Compact Matrix Quantum Group Equivariant Neural Networks
Authors: Edward Pearce-Crump
Venue: ICML 2025 poster (2025)
Keywords: deep learning theory
equivariant neural networks
quantum groups
Primary area: theory->deep_learning

Abstract: Group equivariant neural networks have proven effective in modelling a wide range of tasks where the data lives in a classical geometric space and exhibits well-defined group symmetries. However, these networks are not suitable for learning from data that lives in a non-commutative geometry, described formally by non-commutative $\mathcal{C}^{\ast}$-algebras, since the $\mathcal{C}^{\ast}$-algebra of continuous functions on a compact matrix group is commutative. To address this limitation, we derive the existence of a new type of equivariant neural network, called compact matrix quantum group equivariant neural networks, which encode symmetries that are described by compact matrix quantum groups. We characterise the weight matrices that appear in these neural networks for the easy compact matrix quantum groups, which are defined by set partitions. As a result, we obtain new characterisations of equivariant weight matrices for some compact matrix groups that have not appeared previously in the machine learning literature.

Compact Matrix Quantum Group Equivariant Neural Networks
Edward Pearce-Crump'
Abstract
Maron et al., 2019a; Finzi al., 2021; Villar al., 2021; Pearce-Crump, 2023a;b; 2024; Godfrey et al., 2023). These networks have proven to perform well across a wide range of tasks, including, but not restricted to, particle physics (Bogatskiy et al., 2020), natural language processing (Gor- don et al., 2020), the generation of molecules (Satorras et al., 2021), and computer vision (Chatzipantazis et al., 2023).
Group equivariant neural networks have proven effective in modelling a wide range of tasks where the data lives in a classical geometric space and exhibits well-defined group symmetries. How- ever, these networks are not suitable for learn- ing from data that lives a non-commutative ge- ometry, described formally by non-commutative C* alggeebrs,, since the C' -alalgebra of continuous functions on a compact matrix group is commuta- tive. To address this limitation, we derive the exis- tence of new type of equivariant neural network, called compact matrix quantum group equivariant neural networks, which encode symmetries that are described by compact matrix quantum groups. We characterise the weight matrices that appear in these neural networks for the easy compact matrix quantum groups, which are defined by set parti- tions. As a result, we obtain new characterisations of cquivariant weight matrices for some compact matrix groups that have not appeared previously in the machine learning literature.
However, when we consider tasks that involve learning from data that lives in non-commutative geometry, such as learning properties of random variables in free proba- bility theory ( (Voiculescu, 1985; Voiculescu et al.. 1992), physical phenomena in particle physics (Bhowmick et al., 2014; van Suijlekom, 2014), and properties of exactly solv- able models in statistical mechanics (Podlés, 1995), group equivariant neural network no longer a valid choice of model. This is because conmutamuteeeve geometry, de- veloped by Connes (1995), is formally described using non- commutative C* -'laleebras, which generalize commutative algebras C(X) of continuous functions on compact Haus- dorff spaces X, and one can show that the C -algeba C(G) of continuous functions on compact matrix group G is indeed commutative (Appendix B.4). In particular, in non- commutative geometry, the classical notion of space is replaced by an algebraic structure, thus making traditional group-based symmetries inadequate (Budzyński & Kon- dracki, 1995; Majid, 2000; Sitarz, 2013). Instead, a more general notion of symmetry is required. Quantum groups, which have been developed by number of authors (Jimbo, 1985; Drinfeld, 1986; Woronowiez, 1987; 1991; 1998), are important class of mathematical objects for describing these symmetries.
1. Introduction
Spaces in classical (commutative) geometry are given by sets of points with additional structure, and their symme- tries are formally described by groups. For example, the compact matrix group SO(3) is the group of all rotations of 3 that preserve the origin. A class of neural networks called group equivariant neural networks was conceived to take advantage of group symmetries that are inherent in data by encoding them as an inductive bias in their ar- chitectures. Many group equivariant neural networks have been designed using representations of groups as their layer spaces, and characterisations of the weight matrices that appear in them have been found for number of impor- tant groups (Zaheer et al., 2017; Ravanbakhsh et al., 2017;
Given that there exist symmetries that are not captured by groups, this implies that there is a need for a new type of equivariant neural network. this paper, we focus on quantum groups that were constructed using a topological approach by Woronowiez (1987). Woronowicz generalised compact matrix groups create class of quantum groups called compact matrix quantum groups. We describe and motivate this construction in Section 3, and provide a com- prehensive background on the prerequisite mathematical material, namely, general topology, C* -llgebras, and opera- tor theory, in the Appendix.
Department of Computing, Imperial College London, United Kingdom. Correspondence to: Edward Pearce-Crump <ep1011 @ic.ac.uk>.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
The main contributions of our work are as follows. We

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Compact Matrix Quantum Group Equivariant Neural Networks
broaden the scope of equívariant neural networks beyond classical symmetries by defining new class of neural net- works that encode, an inductive bias, symmetries de- scribed by compact matrix quantum groups. We call these neural networks compact matrix quantum group equivariant neural networks. In fact, we mathematically derive their existence using Woronowicz's version of Tannaka-Krein duality for compact matrix quantum groups (Woronowiez, 1988), which is introduced in Section 5. After proving that these neural networks exist, we characterise their weight matrices for a large class of compact matrix quantum groups known as easy. We focus on casy compact matrix quantum groups not only because they have been studied extensively in the literature (Banica & Speicher, 2009; Banica et al., 2010; Weber, 2013; Freslon & Weber, 2016; Raum & We- ber, 2016; Tarrago & Weber, 2016; Weber, 2017: Tarrago
ing the range of equivariant models. Consequently, their models need to be trained with a custom gradient descent method, whereas our networks can still be trained with stan- dard backpropagation as both the layer spaces and weight matrices remain complex-valued. Crucially, while both ap- proaches can be used to construct group equivariant neural networks, their models do not encode quantum group sym- metry, which is central to our framework. Other works, such as those by Ruhe et al. (2023a;b), use Clifford algebras to develop equivariant neural networks that encode symmetries related to rotations and reflections. However, their approach seems loosely related to ours, in that while both involve non- commutative algebras, their networks capture symmetries in classical geometries whereas ours looks at symmetries in non-commutative geometries. Finally, our approach differs from that of Hoffmann et al. (2020), where they look to learn the algebraic structure from their network. By con- trast, we show in Theorem 7.10 that the algebraic structure for easy compact matrix quantum group equivariant neural networks can be derived mathematically from the weight matrices themselves.
& Weber, 2018; Gromada, 2020; Freslon, 2023), but also because the way in which they are defined, namely, by set partitions, makes it possible to obtain a full characterisa- tion of the equivariant weight matrices for these compact matrix quantum groups. We note that our contributions are primarily theoretical in nature: to demonstrate the practical potential of these neural networks, further work is needed to extend the characterisation that have found for the easy compact matrix quantum groups to the equivariant non-lineat layers so that they can be implemented.
