
In Table 8, we report the best results of cach method with the corresponding hyperparamoters. Different hyperparameters can lead similar results, but report the ones with the least complexity. All experiments for simplex predictions were
single NVIDIA A40 GPU with 48 GB of memory using CUDA 11.5.
Table 8: 2- (Teft) and 3-Simplex (Right) prediction AUC (%) results.
METHODS
AUC
PARAMETERS
METHODS
AUC
PARAMETERS
HARM. MEAN
62.8.2.7
HARM. MEAN
63.6.1.6
ARITH. MEAN
60.8=3.2
ARITH. MEAN
62.2+1.4
GEOM. MEAN
61.7-3.1
Gεом. MEAN
63.1114
MLP-NODE
68.5-1.6
1= 1, 32
MLPP-TRI.
69.0+2.2
= 3, F 32
GNN
93.9=1.0
L= --
GNN SNN-TRE
96.620.5
L= 5, 32, 1
SNN-EDS -NN-EDGE
92.0=1.8
=5 ==
95.1+1.2
1=
PSNN-EDGE
95.6=1.3
PSNN-TRI. 98.1+0.5
1= 32 32, 2,7.
SCNN-EDGE
96.511.5
L = 5, F = 3,,
SCNN-TRI. 98.31004
== 5,
F
BUNCH-NODE
98.3=0.5
K=1,1=4F==3
BUNCH- EDGE
98.5+0.5
-=3,L=4F==
MPSN-NODE
98.120.5
K K=1.-=8P-- 32
MPSN-EDGE
99.210.3
-=3,-= 32
SCONN-NonE
98.7-0.5
==
SCCNN-NODE
99.4+0.3
K=
3,
G.3.6 Complexity
Table 9: (Left) Complexity of the best three methods for 2-simplex prediction. SCCNN with different layers and convolution orders.
(Right) Running time of
METHOD
#PARAMS.
RUNNING (SECONDS EPOCH1)
HYPERPARAMS.
T
T
SCCNN
24288
0.073
L-2 2
0.073
0.082
BUNCH
21728
0.140
L-3 3
0.110
0.130
MPSN
84256
0.028
L=
0.192
0.237
Here report the number of parameters and the running time of SCCNN for 2-simplex prediction one NVIDIA Quadro K2200 with 4GB memory, compared with the two best alternatives. MPSN, compared to convolutional methods, has three times more parameters, analogous to the comparison between message- passing and graph convolutional NNs. We also report the rumning time as the layers and convolution orders increase.
G.3.7 Ablation Study
We perform an ablation study to observe the roles of different components in SCCNNs.
SC Order K We investigate the influence of the SC order K. Table 10 reports the 2-simplex prediction results for == (1 and the 3-simplex prediction results for (1,2,3) We observe that for k-simplex prediction, it does not necessarily guarantee better prediction with higher-order SC, which further
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indicates that a positive simplex could be well encoded by both its faces and other lower-order subsets. For example, in 2-simplex prediction, SC of order one gives better results than SC of order two (similar for Bunch), showing that in this coauthorship complex, triadic collaborations are better encoded by features on nodes than pairwise collaborations. In 3-simplex prediction, SCs of different orders give similar results, showing that tetradic collaborations can be encoded by nodes, as well as by pairwise and triadic collaborations.
Table 10: Prediction results SCCNNs with different SC order K.
