POOLING STRATEGIES FOR SIMPLICIAL CONVOLUTIONAL NETWORKS Domenico Mattia Cinque Claudio Battiloro Paolo Di Lorenzo DIET Department Sapienza University of Rome Via Eudossiana 18 00184 Rome Italy

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POOLING STRATEGIES FOR SIMPLICIAL CONVOLUTIONAL NETWORKS

Domenico Mattia Cinque, Claudio Battiloro, Paolo Di Lorenzo

DIET Department, Sapienza University of Rome, Via Eudossiana 18, 00184, Rome, Italy

E-mail: domenico.cinque98@gmail.com, {claudio.battiloro, paolo.dilorenzo}@uniroma1.it

ABSTRACT

The goal of this paper is to introduce pooling strategies for sim-

plicial convolutional neural networks. Inspired by graph pool-

ing methods, we introduce a general formulation for a simpli-

cial pooling layer that performs: i) local aggregation of sim-

plicial signals; ii) principled selection of sampling sets; iii)

downsampling and simplicial topology adaptation. The gen-

eral layer is then customized to design four different pooling

strategies (i.e., max, top-k, self-attention, and separated top-k)

grounded in the theory of topological signal processing. Also,

we leverage the proposed layers in a hierarchical architecture

that reduce complexity while representing data at different res-

olutions. Numerical results on real data benchmarks (i.e., ﬂow

and graph classiﬁcation) illustrate the advantage of the pro-

posed methods with respect to the state of the art.

Index Terms—Topological signal processing, topological

deep learning, simplicial neural networks, pooling.

1. INTRODUCTION

In the last years, Graph Neural Networks (GNNs) [1–3] have

shown remarkable results in learning tasks involving data de-

ﬁned on irregular domains (e.g., graphs), such as social net-

works, recommender systems, cybersecurity, natural language

processing, genomics, and many more [3, 4]. However, GNNs

are designed to work with graphs, which consider only pairwise

relationships between data. On the contrary, many real-world

phenomena involve multi-way relationships as, e.g., in biologi-

cal or social networks. Some recent works in topological signal

processing [5, 6] have shown that multi-way relationships can

be described using simplicial complexes, which are speciﬁc in-

stances of hyper-graphs with powerful algebraic representation

able to model higher-order interactions among nodes. Conse-

quently, there was also a raising interest in the development of

(deep) neural network architectures able to handle data deﬁned

on topological spaces, as summarized in the sequel.

Related works. Despite its recent birth, many contributions

have been made to the ﬁeld of simplicial deep learning. In

[7], the authors introduced a basic simplicial neural network

(SNN) architecture that performs convolution exploiting high-

order Laplacians without independently exploiting upper and

lower neighbourhoods. In [8], message passing neural net-

works (MPNNs) are adapted to simplicial complexes, with the

aggregation and updating functions taking into account data de-

ﬁned on adjacent simplices, enabling message exchange even

among signals of different orders. The work in [9] exploits

the simplicial ﬁlters introduced in [10] to design a ﬂexible and

low-complexity simplicial convolutional networks (SCNs) with

spectral interpretability. Finally, in [11, 12], simplicial atten-

tional architectures are introduced.

Motivated by the fact that, both in convolutional neural net-

works (CNNs) and in GNNs, the introduction of pooling lay-

ers was proved to be useful for reducing the number of model

parameters while improving the learning performance, in this

work we aim to endow SCNs with pooling strategies. How-

ever, while for CNNs the pooling operation relies on aggrega-

tion based on the natural local neighbourhood provided by the

regular grid domain, even on simpler graph domains the deﬁni-

tion of local patches is not straightforward. Early works tried to

overcome this issue by using graph clustering algorithms such

as GraClus [13] or spectral methods [14] to produce a node

assignment that generalizes the notion of locality present in

regular domains. The most recent trends are instead focused

on differentiable learnable operators that can learn a node as-

signment [15], or simply keep some nodes while discarding the

others [16, 17]. Other works [18] discuss the class of global

pooling methods that reduce the graph to a single vector, ignor-

ing topological information. To the best of our knowledge, no

previous works tackled the problem of pooling for SCNs.

Contribution. The goal of this work is to introduce pooling

strategies for SCNs. Taking inspiration from the select-reduce-

connect (SRC) paradigm [19], we introduce a general simpli-

cial pooling layer that comprises three steps: i) a local aggre-

gation step responsible for providing a meaningful summary of

the input signals; ii) a selection step responsible for selecting

a proper subset of simplices; ﬁnally, iii) a reduction step that

downsamples the input complex and the aggregated signals of

step i) based on the simplices selected in step ii). By tailoring

steps ii) and iii), we introduce four different simplicial pool-

ing layers that generalize the well-known graph pooling strate-

gies. Also, we exploit the proposed simplicial pooling lay-

ers in a jumping knowledge (JK) hierarchical architecture [20],

which aggregates the intermediate embeddings produced by the

simplicial pooling layers to produce the ﬁnal output. Finally,

we assess the performance of the proposed methods on real-

world graph and trajectory classiﬁcation tasks, showing favor-

able comparisons with respect to other techniques in terms of

performance and robustness to compression.

2. BACKGROUND

Simplicial complex and signals. Given a ﬁnite set of vertices

V, a k-simplex Hkis a subset of Vwith cardinality k+ 1. A

face of Hkis a subset with cardinality kand thus a k-simplex

arXiv:2210.05490v1 [eess.SP] 11 Oct 2022

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POOLINGSTRATEGIESFORSIMPLICIALCONVOLUTIONALNETWORKSDomenicoMattiaCinque,ClaudioBattiloro,PaoloDiLorenzoDIETDepartment,SapienzaUniversityofRome,ViaEudossiana18,00184,Rome,ItalyE-mail:domenico.cinque98@gmail.com,fclaudio.battiloro,paolo.dilorenzog@uniroma1.itABSTRACTThegoalofthispaperistointroducepool...

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POOLING STRATEGIES FOR SIMPLICIAL CONVOLUTIONAL NETWORKS Domenico Mattia Cinque Claudio Battiloro Paolo Di Lorenzo DIET Department Sapienza University of Rome Via Eudossiana 18 00184 Rome Italy

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