3. Non-Commueeiive Geometry
In this section, we first introduce non-commutative geom- etry and then motivate the existence of compact matrix quantum groups by studying certain properties of compact matrix groups G(n) S GL(m).
We note as a corollory of our work that the new class of neural networks generalises and encompasses all neural net- works that are equivariant to compact matrix groups, since we show, in Section 3, that compact matrix groups are, and can be expressed as, compact matrix quantum groups. As result, we recover characterisations of the weight matrices that have appeared in Ravanbakhsh et al. (2017); Maron et al. (2019a); Pearce-Crump (2023a; 2024); Godfrey et al. (2023), and obtain characterisations of the equivariant weight matri- ces for three compact matrix groups that have not appeared in the machine learning literature before, namely the hype- roctahedral group H,, the bistochastic group B,, and the unitary group (n).
Two ideas were important in the development of non- commutative geometry. The first was the paradigm shift from thinking in terms of points in space (geometry) to considering functions on spaces (algebra) as the primary method for solving problems in geometry and topology (Budzyáski & Kondracki, 1995; Majid, 2000; Sitarz, 2013). Indeed, whilst one can show (Appendix B.4) that the alge- bra C(X) of continuous, complex
functions on a compact Hausdorff space X is a commutative, unital, C+- algebra, famous theorem in C* allebbra theory by Gelfand & Naimark (1943) shows that every commutative, unital, C* -llaeaabra is, in fact, an algebra of continuous functions C(X) for some compact Hausdorff space X. In this way the geometric notion of a (compact Hausdorff) space X is dualised to the algebra of continuous functions on X.
A word on notation: we use [n] to refer to the set of elements f1...., n), and, for any set A, we denote the set of n X n matrices with entries in A by (A).
2. Related Work
The second important development was the abstraction of spaces entirely by allowing unital C*-aageeras to be non- commutative (Jimbo, 1985; Drinfeld, 1986; Woronowicz, 1987). In doing so, this created a notion of geometry through these algebras, namely by treating non-commutative C*- algebras as if they were algebras of continuous functions on -xintaxist non-commutative spaces, as an abstract ex- tension of the theorem by Gelfand & Naimark (1943). Con- sequently, non-commutative C* algebras the foundation of non-commutative geometry, and, under this framework.
A few prior related works (Hashimoto et al., 2022; Hataya & Hashimoto, 2023; Hashimoto et al., 2024) use -llgebaas to design new machine learning architectures, though their approach is motivated by different objectives. While they generalize complex-vaie vaauued layer spaces and parameters to function-valued ones, our approach extends the concept of group symmetry in group equivariant neural networks to certain type of quantum group symmetry, thus expand-
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Compact Matrix Quantum Group Equivariant Neural Networks
the notion of space (of points) is no longer meaningful.
Note that for compact matrix group G(n) C GL(n), points 1 and 2 of Definition 3.1 make sense. Indeed, if we define functions U. G(n) + C for i, E [n] such that (وا 9t.j, then, by the Woone-Weirsssrass Theo- rem (Appendix B.6) and the compactness of G(z), the "ig generate (r)().. Also, the matrices u == (u:5) and u are invertible since u (4) = u(g-t), Furthermore, by defining the comultiplication map A for C(G(n)) by
But with these developments, new notion of symmetry for non-commutative C" -lggebrass was needed, since, for any compact matrix group Gi(n) S GL(m), one can show (Appendix B.4) that the algebra ((G(n)) is a commutative, unital C -llgebra.
However, we can study the properties of C(G(n)) to mo- tivate the definition of compact matrix quantum group, as given by Woronowicz (1987), to describe this new type of symmetry. Firstly, G(n) is not just compact Haus- dorff' space, it is a group, and so it comes with group law G(n) x Ginn) - G(n) where (g1. + 9192- We can express this group law the language of C* -llalgebras as fol- lows: define A C(G(n)) + C(G(n) x G(n)) under the mapping/ f + ((g):9)) + f(g192)). Furthermore, since C(G(n) x G(n) is -isomorphic to ((G(n)) @ C(G(n)) under ((91 + (11)))((02) = fi @ f2, we see that : C(G(m)) + ((G(n) @ C(G(n))
) h) = (g.
we obtain the following result:
Proposition 3.3 (Appendix: Section C). Compact matrix groups G(n) S GL(n) form a special class of compact matrix quantum groups.
Moreover, Woronowicz (1987, Theorem 1.5) also showed, using the theorem by Gelfand & Naimark (1943), that if (A, u) is a commutative compact matrix quantum group for some positive integer n, then A z C(G(n)) for some compact matrix group G(n)) C GL(m). Combining this with Proposition 3.3, we obtain the Fundamental Theorem of Compact Matrix Quantum Groups:
(1)
is the expression of the group law of G(n) in the language of C* -lagebbba...
We generalise this construction by replacing C(G(n)) by a (potentially non-commutative) -llalgebra A, to define compact matrix quantum group in the style of Woronowiez (1987).
Theorem 3.4 (Fundamental Theorem of Compact Matrix Quantum Groups). Lei (A, u) be a compact matrix quanium group some n E N. Then A is commutative if and only if A = C(G(n)) for some compact matrix group G(n) S GL(n).
Definition 3.1. Let A be unital C*+ algebba, and let u.j E A, for all i,j E [n], for some positive integer n. Let u be the n x n matrix whose (inj)-entry is ui.j. that is, u is an element of the unital C* -l-aagebra (A). The pair (A, u) is said to be a compact matrix quantum group if
Hence, the Fundamental Theorem shows that if (G(n), u) is non-commutative, then we obtain "true" compact matrix quantum groups. Bearing this theorem in mind, convention dictates that we denote A by C(G(m)), even if A is non- commutative, and so we often refer to G(n), or sometimes the pair (G(n), u), as the compact matrix quantum group.
1. A is the universal C* -alalgebra C* (u4,,, 1 S i,J $ n).
2. u and u (uy,)) are invertible matrices, and
We now provide an example of a "true" compact matrix quantum group to show that they do indeed exist.
3. the comultiplication map A : + A 8min A defined by
Example 3.5. Recall that each element of the symmetric group Sn, considered as a subgroup of GL(n), can be thought of as an n x n matrix having exactly one entry of in each row and in each column, with all other en- tries being O. Writing the elements of g E Sn as giy. this condition can be expressed as the following relations:
A(us.)) = Σ Wiik 2 WA.j
(2)
is a t-homomorphism.
Remark 3.2. In motivating Definition 3.1. we did not spec- ify the tensor product in ((G(n)) @ C(G(n)) for con- structing A because C(G(m)) is a nuclear C*-algebra (Courtney & Gillaspy, 2023, Proposition 10.10), implying that all norms that complete the algebraic tensor product C(G(n)) O C(G(n)) into -"-algebra give the same one. However, in general, the algebraic tensor product of two C"- algebras, A @ B. can be completed in many ways to form a C algebra. One method is to complete with respect to the minimal norm I| giving the C* -algebr A &min B. We review these concepts in Section B of the Appendix.