METHOD
3-SIMPLEX
PARAMETERS
METHOD
2-SIMPLEX
PARAMETERS
SCCNN-NODE
99.3.10.3
==1,
32,
SCCNN-NODE
08.7
K ==,,,=2,f=3,,=
SCCNN-NODE
99.310.2
=== 32,
SCCNN-NODE
98.410..5
==2,==2f==377=2
OCCNN-NOME
99.4+0.3
==
BUNCH-NODE
98.3=0.4
K=1,-=4,F=*
MPSN-NODE 96. 96.01112
=,1,==3,==0 ==
BUNCH-NODE
98.0=0.4
==
MPSN-NODE 98.2+-0.8
MPSN-NODE
94.511.5
==1,L=3,P 32
MPSN-NODE
98.1+0.5
ACCNN-EneH
98.910.3
K ==1,L=3,F =
SCCNN-EDeN 99.2.10.4
K= 2, K=2,L=5,F7 = 32,T=
SCCNN-Fner 97.9+0.99
3,
SCCNN- -OCNN-EDGE
99.0#1.0
K ==
SCCNN-EDGR 95.9#1.0
==2,= =2,1"====
MPSN-EDGE
96,31111
K=1,- - 3,
BUNCH- EDGE
97.311.1
K=1,L=4,P=
MPSN-ENS EDGE
98.3++0.8
3,2,133F=332
BUNCH- EDGE 94.611.2
K= ==2,L4,77 -
MPSN-EDGE 94.1+2.4
K 3,
SCCNN-TRI
97.9+0.7
K 2, T= F,F -2,
MPSN-EDGE 97.011.2
K 2. 2, 16
SCCNN-TRI. 97.4+0.9
K=
32,7
MPSN-TRI.
99.1+0.2
K= 3,
With a focus on 2-simplex prediction with SCCNN-Node of order one, to avoid overcrowded settings, we study how each component of an SCCNN influences the prediction. We consider the following settings without: 1) "Edge-to-Node", where the projection Yo,u from edge to node not included, equivalent to GNN;; 2) "Node-to-Node", where for node output, we have x o HH... R1, _
Missing Components in SCCNN
3) "Node-to-Edge", where the projection æı,d from node to edge is not included, i.c., we have æt = o H ( 1-1); and 4) "Edge-lo-Eae Edge", where for edge output, we have æ = 1,d Ri,a R
Table 11: 2-Simplex prediction (SCONN-Node without certain components or with limited inputs).
Missing Component
AUC
Parameters
Missing Input
AUC
Parameters
98.7+0.5
L= 2, F 32, =
98.710.5
L 2, F 32, T I= 2, F 32, T 4
Edge-to-Node
93.9+0.8
I 5, F.F= F= =332,7
Node input
98.2+0.5
Node-to-Node
98.7+0.4
L=4,F F. =
Edge input
98.1+0.4
I= F 32, T =3
Edge-to-Edgee Edge
98.511.0
L ==3,F F =32,
L = F,F F- 32,T = Node, Edge inputs
50.0=0.0
Node-to-Eie Edde
98,810.3
From the results in Table 11 (Teft), we see that "No Edge-to-Node", i.e., GNN, gives much worse results as it leverages no information on edges with limited expressive power. For cases with other components missing, a similar performance can be achieved, however, at cost of the model complexity, with either a higher convolution order or a larger mumber layers L, while the latter turn degrades the stability of the SCCNNs, as discussed Soction SCCNNs with certain inter-simplicial couplings pruned/ /aissdmmssnngggg can be powerful as well (this is similarly shown by (Bodnar et al., 2021b, Thm. 6)), but if we did not consider certain component, it comes with a cost of complexity which might degrade the model stability if more number layers are required.
Limited Input We study the influence of limited input data for model SCCNN-Node of order two. Specifically, we consider the input on either nodes or edges is missing. From Table 11, we see that the prediction performance does not deteriorate at cost of the model complexity (higher convolution orders) when certain part of the input missing except with full zeros as input. This ability of learning from limited data shows the robustness of SCCNNs.
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G.3.8 Stability Analysis
We then perform a stability analysis of SCCNNs. We artificially add perturbations to the normalization matrices when defining the Hogde Laplacians, which resemble the weights of simplices. We consider small perturbations Eg on node weights which is a diagonal matrix following that || S co/2. We generate its diagonal entries from uniform distribution ////2,co/// with co [0, 1], which represents one degree of deviation of the node weigths from the true ones. Similarly, perturbations on edge weights and triangle weights are applied to study the stability. In a SCCNN-Node for 2-simplex prediction of K =2, we measure the distance between the simplicial outputs with and without perturbations on nodes, edges, and triangles, i.e., || -/ Il, for l = 0,0,,,, 1,2.
