In this section, we provide a brief introduction to the concepts from Group Theory which we need in our derivations. A group is a pair (G,)containing a set Gand a binary operation: G G! G,(h,g) 7! h g which satisfies the group axioms: Associativity: 8a,b,c 2 Ga (b c)=( a b) c Identity: 9e 2 G: 8g 2 Gg e = e g = g Inverse: 8g 2 G 9g 1 2 G: g g 1 = g 1 g = e The operation is the group law of G. The inverse elements g 1 of an element g, and the identity element e are unique. In addition, if the group law is also commutative, the group G is an abelian group. To simplify the notation, we commonly write ab instead of a b. It is also common to denote the group (G,) just with the name of its underlying set G. The order of a group G is the cardinality of its set and is indicated by |G|. A group G is finite when |G|2 N, i.e., when it has a finite number of elements. A compact group is a group that is also a compact topological space with continuous group operation. Given a group G, its action on a set X is a map . A simple example of group action is the group law itself: G G! Gwhich defines an action of G on its own elements (X = G). Another important action is the one defined on signals overs the group G. Given a signal x: G! R, the action of an element g 2 G maps x 7! g.x, [g.x](h):= x(g 1h).

A.1 Cellular WLResults In this section, we assume basic familiarity with the WL test and its higher-order variants. For an introduction to these topics, we refer the reader to the survey of Sato [62]. We begin by introducing a few useful concepts. A cellular colouring is a map c that maps a cell complex X and one of its cells to a colour from a fixed colour palette. Let X,Y be two regular cell complexes and c a cellular colouring. We say that X,Y are c-similar, denoted by cX = cY, if the number of cells in X coloured with a given colour equals the number of cells in Y with the same colour. Otherwise, we have cX 6= cY . We emphasise that in this paper we are interested only in colourings c with the property that any two isomorphic cell complexes are c-similar. A cellular colouring c refines a cellular colouring d, denoted by c v d, if for all cell complexes X and Y and all 2 PX and 2 PY, cX = cY implies dX = dY . Additionally, if d v c, we say the two colourings are equivalent and we represent it by c d. We state the following result from Bodnar et al. [8] about simplicial colourings, which we translate here directly to cell complexes. The proof is however, identical, and we refer the reader to their work for that. Let X,Y be any regular cellular complexes with A PX and B PY . Consider two cellular colourings c,d such that c v d.

Graph Neural Networks (GNNs) are limited in their expressive power, struggle with long-range interactions and lack a principled way to model higher-order structures. These problems can be attributed to the strong coupling between the computational graph and the input graph structure. The recently proposed Message Passing Simplicial Networks naturally decouple these elements by performing message passing on the clique complex of the graph. Nevertheless, these models can be severely constrained by the rigid combinatorial structure of Simplicial Complexes (SCs). In this work, we extend recent theoretical results on SCs to regular Cell Complexes, topological objects that flexibly subsume SCs and graphs.

Itconsists4-regular 10 different passingapproaches Table 1: Classification Method Mean MP-GNNs 0.000 10.000 RingGNN 0.000 10.000 3WLGNN 10.916 30.000

Abstract: We are perceiving and communicating with the world in a multisensory manner, where different information sources are sophisticatedly processed and interpreted by separate parts of the human brain to constitute a complex, yet harmonious and unified sensing system. To endow the machines with true intelligence, the multimodal machine learning that incorporates data from various modalities has become an increasingly popular research area with emerging technical advances in recent years. In this paper, we present a survey on multimodal machine learning from a novel perspective considering not only the purely technical aspects but also the nature of different data modalities. We analyze the commonness and uniqueness of each data format ranging from vision, audio, text and others, and then present the technical development categorized by the combination of Vision X, where the vision data play a fundamental role in most multimodal learning works. We investigate the existing literature on multimodal learning from both the representation learning and downstream application levels, and provide an additional comparison in the light of their technical connections with the data nature, e.g., the semantic consistency between image objects and textual descriptions, or the rhythm correspondence between video dance moves and musical beats.

The operating room (OR) is a dynamic and complex environment consisting of a multidisciplinary team working together in a high take environment to provide safe and efficient patient care. Additionally, surgeons are frequently exposed to multiple psycho-organisational stressors that may cause negative repercussions on their immediate technical performance and long-term health. Many factors can therefore contribute to increasing the Cognitive Workload (CWL) such as temporal pressures, unfamiliar anatomy or distractions in the OR. In this paper, a cascade of two machine learning approaches is suggested for the multimodal recognition of CWL in four different surgical task conditions. Firstly, a model based on the concept of transfer learning is used to identify if a surgeon is experiencing any CWL. Secondly, a Convolutional Neural Network (CNN) uses this information to identify different degrees of CWL associated to each surgical task. The suggested multimodal approach considers adjacent signals from electroencephalogram (EEG), functional near-infrared spectroscopy (fNIRS) and eye pupil diameter. The concatenation of signals allows complex correlations in terms of time (temporal) and channel location (spatial). Data collection was performed by a Multi-sensing AI Environment for Surgical Task & Role Optimisation platform (MAESTRO) developed at the Hamlyn Centre, Imperial College London. To compare the performance of the proposed methodology, a number of state-of-art machine learning techniques have been implemented. The tests show that the proposed model has a precision of 93%.

Graph Neural Networks (GNNs) are limited in their expressive power, struggle with long-range interactions and lack a principled way to model higher-order structures. These problems can be attributed to the strong coupling between the computational graph and the input graph structure. The recently proposed Message Passing Simplicial Networks naturally decouple these elements by performing message passing on the clique complex of the graph. Nevertheless, these models are severely constrained by the rigid combinatorial structure of Simplicial Complexes (SCs). In this work, we extend recent theoretical results on SCs to regular Cell Complexes, topological objects that flexibly subsume SCs and graphs. We show that this generalisation provides a powerful set of graph ``lifting'' transformations, each leading to a unique hierarchical message passing procedure. The resulting methods, which we collectively call CW Networks (CWNs), are strictly more powerful than the WL test and, in certain cases, not less powerful than the 3-WL test. In particular, we demonstrate the effectiveness of one such scheme, based on rings, when applied to molecular graph problems. The proposed architecture benefits from provably larger expressivity than commonly used GNNs, principled modelling of higher-order signals and from compressing the distances between nodes. We demonstrate that our model achieves state-of-the-art results on a variety of molecular datasets.

We initiate the work towards a comprehensive picture of the smoothed satisfaction of voting axioms, to provide a finer and more realistic foundation for comparing voting rules. We adopt the smoothed social choice framework, where an adversary chooses arbitrarily correlated "ground truth" preferences for the agents, on top of which random noises are added. We focus on characterizing the smoothed satisfaction of two well-studied voting axioms: Condorcet criterion and participation. We prove that for any fixed number of alternatives, when the number of voters $n$ is sufficiently large, the smoothed satisfaction of the Condorcet criterion under a wide range of voting rules is $1$, $1-\exp(-\Theta(n))$, $\Theta(n^{-0.5})$, $ \exp(-\Theta(n))$, or being $\Theta(1)$ and $1-\Theta(1)$ at the same time; and the smoothed satisfaction of participation is $1-\Theta(n^{-0.5})$. Our results address open questions by Berg and Lepelley in 1994 for these rules, and also confirm the following high-level message: the Condorcet criterion is a bigger concern than participation under realistic models.