Jh, = =g. = -i
9o.t =
Ih. = 1
(4)
1=1
1=1
and
9i.ghh. 9k.19i.
(5)
By the universal property of universal C* C*lalgebras (Ap- pendix B.24), we have that C(S.) is the universal C*. algebra C* R) with generators E given by {u., 1 S
i S n) and relations R given in (4) and (5) (replacing gi.
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Compact Matrix Quantum Group Equivariant Neural Networks
with Wizz). Wang (1998) defined the symmetric quantum group as compact matrix quantum group S by liberating C(S)) of its commutativity relations to obtain C(S)) as the universal C* -llgebaa C* (E R) where E is the same as for Sn but now the set of relations R is given by (4) only. As a result, the symmetric quantum group St is our first example of a compact matrix quantum group that is not a group.
complex conjugation of representations, starting with the following definition.
Definition 4.4. Let v E MM(C((((n)) and wo e Mp Mp(C(G(n))). Then the tensor product v @ w e Mmp(C(((n)) (z)) simply the Kronecker product of ma- trices. and the complex conjugate is = (vi,) Mm (C((
Proposition 4.5 (Appendix: Section C). If w are repre- sentations of a compact matrix quantim group (G(n), u), then both the tensor product and complex conjugate are also representations of ((z).
4. Representation Theory of Compact Matrix Quantum Groups
Analogous to compact matrix groups, compact matrix quan- tum groups have representation theory. We now moti- vate the definition of a representation of a compact matrix quantum group. Recall that for a compact matrix group G(n)) an m-dimensional representation is a choice of m- dimensional vector space V over C and group homomor- phism PV : G(n) + GLVV.. By choosing a basis of V, we can replace (V) by GL(m). The representation pV can itself be represented by functions Vi, C(G(n)) for
The following lemma will be useful when it comes to con- sidering the fundamental representation of a compact matrix quantum group.
Lemma 4.6 (Appendix: Section C). Let (G(n), u) be compact matrix quantum group. Then u. T is unitary if and only if u is unitary.
We can also define the analogous concept of equivariance for compact matrix quantum groups.
1 S i.j S that map g E G(n) to the (i, entry of Pv (g), since PV is continuous by the compactness of G(m). We claim the following result.
Definition 4.7. Let v E Mm(C(G((n)) and w E (n(C(G(n)) be representations of a compact matrix quan- tum group G(n). A map & : C - C is said to be G(n)-qquivariant (also known as an intertwiner) if Φbu = wv. The set of all such linear maps is denoted by Homa(n) w)) and it is, in fact, vector space.
Lemma 4.1. If A is the comultiplication on C(G(n)) de- fined in (54), then the homomorphism property of the repre- sentation satisfies
Remark 4.8. It is worth noting that the expression dv = we given in Definition 4.7 has a different interpretation relative to its sister definition for group equivariance. The represen- tation V E Mn (C(G(()) coucts on the space Cm in the following sense: if x E C then vz E C & ((G())). Hence can be thought of as a map Cm + Cm & (()( or as an element of Mm (C) @ C(G(n)). In fact, any rep- resentation on a vector space V can be described by a map V + V & C(G(n)). Consequently, & E Homa (v, w) if and only if satisfies the commutative square:
(e, = (Dor
Di. viu @ ukis h)
(6)
Proof. Indeed, we have that A1si h) is equal to
(ig(gh) = (ovigh))is =
Σ
Σ (1)(()) =
Σ 0i.k @ Tkg
h0)
LI
C" 2 C(G(n)
As result, we have the following definition.
Definition 4.2. Let (G(n), u) be compact matrix quantum group. Then an m-dimensional representation of G(n) is an element e E Mm (C(G())())) such that (2) holds. If a representation v of G(n) has a matrix inverse, we say that e is a non-degenerate representation, and if o is unitary, that is, vo- 1 2 v, then we call v a unitary representation. Remark 4.3. The matrix u given in the definition of com- pact matrix quantum group (G(n), u) is an n-dimensional representation called the fundamental representation of G'n). and by Definition 3. it is non-degenerate represen- tation of G(n).
CP CP 8 C(G(n))
where the right-hand arrow maps x & f ox + S. Definition 4.9. We say that two representations v Mm, (C(G(()) and w E Mc.(C(((n)) are equivalent if there is an invertible linear map 6 C. + C that intertwines v and w.
One can show (Gromada, 2020, Proposition 2.2.3) that ev- ery non-degenerate representation is equivalent to a unitary representation. Hence, going forward, we always choose the fundamental representation u of a compact matrix quantum group and its transpose to be unitary. By Lemma 4.6, we therefore have that both u and are unitary.
Similar to groups, we can also define the tensor product and
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Compact Matrix Quantum Group Equivariant Neural Networks
5. Kannaka-Krein Duality for Two-Coloured Representation Categories
3. If de E C(wk: wi), then o' C(wi, wx).
4. For every word wo (having some length k)), we have 1 E C(w, w). and
Woronowicz's version of Kennaka-Krren duality is an impor- tant result in the theory of compact matrix quantum groups. Informally, the duality says that we can construct a com- pact matrix quantum group just by knowing its fundamental representation category. We will use this duality to derive compact matrix quantum group equivariant neural networks in Section 6. We begin with some definitions.
5. The map R : 1:1 H E e, @ &e is in both C(Ø, .o) and C(2, mo).
Definition 5.6. A one-coloured representation category is a two-coloured representation category where u' = u.. Hence, the words correspond precisely to natural numbers because, for any natural number k, there is only one word of length k in this case.
Definition 5.1. Consider the two-coloured set (o. con- sisting of a white point and black point. For any non- negative integer k, we can construct a word w of length k as a string of & colours from for o.. If k = O, we define 2 to be the empty word. The concatenation of two words, wi, w2, is the concatenation of their two strings and is writ- words, + by first defining it on the individual colours by 5 = == o, and then applying it element wise to the word w. If w is a word, then its involution w" is the word read backwards together with its colours inverted.
The following two theorems describe the relationship be- tween two-coloured representation categories and compact matrix quantum groups.
ten as wi W22 We define homomorphism
Theorem 5.7 (Appendix: Section C). If (G(n), u) is a compact matrix quantum group, then FundRepa(s) is a two-coloured representation category.
Duality, (Woronowiez, 1988).. Let C be a two-coloured representa- tion category. Then there exists unique compact matrix quantum group (G(n), u) such that (undRePG(()) = C.
Theorem 5.8 Cerronowwcz-Tannnke-a
Example 5.2. If Wk = and wy =
is a word of length is word of length 5, then We wy and w
Remark 5.9 (Appendix C.1). There is a second part to The- orem 5.8, which states how to construct the unique compact matrix quantum group (G(n), u) from C. We provide this part in the Appendix for brevity.