Stability dependence We first show the stability mutual dependence between different simplices in Fig. 11. We see that under perturbation on node weights, the triangle output is not influenced until the number of layers becomes two; likewise, the node output is not influenced by perturbations on triangle weights with onc-layer SCCNN. Also, a onc-layer SCCNN under perturbations edge weights will cause outputs nodes, edges, triangles perturbed. Lastly, we observe that the samne degree of perturbations added to different simplices causes different degrees of instability, owing to the number n& of k-simplices in the stability bound. Since No < N1 < N2, the perturbations on node weights cause less instability than those on edge and triangle weights.
10'
wale
nige
sdge
11-3
10
(a) Node pert.L = 1 (b) Node pert.L = 2 (c) Edge pert. = (d) Tri. pert. =
(e) Tri. pert. = 2
Figure 11: The stabilities of different simplicial outputs are dependent on each other.
Number of Layers Fig. 12 shows that the stability of SCCNNs degrades as the number of layers increases as studied in Theorem 24. As the NN deepens, the stability deteriorates, which corresponds to our analysis of using shallow layers.
socs
vide otig
stge
edge
D0
10*
161
16-
161
39'7
19
18
161
169
(a) 1
(b) = 2
(c) L
(d) L
(e) 5
Figure 12: The relative difference of SCCNN outputs with and without perturbations in terms of different. numbers of layers. We consider perturbations on edge weights.
G.4 Additional details on Trajectory prediction
G.4.1 Problem Formulation
A trajectory of length m can be modeled as a sequence of nodes [vo, Ui...., 2m-1] Um-1] in an SC. The task is to predict the next node vm from the neighbors of um-1, Nym... The algorithm in Roddenberry et al. (2021) first represents the trajectory equivalently as sequence of oriented edges [[ 81], [V,, og]...., (m.. Pm]]]]. Then, an edge flow æi defined, whose value on edge is [æa]e- 1 if edge is traversed by the trajectory
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in forward direction, [2ile- -1 if edge is traversed in backward direction by the trajectory, and [aa]e- = 0, otherwise.
With the trajectory flow 21 as the input, together with zero inputs on the nodes and triangles, an SCCNN of order two is used to generate a representation of of the trajectory, which is the output on edges. This is followed by a projection step = = B, Waf, where the output first passed through a linear transformation via W, then projected into the node space via B Lastly, distribution over the candidate nodes Ne.. is computed via softmax operation, n, softmax((= J E N The best candidate is selected Um = argmax, R. We refer to Roddenberry & Segarra (2019, Alg. S-2) for more details.
Given that an SCCNN of order two generates outputs also on nodes, we can directly apply the node feature output #6 to compute a distribution over the candidate nodes Nr... without the projection step. We refer to this as SCCNN-Node, and the method of using the edge features with the projection step as SCCNN-Edge.
G.4.2 Model
In this experiment, we consider the following methods: 1) PSNN by Roddenberry el al. (2021); 2) SNN by Ebli et al. (2020); 3) SCNN by Yang et al. (2022a) where we consider different lower and upper convolution orders Ta. Tui and 4) Bunch by Bunch et al. (2020) where we consider both the node features and edge features, namely, Bunch-Node and Sunch-Edge.
Synthetic Data Following the procedure in Schaub et al. (2020), we generate 1000 trajectories as follows. First, we create an SC with two "holes" by uniformly drawing 400 random points in the unit square, and then a Delaunay triangulation is applied to obtain a mesh, followed by the removal of nodes and edges in two regions. To generate a trajectory, we consider a starting point at random in the lower-left corner, and then connect it via shortest path to a random point in the upper left, center, or lower-right region, which is connected to another random point in the upper-right corner via a shortest path.