We can use words of colours create tensor products of the fundamental representation of compact matrix quantum group as follows.
Definition 5.3. Let (G(n), u) be a compact matrix quantum group. Let u² == u and u² == ū. Then, for any word w formed from the two-coloured set {o, ) we define Sw to be the corresponding tensor product of representations u and u..
6. Compact Matrix Quantum Group Equivariant Neural Networks
In this section we present the major contribution of this paper: we derive a new class of neural networks which we term compact matrix quantum group equivariant neural net- works using Woronowicz's s formulation of Tannaka-Krein duality. In the following definition, when referring to word of length ki, we write wy instead of wh, to suppress multiple subscripts.
Moreover, if Wk and wi are words of lengths k, l re- spectively, then we define FundRepa (wk, wy) to be Homo (ww (.)), that is, the vector space
(Ø: (C") + (Cm)31 pn Owr = u Sur x} (8)
Example 5.4. If w is the word o o, then u 0w u 2 i 7 a 2 u.
Definition 6.1. Let (G(n), u) be a compact matrix quantum group with fundamental representation u E (,,(C(((n)) A compact matrix quantum group equivariant neural network for G(n) is function fNN that is composition of some I. layer functions
We can also construct so-called representation categories using words of colours as follows.
Definition 5.5. two-coloured representation category C is collection C(w,, wi). where Wk, wy are two words constructed from {o, } of lengths k, I respectively, that are subspaces of the set of all linear maps (Cn)ak + (C")@l
fNN = fL f..... fo.....f
such that the 1hh layer function is equivariant to G(n) and is function between representation spaces of G(n) under tensor products and complex conjugations of u
and which satisfy the following axioms.
1. If of E C(wk, wz). &2 E C(wy, Wm). then Φ1 @ t2 E C(wk- - Wy: wi wm).
fi : ((C (CC))Rh-, , sw--1) ((C)) @h u @w)
(10)
that is itself composition
2. If di E Clwk, wi), 02 E C(wi, wm), then d2 o 01 E C(wn; W..).
fi = 0
(11)
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Compact Matrix Quantum Group Equivariant Neural Networks
of learnable, linear function in MundReeptss wi)
correspond to the connected components of the set partition
Φ1 : ((C")8ki-,, 1)) + ((C")ak , )
(12)
Definition 7.2. A (two-coloured) (wx, wi ma)-ptrtiiion di- agram is a (k, O)partition diagram together with two- coloured words wkk Wy of lengths k., I respectively such that the vertices on the top row have the same colours as the word wy, from left to right, and the vertices on the bottom row have the same colours as the word wi, from left right. We use the same notation dæ to refer to the two-coloured version, since wμ. and wy will be known in making reference to the diagram. We define the two-coloured partition vec- tor space P (n) to be the C-linear span of the set of all (wk, wi) ( -partition diagrams.
that is equivariant to G(n), together with a fixed, non-linear activation function
oy ((C*)hh ,ww) @w)) + ((Cr)okk u ewi)
(13)
that is also equivariant to G(n). By choosing the standard basis of C", the learnable, linear functions é that are equiv- ariant to G(n) become matrices that have weights that can be learned. As usual, we call these matrices (equivariant) weight matrices.
Remark 6.2. A compact matrix quantum group equivari- ant neural network is well-defined since the fundamental representation category (undRepa(s) of a compact matrix quantum group (G))), u) a two-coloured representation category by Theorem 5.7.
Example 7.3. Suppose that i is the set partition of [5 66 6]
given by
1,, 2,3
4,8
5, 9
10
11}
(14)
Then
We immediately obtain the following corollary for compact matrix groups G(n) S GL(m).
(15)
Corollary 6.3. Let SNN be a compact matrix quantum group equivariant neural network for a compact matrix group (G(n), u). If all of the words that are used in the network only consist of the white point o, then fNN is, in fact, a compact matrix group equivariant neural network for G(n). Moreover, every compact matrix group equiv- ariant neural network is a compact matrix quantum group equivariant neural network.
0
10
is the two-coloured (wk, wt) partitiio diagram d₂ where Wy is the word .ooo and wy is the word
We can now define a category for two-coloured partition diagrams, follows:
Definition 7.4. P(n) is the category whose objects are the two-coloured words and, for any pair of objects wh and W1, the morphism space Homp( (wy, wy) is defined to be Pwg (n).
7. Easy Compact Matrix Quantum Groups
Having just derived the existence of compact matrix quan- tum group equivariant neural network, we show, in this sec- tion, that we can characterise the weight matrices that appear in these neural networks for the so-called "casy" compact matrix quantum groups. A compact matrix quantum group said be easy if its two-coloured representation category is determined under a functor by so-called two-coloured category of partitions. We first explore two-coloured cat- egories of partitions, then define the functor, and finally construct both the easy compact matrix quantum group and characterise its equivariant weight matrices.
We provide definitions of the vertical, horizontal and involu- tion operations on morphisms in Section C of the Appendix. The unit object is the empty word Ø.
One can show the following result. Proposition 7.5 (Appendix: Section C). P(n)) is a strict C-linear monoidal category.
We achieve our first goal with the following definition. Definition 7.6. A two-coloured category of partitions ((n)) is any subcategory of P(n) such that
Our first goal is to define a two-coloured category of parti- tions. We need a few preliminary definitions.
1. if uk, wi are objects in K(m),, then (on wx) is a subspace of Homp(r) (wy, wr).
Definition 7.1. For all non negative integers I and k. a set partition ग of [1 + k) is a partition of the set + k] into disjoint union of subsets, each of which is called block. A (one-coloured) (k:, -parrtiin diagram dx represents set partition π as a diagram, where we place / white nodes on the top row, labelled left to right by 1,...,7, 1, where we place l white nodes on the bottom row, labelled left to right by + 1,..., I + k. and the edges between the vertices
2. the identity partition diagram is element of (omk(n)(o,0))
3. the top-row pair partition diagram corresponding to the set partition {1, 2} of {1,2} superimposed, on the one hand, with the word oe, and on the other, with the word oo, is in Hom (Ø, oo) and Homx .o), respectively, and
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Compact Matrix Quantum Group Equivariant Neural Networks
4. the morphism spaces are closed under the vertical, hor- izontal and involution operations.
Consequently, we obtain the following key result, which makes it possible for construct two-coloured rep- resentation category from any two-coloured category of partitions.
By Proposition 7.5, it is immediate that a two-coloured category of partitions (C(n) is a strict C-linear monoidal category.
Theorem 7.9. For any two-coloured category of parti- tions K(n),, can form a category (x(n) whose objects are the same as those in K(n)), and, for any pair of ob- jects wk and wy, the morphism space (o) (wk, wi) is the image of Homx(()) (Wk, wi) under d + Dz, that is Homex(n) (wy, wi) is defined to
We now introduce the critical functor for our purposes: we show that this functor takes two-coloured (wk, wy)-partiiion diagrams d, that live in two-coloured category of parti- tions (n) to linear maps du. (C")@x + (C*))r that live in two-coloured representation category. By choosing the standard basis for CC,, we obtain n' x n² matrices instead, which are denoted by D in what follows.
{ dx E Homm()) wi}}
(21)
Then the map a + Dx defines a strict C-linear monoidal functor
Definition 7.7. Suppose that dz a (we, wt)-partition dia- gram, where Wky Wt are two-coloured words of lengths k, respectively. We define D as follows.
θ K(() + Cx(nn)
(22)
such that Cx(n) is a two-coloured representation category. [ppenddix: Section CJ.
Associate the indices i, i2 with the vertices in the top row of d, and J2........ with the vertices in the bottom row of dz. Then, if 5 J)) is defined to be the set of tuples () E [n]*** such that
Hence we can use Woronowicz-Tannann Tannaka- Krein duality given in Theorem 5.8 to not only construct the easy compact matrix quantum group but also to characterise the weight matrices that appear in the neural network that was derived in Definition 6. as follows.
if xr y are in the same block of then i, = in (16)
(where we have momentarily replaced the elements of J by 4+m == Im for all [k]), we define
Theorem 7.10. A two-coloured category of partitions K(n) determines a unique compact matrix quantum group (G(n), u). We call these compact matrix quantum groups easy, and they can be constructed as the universal C*- algebra C* R),. where E {u E [n]} and R = uww = -u D,, for all d, E K(n)(WRk, wi)}- The weight matrix W for the learnable linear function (12) in the neural network that is equivariant to G(n) is given
Dx Σ
8x (I,
(17)
JEIM
where
8n.(1.3) = [I
if e
(18)
0 otherwise
and Er.J is the n' X n matrix having a 1 in the (I position and is ( elsewhere. We extend the definition of the map dz + Dr linearly to P (n).
by
W
Σ
Wr Wr Dr
(23)
x|de ((())(()11))))))
Example 7.8. If dz is the two-coloured (o (......, (...0)- partition diagram given in Example 7.3, then, if n = 3. for example, we see from
8. Examples
2
2
3
We noted in the Introduction that easy compact matrix quan- tum groups have been well studied in the literature and many classes of them have been characterised in full. We intro- duce a few examples, characterise their weight matrices, and refer the reader to the referenced works for more details.
(19)
0
1
2
1
3
2
that the (1, 2, 2, 1,3 1,2,
1, 3,2, 2, 2) -entry of D2 is 1,
Note in the following that it is enough to provide the span- ning set of matrices (21) for easy compact matrix quan- tum group, since, by (23), the weight matrix is a weighted linear combination of these matrices.
whereas, we see that, from
3
(20)
Firstly, we are able to characterise the equivariant weight matrices for some compact matrix groups that have not appeared previously in the machine learning literature by combining Theorem 3.4 and Corollary 6.3. Namely, if we
2 2
3
the (1,1,2,1,3 | 222 of D, is 0.
7

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Compact Matrix Quantum Group Equivariant Neural Networks
Procedure 1: How to Calculate the Weight Matrix of an Equivariant Linear Layer Function from ((C")SR, u ((C*)ot u ( for an Easy Compact Matrix Quantum Group (G(n),
Assume that K(n)) is the two-coloured category of partitions that defines the easy compact matrix quantum group (G(n), u) as in Theorem 7.10.
Perform the following steps:
1. Calculate all of the two-coloured (wk, parriiito diagrams d that live in K(n)). These diagrams form basis of the morphism space Hommy()))) (wC,, wr).
2. Apply the function d, + D= each diagram obtain its associated spanning matrix D
3. Attach a weight W/ E C to each matrix Dx.
4. Finally, calculate Σ Wx Dr to give the overall weight matrix.
consider one-coloured partition categories that contain the swap partition diagram
crossing (k, I) partition diagrams (i.e. no two connected components cross).
Example 8.4. (Wang, 1995) originally discovered the or- thogonal quantum group C(O(n)+) as a universal C*- algebra. Banica & Speicher (2009) showed that the span- ning set of matrices (21) for C(O(n)+) is the image der d= r D2 of all non-crossing (k, -)pattition diagrams whose blocks come in pairs, and also recovered C(O(n)+) itself in this way (using the result of Theorem 7.10).
(24)
then, by Theorem 7.10, we obtain easy compact matrix groups (as the Cd -algebr is commutative) and spanning sets of matrices (21), which allow us to characterise their weight matrices by (23). The groups themselves were char- acterised by Banica Speicher (2009). We highlight two examples here that new, and provide more examples in the Appendix that recover characterisations that were known to the machine learning community before.
Finally, for two-coloured partition categories, the most im- portant examples are the unitary group (n) and the uni- tary quantum group (n) These depend the two- coloured words wy and wy that chosen, but Tarrago & Weber (2016; 2018) showed that, for (n). the spanning set of matrices is the image under dx + D of all (wk, wy). partition diagrams whose blocks come in pairs such that two vertices of block are in the same row, then they have different colours, otherwise they have the same colours, and for U (m)+ (defined by Theorem 7.10), it is the same except all crossing partition diagrams first removed.
Example 8.1. The first is the hyperoctahedral group H,, the symmetry group of the hypercube. Banica Speicher (2009) showed that the spanning set of matrices (21) is the image under d, + Dx of all (parttinn diagrams whose blocks have even size.
Example 8.2. The second is the bistochastic group B., which is the group of orthogonal matrices having 1 in each row and column. Again, Banica & Speicher (2009) showed that the spanning set of matrices (21) is the image under d, + Dj of all (k: ()-partition diagrams whose blocks have size one or two.
9. Conclusion
We have derived the existence of compact matrix quantum group equivariant neural networks for learning from data that lives in non-commutative geometry and has symme- tries that are described by compact matrix quantum groups. We have characterised the linear layers of these neural net- works for the easy compact matrix quantum groups. As future work, we suggest that it would be good to extend this characterisation to the equivariant non-linear layers, which would enable us to demonstrate in practice what these neural networks promise in theory.
Secondly, in the case where the partition categories are one- coloured but they do not contain the swap partition diagram (24), we obtain true compact matrix quantum groups (as the -'-algebra is non-commutative). Again we highlight two examples.
Example 8.3. Wang (1998) discovered the symmetric quan- tum group C(S which we introduced in Section 3. Ban- ica & Speicher (2009) showed that the spanning set of matri- ces (21) for C(S+) ) is the image under dz +> D2 of all non-
8

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Compact Matrix Quantum Group Equivariant Neural Networks
Impact Statement
Brauer, R. On Algebras Which Are Connected with the Semisimple Continuous Groups. Ann. Math., 38:857- 872, 1937. ISSN 0003486X. https:::: /doi. or g/10. 2307/1968843
This paper presents work whose goal to advance the field of Machine Learning. There are many potential societal consequences of our work, none which feel must be specifically highlighted here.