We consider the random walk Hodge Laplacians Schaub et al. (2020). For Bunch method, set the shifting matrices as the simplicial adjacency matrices defined in Bunch et al. (2020). We consider different NNs with three intermediate layers where each layer contains F' = 16 intermediate features. The tanh nonlinearity is used such that the orientation equivariance holds. The final projection n generates a node feature of dimension one. In the 1000-epoch training, we use the cross-entropy loss function between the output d and the true candidate and we consider an adam optimizer with learning rate of 0.001 and a batch size 100. To avoid overfitting, apply a weight decay of 5 10-6 and carly stopping.
As done in Roddenberry et al. (2021), besides the standard trajectory prediction task, we also perform a reverse task where the training set remains the same but the direction of the trajectories in the test set is reversed and generalization task where the training set contains trajectories rumning along the upper left region and the test set contains trajectories around the other region. We evaluate the correct prediction ratio by averaging the performance over 10 different data generations.
Real Data We also consider the Global Drifter Program 3 localized around Madagascar. It consists of occan drifters whose coordinates are logged every 12 hours. An SC can then be created as Schaub et al. (2020) by treating each mesh as node, connecting adjacent meshes via an edge and filling the triangles, where the "hole" is yielded by the island. Following the process in Roddenberry et al. (2021), it results in 200 trajectories and we use 180 of them for training. In the training, batch size of 10 is used and no weight decay is used. The rest experiment setup remains the same as the synthetic case.
G.4.3 Results
We report the prediction accuracy of different tasks for both datasets in Table 12. We first investigate the effects of applying higher-order SCFs in the simplicial convolution and accounting for the lower and upper contributions. From the standard accuracy for both datasets, we observe that increasing the convolution orders improves the prediction accuracy, c.g., SCNNs become better the orders Ta, Tu increase and perform
3http://wwe aoml. got/e//////vvid/////.
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always better than PSNN, and SCCNNs better than Bunch. Also, differentiating the lower and upper convolutions does help improve the performance as SCNN of orders Ta Tu = performs better than SNN of 3.
However, accounting for the node and triangle contributions in SCCNNs does not help the prediction compared the SCNNs, likewise for Bunch compared to PSNN. This due the zero node and triangle inputs because there are no available node and triangle features. Similarly, the prediction directly via the node output features is not accurate compared to projection from edge features.
Moreover, we also observe that the performance of SCCNNs that are trained with the same data does not deteriorate in the reverse task because the orientation equivariance ensures SCCNNs to be unaffected by the orientations of the simplicial data. Lastly, we see that, like other NNs on SCs, SCCNNs have good transferability to the unseen data.
Table 12: Trajectory Prediction Accuracy. (Left): Synthetic trajectory in the standard, reverse and gener- alization tasks. (Right): Ocean drifter trajectorics. For SCCNNs, we set the lower and upper convolution orders Ta, Tu the same as T.
METHODS
STANDARD
REVERSE
GENERALIZATION
PARAMETERS
STANDARD
PARAMETERS
PSNN
63.143.1
58.4+3.9
55.34:2.5
49.0+8.0
SCNN
65.643.4
56.626.0
56.113.6
Ta Tu
52.5=9.8
Ta T.
SCNN
66.5#5.8
57.7#5.4
60.6+4.0
Ta = Tu =
52.5=7.2
Ta Tu 3
SCNN
67.3.1..3
56.9144.8
59.414.22
Ta T,
52.518.7
Ta T.
SCNN
67.711.7
55.315.3
61.213.2
Ta T. =
53.017.8
Ta T. 5
SNN
65.5++2.4
53.6416.1
59.5+3.77
52.5+6.00
== 3
BUNCH-NODE
35.413.4
38.114.66
29.013.0
35.015.9
MONCH-EDGE
62.3+4.0
59.6+6.11
53.9+3.11
46.0+6.2
CCCNN-NoE 46.8+7.3
44.5+8.2
31.945.0
40.5+4.7
==1
SCCNN- EEDGE 64.643.9
57.246.3
54.0#3.0
T =
52.5=7.2
T -
SCCNN-NODE 43.549.6
44.447.6
32.842.6
T=
45.5+4.7
T
SCCNN-EDGE 65.2144.1
58.9144.1
56.81224
T 2
54.547.9
T=
G.4.4 Convolution Order and Integral Lipschitz Property
Here, to illustrate that the integral Lipschitz properly of the SCFs helps the stability of NNs on SCs, we consider the effect of regularizer rı in Eq. (16) against perturbations in PSNNs and SCNNs with different. Ta and Tú for the standard synthetic trajectory prediction. The regularization weight 7iL is set as 5 10-4 and the number of samples to approximate the frequencies is set such that the sampling interval is 0.01.
Fig. 13 shows the prediction accuracy and the relative distance between the edge outputs of the NNs trained with and without the integral Lipschitz regularizer terms of different levels of perturbations. We see that the integral Lipschitz regularizer helps the stability of the NNs, especially for large SCF orders, where the edge output is less influenced by the perturbations compared to without the regularizer. Meanwhile, SCNN with higher-order SCFs, c.g., Ta T, achieves better prediction than PSNN (with one-step simplicial shifting), while maintaining good stability with its output not influenced by perturbations drastically.
We also measure the lower and upper integral Lipschitz constants of the trained NNs across different layers and features, given by maxxa. A (Ax,a)) and max | shown Fig. 14. We see that the SCNN trained with 7n. indeed has smaller integral Lipschitz constants than the one trained without the regularizer, thus, better stability, especially for NNs with higher-order SCFs.
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0.5
0.5
0.5-
0.5
01.5
0.5
0.4
with
TIL
without Pi..
0.4
without FtL
without TTL
0.44
mitiout FTL
2.0
2.2
2.0
1.5
without is.
1.5-
without
without
without
1.0
0.5
0.5
0.0
0.3
0.0
0.10
0.25
0.50
0.75
1.00
0.00
0.25
0.50
0.75
1.00
0.03
0.35
0.50
0.70
1.00
0.00
0.25
1.50
0.75
1.00
(a) PSNN
(b) SCNN13
(c) SCNN31
(d) SCNN55
Figure 13: Effect the integral Lipschitz regularizer VIL in the task of synthetic trajectory prediction against. different levels e of random perturbations on Lia and Lı We show the accuracy (Top row) and the relative distance between the edge output (Bottom row) for different NNs on SCs with and without TnL- SCNN13 is the SCNN with Ta and T, 3.
0.5
Cle Cla Cie Ci. Cie C
Ola Cl. Cis Ci. Ct. Cf.
Ola Cl Cis Ct Ci C
d Cr Cis CLe cia Ci.
(a) SCNN31, without Vn.
(b) SCNN31, with VTL
(c) SCNN55, without VTL
(d) SCNN55, with rn
Figure 14: The integral Lipschitz constants of SCFs each layer of the trained SCNNs with and without the integral Lipschitz regularizer riL- We symbols d and oe to denote the lower and upper integral Lipschitz constants at layer I. Regularizer rı promotes the integral Lipschitz property, thus, the stability, especially for NNs with large SCF orders.