Budzyáski, R. and Kondracki, W. Quantum Spaces: Notes and Comments on Lecture by L. Woronowicz. Ba- nach Center Publications, 34(1):37-455 1995.
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A. General Topology
We introduce some concepts from general topology - also known as point set topology - that are used in the theory of C** -'llagebras (Appendix: Section B). General topology provides an abstract mathematical framework for studying the properties of wide range of mathematical structures. The mathematical structures in general topology are the so-called topological spaces, namely, sets containing family of subsets that satisfy some nice properties, together with the so-called continuous functions, which are the structure- preeturepreeeiiing maps between topological spaces. The family of subsets that is chosen for a set is called a topology, and set can have different topologies assigned to it. In this way the notion of a continuous function depends on the topology that is assigned to the domain and the codomain of the function. Our treatment is only an overview of the main ideas and results: we will mostly state these without proof and so we refer the reader to any standard reference, such as Bourbaki (1995) or Munkres (2018), for more details.
We define the first key concept of general topology. A topology on a set X is family of subsets tx of X such that
1. Both the empty set and X are elements of TX:
2. Any union of elements of rx is an element of TX:
3. Any intersection of finitely many elements of 7x is an element of Tx.
We call a set X together with a topology 7x on X topological space, written as pair Tx). When the topology of a topological space (X, Tx) is understood, we often just refer to the topologiaal space as X. We sometimes call the elements of X points. We call the elements of tx the open sets of (X, Tx). A subset of (X, ) is said to be closed if its complement is an open set. Note that by changing the topology that is associated with a set, the definition of the open sets changes.
We said that the maps of interest between topological spaces are the continuous functions, which are defined as follows. A function f (X,Tx) + (Y, ry) between topological spaces is said to be continuous if f is a function on the underlying sets such that is an open set of Y. then its preimage, f (U), ) an open set of X. One can show using this definition that f is continuous if and only if the preimage of closed set in Y is a closed set in X. The continuous functions are important because they preserve certain properties of topological spaces, many of which will be defined in what follows.
One useful notion that helps to define topologies on sets is that of a base, or sometimes basis. A base B for topological space (X, Tx) is a collection of open sets in rx such that every open set can be written as a union of elements in B. We say that B generates the topology Tx-
An important class of topological spaces are metric spaces, which come with notion of distance between any pair of points in the set. A metric space is set X together with a function dx X x X + R, called a metric, such that the following axioms hold:
1. dx 2 for all y, y E X and dx (x, 0 if and only if = y,
2. dx(x,)) = .) for a,y E X;
3. dx(x))) I dx dx(2,)) v) for all x. 2. X.
metric space often denoted by the pair (X, dx) or simply by X when the metric dx is understood. Some of the most important sets in metric space are the open balls. For any point x in a metric space (X, dx) and for any real number >>00 0, the open ball of radius around z, written (xx, e), is defined to be the set {v E X dx y) < e}. When metric space is given a topology that is induced by the metric, known as the metric topology, then the metric space together with the metric topology becomes a topological space. The metric topology is generated by the base that is given by all of the open balls that are defined by the metric.
A particularly important class of metric spaces are the normed vector spaces, which are commonly refered to simply as normed spaces. To define them, we first need to define the concept of norm on a vector space. A norm on a complex vector space X is function ||+|| X + IR such that
1. ||^||| 2 0 for all x E X and if ||^||| = 0 then æ = 0:
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2. || s [I] || for all X and À C:
3. || + ull - |||||| + llvll for all x, E X;
If x is an algebra, then a norm on X must also satisfy the submultiplicatiiity axiom, namely
4. | || - |||||||||| |vl| for all 2, e
A normed space is a complex vector space (or algebra) X that is equipped with norm ||:|| X -> IR. Normed spaces X become metric spaces under the metric that is induced by the norm, namely
dx(2, v) = || - vll for all 2, E X
(25)
In partieular, one can show that dx satisfies the axioms of metric on X.
One nice property for topological space to have is the Hausdorff property. The Hausdorff property defines notion of separation between points in a topological space that will be particularly useful when we consider limits of sequences below. To define the Hausdorff property, we first need the definition of a neighbourhood. A neighbourhood of a point p in a topological space (X, Tx) is a subset V of X such that V contains an open set U that contains p. that is, p E U C V S X. Note that V does not necessarily need to be an open set itself. We say that a topological space (X, rx) is Hausdorff if, for any two distinct points v. y E X, there exists neighbourhood V of x and neighbourhood W of y such that V and W are disjoint. In particular, one can show that every metric space (X, dx) is Hausdorff.
We now look at the closure of a set in topological space Tx). Every subset V of a topological space (X, rx) has a closure which is denoted by V: it is the intersection of all closed sets in (X, Tx) that contain V. This definition is not particularly helpful, so to make more useful we need the following definition of a limit point. For a subset V of a topological space (X, Tx). a limit point of V is a point æ E X such that every neighbourhood of x contains a point of V that is not x. One can then show that the closure of V is the union of V and the set containing all limit points of V. We say that a subset V of topological space (X, -x) dense in X if its closure is X, that is, V = X. Moreover, one can show that the closure of a closed set is the set itself. Hence if V is dense in X and V is closed, then V = X. All of these definitions apply immediately in the case where X is metric space.
We continue now with metric spaces and consider sequences of points in metric space. A sequence (2x,) in a metric space (X, dx) ) is said to be Cauchy if for all > 0, there exists a positive integer N such that for all m, n 2 N, dx (Im. v,,) < c. A sequence (xn) in a metric space (X, .xx)
) is said to converge to a limit x E X if for all e > O. there exists a positive integer N such that for all n 2 N, dx (r) < e. In partieular, one can show that if a sequence converges space, then the limit must be unique. This a direct consequence of a metric space being Hausdorff. Moreover, continuous functions preserve the convergence of sequences, in that if f (X,dx) dx) (Y,dy) is continuous function between metric spaces and (In) is a sequence that converges to x E X, then the sequence (f(xr)) in Y converges to f(x) E Y. Particularly nice metric spaces are those that are said to be complete: a metric space (X, dx) is complete if every Cauchy sequence in (X, dx) converges. In particular, a normed space that is complete with the metric induced by the norm is known as a Banach algebra, which is the starting point for the material that appears in Section B of the Appendix. One further useful result that a subset of complete metric space complete if and only if is closed.
metric
A notable point is that closed sets have an alternative characterisation when X is a metric space. Firstly, note that if (Ix) is sequence that lives in subset V of X that converges, then its limit z is not necessarily in V (but it is in X). However, one can show that subset V of a metric space (X, dx) ) is closed if and only if every sequence (rx,) in V converges to a limit in V.
We said that complete metric spaces are particularly nice: they allow us to solve problems iteratively using Cauchy sequences, since we know that they have a limit. However, even if a metric space is incomplete, one can create a complete metric space from in a process known as completion. We first need the following definition. map i (X,dx) - (Y, dy) is said to be an isometry if it satisfies
(26) for all E1, 22 E X. A surjective isometry between metric spaces is called an isomorphism of metric spaces, or simply an isomorphism. Hence, we have that a metric space (X, dx) is called the completion of a metric space (X, dx) if the following conditions are satisfied:
dy (i(z1),
(x1,
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there is isometry i: X + X
the image i(X) is dense in X, and
the space (X,dx) is complete.
In fact, one can show that every metric space has unique completion, up isomorphism. In particular, every incomplete metric space can be completed.
One of the most important concepts in general topology is compactness, which is a notion of finiteness for topological spaces. In general topology, compact topological spaces play an analogous role to finite dimensional vector spaces in the theory of vector spaces or to finite sets in the theory of sets, The motivation for compactness of a topological space X comes from wanting to understand the conditions that X needs to satisfy such that any continuous map f X - C is bounded. Recall that a function f: X + C is bounded if there exists non-negative real number M such that [f(r)) S M for all E X.
One way in which a continuous function f X - C will be bounded is if it is bounded on a finite number of subsets of X whose union is X itself. This provides sufficient motivation for the following definitions. If (X, Tx) is a topological space, then a cover of X is a family (Ut)tes of subsets of X such that U.er Ug = X. The cover is finite if the indexing set / finite, and the cover is open if U, open for each E T. If (Ut)tes is a cover of X, then we say that (Uz)je) ) is a subcover of (Ut)er if J C I and (Uz)seJ is itself cover of X. Consequently, a topological space (X,Tx is compact if every open cover of X has a finite subcover. Hence, every continuous function f X + C on a compact topological space X is bounded. We study CX)), the space of continuous functions f X - C, in the case where X is compact Hausdorff space in Section B. It is also worth noting that every finite topological space is compact since any open cover must already be finite.
One can also show that every closed subspace of a compact space is compact, where the subspace is given the subspace topology: if V is a subspace of a topological space (X, rx), then the subspace topology Ty on V is defined by (U n V U E Tx Although it is not true that every compact subspace of compact space closed, we do have that every compact subspace of Hausdorff space is closed. Moreover, if (X; Tx) ) is compact and f (x, + (Y, is continuous function, then f(X) is compact. And if (X, is compact and f X + C is a continuous function, then (XX is bounded. For metric spaces (X,dx) dx), one can show that compactness is equivalent to sequential compactness, which means that every sequence in X has a convergent subsequence. Furthermore, if the metric space is a Euclidean space, say IR" or Cr, then one can show that a subspace of a Euclidean space is compact if and only if that subspace is closed and bounded.
Some of the most important compact topological spaces that we consider are compact groups. Firstly, a topological group a topological space (G, TG) that has group structure defined on it such that the group operations of multiplication and inversion
(g. h)) + gh, g 9
(27)
are both continuous functions. A topological group is called a compact group if the underlying topological space is compact. Note that we always choose our topological groups to be Hausdorff, and so this property is always assumed when we refer to a topological group. Note also that, in the theory of topological groups, a subgroup of topological group is always assumed be closed. In particular, closed subgroup of a compact group is compact group.
In the main paper, we study compact matrix groups. The most notable example of a compact matrix group is the orthogonal group ((n) of n X matrices M such that M + M M = I. To show that ((n) is compact, it is enough to show that it is closed and bounded, since ((n) is a subset of the Fuclidean space RRxn, Given that one can show that singleton sets in a metric space are closed and that polynomials are continuous functions, (n) is closed since is the preimage of the singleton set {/} under the continuous map ʃ : Raxx + Ruxx that is given by M +
M T M. O(n) is bounded since the Euclidean norm of an orthogonal n X n matrix is vn. Moreover, any finite group - in particular, the symmetric group Sn, is compact.
B. An Overview of C* -'-Algebras and Operator Theory
We provide an overview of the key results that appear in the theory of C* algebras, which is an area of mathematics that lies at the intersection of algebra and topology. The theory has its roots in the development of quantum mechanics in the 1920s, where the non-commutativity of bounded, linear operators on a Hilbert space is used to model physical observables. Gelfand
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Compact Matrix Quantum Group Equivariant Neural Networks
& Naimark (1943) developed abstract definition of C* -"-algebras that did not require the of bounded, linear operators on Hilbert space. Forty-five years later, Woronowiez (1987; 1988) used -aaaleebras develop compact matrix quantum groups. Since proper treatment of the theory of C** -allebbaass would amount to several hundreds of pages, focus solely on the key concepts that also appear in the theory of compact matrix quantum groups given in Section 3; notably, tensor products of C -alalbabras and the construction of the universal C* -algeebra from a set of generators and relations on those generators. When we quote a result without proof, we provide a reference to the literature where its proof can be found. We also rely on the background material on general topology that appeared Section A throughout. For more comprehensive treatment on -llalgebaas, we recommend any of the following references: Blackadar (2006); Courtney & Gillaspy (2023); Gromada (2020); Lin (2001); Maaßen (2021); Speicher & Weber (2020) and Rørdam et al. (2000).
Definition B.1. We build up the definition of C* -'llagebra with the following definitions.
An algebra A over C is a complex vector space together with a bilinear, associative multiplication A x A + A satisfying X(ry) = (Az)y x ((yy) for x; y E A and E C.
An algebra A is unital if it contains an element E A, called the unit, such that 1r x1 = z for all E A.
A subset B S A is called a subalgebra if B is an algebra with respect the same operations.
A subset I C A is called ideal if 7 is subalgebra of A that is invariant with respect both left and right multiplication by elements of A.
A normed algebra is algebra A that has norm ||-1| A C such that it is submultiplicative: ||ay||
|a||| Illl for
all x, y E A.
A Banach algebra is a normed algehra A which is complete with respect the metric induced by the norm ||-I1-
An involution on an algebra an antilinear map" " +A A A such that (a*)* and (ry)) y²x" for all x, y E A.
A -lalgebra is an algebra A that has an involution. Similarly, we define * -suballeebra and --ideal.
A Banach * -algeerra is a Banach algebra A that has an involution.
Consequently, we have that
Definition B.2. A C* -llalgebra is Banach "algebra that satisfies the C*-identity: || x/| ||>|| for all e A.
Note that closed subalgebba of a C* -llaagebra is also C'*aalgeba.
Definition B.3. We say that element of C*-algebra A is self-adjoint a" a, and, if A is unital, we say that is unitary if aaa = aa" =
Example B.4. We give some examples of C* -aggebras.