46


Reviews:
[{"review_id": "t36q6ODV7W", "reviewer": "Reviewer_bo12", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Overall, I think it is a strong contribution, and I feel quite positive about this work. The writing is clear (although it could be better, see 'Requested Changes'), the proposed generalization of SCFs is natural, its theoretical guarantees are nice and important, and numerical experiments are conducted to investigate the practical behavior of the network. To me, this work would be a good fit to TMLR.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In this article, the authors propose a new generalization of simplicial complex convolutional neural networks (SCCNNs), which encompasses previous approaches from the literature, and, most importantly, satisfy key theoretical properties related to the discrete Hodge decomposition theorem of flows on simplicial complexes. As these flows can always be written uniquely as a linear combination of a harmonic, div-free and curl-free part, the authors propose to extend the usual simplicial convolutional filters (SCFs), which are intuitively polynomial functions of the up and down Laplacians, with similar polynomial functions of the projections from higher and lower dimensional flows. This uncoupling of the up and down Laplacians, incorporation of the lower and higher dimensional signals, and higher order convolutions (quantified by the polynomial degrees) allows to make use of the Hodge decomposition, as all Hodge subspaces are preserved under the operation, while keeping high discriminativity and expression levels. Moreover, the authors relate their operations to the Dirichlet energy of the signal (which do not necessarily decrease to zero, as in a traditional SCF, thanks to the projections), and provide a theoretical upper stability bound upon perturbations of both the Laplacians and simplicial flows, which depends on the integral Lipschitz constants of the SCFs. Finally, they provide several experiments showcasing the different properties of their SCCNNs.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "There are a few things and questions that I would like the authors to clarify or take into account:\n\n- I did not understand the paragraph on information spillage at the end of page 8 (just before Proposition 13) and just before Section 6. I do not see why the nonlinearity plays a role in it, and I do not understand the connection with Figure 3(b). I think this should be discussed in greater details.\n\n- In Section 3.2, if the goal is to avoid over-smoothing, wouldn't a lower bound on $D(x_k)$ be more interesting than an upper bound in Proposition 6?\n\n- I am not sure how to interpret Assumption 21. What would be the values of $r_{k,d}$ and $r_{k,u}$ for a simplicial complex with $n$ simplices? Is there a way to control these values in practice?\n\n- It would be nice to connect Theorem 24 more explicitly with the parameters  $w_{k,d/u,t}$, and $w'_{k,d/u,t}$. How should I tune these parameters in practice if I want a tight control on, e.g., the integral Lipschitz constants? Overall, it would be useful to make the discussion just before Section 6 more formal by quantifying it with these weights.\n\n- In the experiment in Section 6.1, it is claimed that SNNs perform poorly because of a shared eigenvalue in the curl and gradient spaces. Is it just a guess, or has it really been observed numerically? I have the impression that those spaces sharing the exact same eigenvalue would never happen generically.\n\n- The text should be proofread, there are several minor typos.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "LgJLoScJU0", "reviewer": "Reviewer_sSBr", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "I have made a serious attempt to read this article, which seems interesting and well-polished, but I'm afraid that even after looking up several concepts, I still find it difficult to follow and I seriously suspect that this is due to a lack of expertise on my side, rather than the writing of the article, since the article is out of my expertise - for example I am not familiar with most of the cited literature. Hence I am afraid that I am not able to offer a useful review of the paper. I am sorry I cannot be of more help.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The present paper studies learning from data defined on simplicial complexes, such as nodes, edges, triangles, etc. The key idea is to leverage the Hodge theorem that decomposes simplicial data into three orthogonal characteristic subspaces, such as the identifiable gradient, curl and harmonic components of edge flows. \n\nThe main contributions of the paper are then threefold: (1) proposing a unified network architecture called Simplicial Complex Convolutional Neural Network (SCCNN) that uncouples lower and upper simplicial adjacencies, accounts for inter-simplicial couplings, and performs higher-order convolutions to mitigate oversmoothing; (2) characterising the spectral behaviour and expressive power of SCCNN using Hodge decomposition, enabling effective and rational learning on simplicial complexes; and (3) providing a theoretical stability bound for SCCNN outputs against perturbations in simplicial connections, guiding the design of robust convolutional architectures. These contributions are validated through various simplicial tasks, demonstrating the effectiveness of Hodge-aware learning and stability analysis.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "n/a", "broader_impact_concerns": "n/a", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "m3J0O1CpvZ", "reviewer": "Reviewer_yPbz", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: The paper is very mathematical and this is not my field; still, I found the exposition clear and illustrations very helpful and well done. The suggested architecture makes sense. I cannot fully judge on the mathematical theory as it went too far beyond my knowledge and I could not follow everything.\n\nWeaknesses: The experimental section requires more details -- I could not fully understand the experimental setup.\n\nOverall, I think the paper can be accepted to TMLR after a straightforward revision.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "In their paper \"Hodge-aware convolutional learning on simplicial complexes\", the authors suggest a new architecture (SCCNN) generalizing graph convolutional neural networks (GNN) from graphs to more general simplicial complexes. They argue that SCCNN have good theoretical properties and show that in experiments, they outperform some existing convolutional architectures on simplicial complexes.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "MAJOR COMMENTS\n\n* Description of experiments in Section 6 should be MUCH MORE DETAILED. I cannot follow the tasks and the experimental setups. For example in 6.1: what exactly is the dataset? What is K, are there any inputs beyond x_1? What is the data size? What is the training set and the test set? What is being predicted exactly? What exactly is the loss function of the network? Please really spell all of it out in detail.\n\n* The same in 6.3: what are you predicting? \"Predicting collaborations\" -- what does it mean? If the simplex is present or not? How can the network make this prediction? If you remove one simplex from the traning data, then how can the network \"fill it in\"? Are you masking out simplexes? So what does the network get as input for that simplex then? Please explain the train/test splits and the loss function etc.\n\n* Tables should have in bold not only max values, but everything that is not  signif different from the max value. E.g. in Table 2 it seems MPSN peforms as good as SCCNN given the uncertainties. So make it bold too. Same in other tables, esp. Table 4. By the way, what are the uncertainties? Std over multiple runs? How many runs? What was randomized? Please specify.\n\n* Table 3: Would be good to have a column for performance at L=2, T=2. If I understand the logic correctly, the performance should be lower for many ablations.\n\n* Are GNNs a special case of SCCNNs? Can you make an explicit comment about it?\n\n* Please include data/code availability statement even though it cannot contain a link yet due to anonymity.\n\n\nMINOR COMMENTS\n\n* Section 2.2, \"Simplicial signals\": I did not undersatnd the technical defition of \"alternating map\". If f: S^k -> R^n, then f acts on S^k which is the set of all k-simplices defined above without orientation. I.e. orientation is not part of S^k, right? Then what does \"alternating\" mean?\n\n* Section 2.2: the first time you say \"edge flow\", this term is not defined. Maybe state that \"simplicial signal x_1 is also called edge flow\".\n\n* Equation 4: this equation contains lots of terms that are partially defined in Equation 5 below and partially in the subsequent text (H_k,d etc.). This makes it difficult to parse. I suggest to define all terms in a \\gather environment together with Eq 5.\n\n* Section 3.1, \"Complexity\" or earlier in \"Example 3\". I think it would be helpful to be explicit about the number of parameters, e.g. in a specific situation T_d = T_u = 2, and k=1. If I understand correctly, there are only VERY FEW parameters, like maybe 2*2*2 = 8 if counted correctly? That's not what one expects from a neural network \"layer\". Of course this is because you only consider d=1 dimensional features. But to be honest, I am even wondering if it is possible to learn anything useful at all, if original \"features\" are scalars...\n\n* Section 5: I did not understand the sentences between Definitions 17 and 18. \n\n* Section 5: I would like to see some comments on why the specific form of the perturbation in Definition 17 makes sense.\n\n* Theorem 24 formulation is VERY complex and hard to parse. Can you provide some intuition about what it implies? I cannot even easily see if the bound is linear/quadratic/cubic in epsilon and r terms...\n\n* Table 1: What is \"interpolation\" column?\n\n* Table 1: Random Noise, total arbitrage column: SCCNN id should be bold, not SCCNN tanh. Same in the interpolation. Also in Curl Noise something should be bold.\n\n* Typos: \"generalzing\", \"arbitray\", \"messgae\"", "broader_impact_concerns": "No concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The reviewers were unanimous in recommending acceptance of the manuscript. They recognized its methodological, theoretical and experimental contributions to the problem of learning from simplicial complexes with neural networks. The authors addressed the reviewer's requests for changes for the revised version of the manuscript.
Decision: Accept (poster)