1. The complex numbers form a unital C* allgebaa..
2. Let H. be a complex Hilbert space with inner product denoted by (s-). The collection bounded linear operators B(H) is a unital C*-aagebra, where the norm is the operator norm
|T|| = sup [Tell
H. || - 1}
(28)
for all T B(H).. Note that multiplication is given by the composition of operators, and that the involution is the adjoint, defined with respect to the inner product by (T'e, v, w,w) (v. Tw) for all T E B(H) and u, H. Also, see that B(H) is non-commutative.
3. For any positive integer n, let (C) denote the set of matrices with elements in C. Then M,(C) is unital C*- -alalgebra, where the norm is the operator norm
M sup [Mvl e C" 1|ol1 <
(29)
where 11-11, is the Euclidean norm on C", and the involution is given by the conjugate transpose. This is a special case of the previous example, where we have chosen = C" and picked a basis to represent linear transformations as matrices.
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Compact Matrix Quantum Group Equivariant Neural Networks
4. Let X be compact Hausdorff space. Then the space of continuous functions on X,
c(X) = ( : - CI
continuous}
(30)
is a unital -alaegebra where the norm is the supremum norm
| = sup(l
(31)
such that (f + g(a)
+(()),
f())))))))))))))) == and [*)x = f(x) for all
fig E C(X), x E X and À C. Note, in particular, that C(X) commutative. The supremum norm indeed norm
since the compactness of X implies that every continuous function f X C bounded, and so |l oo for all
f E C(X). The norm is also submultiplicative because
Ifg(=)) (f(x))(=)) - (f()))l(()) - ll 1
(32)
which implies that ol < ol || Similar reasoning on the fact that fx(x lf()) shows that the supremum norm satisfies the C C"-identity. Finally, the supremum norm is complete because one can show that any Cauchy sequence of continuous functions on converges uniformly on X such that the limit also continuous function on X.
An important theorem for the -alallebra of continuous functions on a compact Hausdorff space X is the Stone- Weierstrass theorem. Its proof can be found in Speicher & Weber (2020, Theorem 3.3). First we need the following definition.
Definition B.5. Let A be an algebra of functions from some set X to the complex numbers C. We say that A separates points in X if, for all x + y E X, there is a function ʃ E A such that ((x) # (w)).
Consequently, we have that Theorem B.6 (Stone- Weierstrass). Let X be compact Hausdorff space. If A is unital "-subalgebra of C(X) such that A separates points in X, then A is dense in C(X). In particular, if A is closed, then A = C(X).
We also have homomorphisms on -"~algebras:
Definition B.7. If A, B are algebras, then linear map Φb + B such that d(zy) ((x))(v) for all xr A is called an algebra homomorphism.
If A, B are -lalgebras, then if Φ A + B is an algebra homomorphism that preserves the involution operation, that is, <(**) = (x))* for all æ E A, then @ is called a "-homomorphism.
If A, B are *-algebras, then bijective --homomorphism é + B is called a "-isomorphism.
We use the properties of B(H) to obtain representations C' -lllagebras:
Definition B.8. A representation of an algebra A is a choice of complex Hilbert space 7. and an algebra homomorphism π - B(H).
t-representation of a "-algebra A is a representation of that preserves the involution operation.
representation said be faithful if it is injective.
There another important theorem relating to the unital -llaleebra of continuous functions on a compact Hausdorff space X. This the Gelfand-Naimark Theorem that we referenced in Section 3, and since it the first of two theorems by Gelfand and Naimark that appears in this section, will call it Gelfand-Naimark I. Its proof can be found in Blackadar (2006, Theorem II2.2..4).
Theorem B.9 (Gelfand-Naimate I), Let A be a commutative, unital C -l-algebra. Then A is -isomorppiic C(X), where
X = (Φ: A + C is non-zero" -homomorphisn)
(33)
a compact Hausdorff space. The -isomorphism is given by f A C(X) where f(a) X + for all a E is such that [(a][o] = ((a) for all o E X.
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We now look tensor products of -llalebbras. First, we have the following definition.
Definition B.10. Let A, B be "aagebras. The algebraic tensor product, written A @ B, is defined be the algebraic tensor product of the vector spaces A and B together with multiplication operation that is given by
(a1 ② (1) @ b2) 01 @ 6162
(34)
and an involution operation that is given by
(a a" 0
(35)
In particular, A @ B is "aaleebra.
To define a tensor product on -llalgebras A and B such that the result is also a -'-algebra, we need to be able to define a norm ||||| that satisfies the -"-identity on A @ B, and then complete (A o B, II-II) to obtain a C* -algebrr.. Any norm that satisfies the C -ididentity is said to be a -orm. Although it can be shown, as a consequence of the Delfand-Naimark -Seall Theorem, that C* -norms on algebraic tensor products of -*~algebras always exist - see the definition of the spatial norm below one of the main difficulties behind this construction that in many cases there can be more than one C*-norm that can be associated with A @ B (Courtney & Gillaspy, 2023), and so in those cases there are many possible definitions of tensor product on C* -ollaaebras A and B such that the result is a -'lalgebra. Hence, in what follows, we choose only to outline the main ideas that lead to the construction of particular tensor product called the minimal tensor product since this tensor product will be used in the definition of a compact matrix quantum group.
We start with another theorem by -Naimar,,,, which is technically a corollary of the Theorem. Its proof can be found in Speicher & Weber (2020, Theorem 5.19).
-eaaal
Theorem B.11 GGelfand-Naimate II). Every C* -aleebra A has a faithful * "rreressentatinn π : A - B(H) on some Hilbert space H.
Hence, for C'-algebras A and B, we have faithful "-representations TA A + B(HA) and uB : B B((H)). One can show (Courtney & Gillaspy. 2023, Corollary 11.15) that this induces faithful * "-representation TA @ TT :AQ A B + B(HA @ HB) such that A ④ (a ④ b) = (a) @ FB (b) for all E A and b E B.
Hence we can define a norm ||. on A @ B, known as the spatial norm, that is given by
|=||_
A ② [2)))))|(()
(36)
which is a C* C"-norm by the fact that the norm on B(H & HB)
) a -o-morm together with the injectivity of A ② n
Takesaki (2002) showed that the spatial norm is independent of the faithful "-representation, and that is the minimal C*-norm on A O B; that is, for all -o-norms y on A @ B, we have that
||||| S 7(x)
(37)
for x E A @ B. Hence we call ||-||, the minimum norm, and write it as ||=]|lin'
Consequently, the completion of A @ B, for Ch-algebras A and B, with respect to the minimal norm || ||=||...nn is called the minimal tensor product and is denoted by A @min B.
One can also show (Courtney & Gillaspy, 2023, Corollary 11.29) that for a pair of "-homomorphisms Φ1 Ai + B₁ and Φ2 : A2 > B2 on "-algebras A1, A2, B1, B2, the algebraic tensor product & @ Φ2 A @ A2 + B1 @ B2 extends to *-homomorphism Φ1 @min Φ2 Ai 8min A2 + B1 8min B2.
We now focus on Mn (A), the set of n x n matrices with entries in a C" -"lalgebra A, that is
(A(4) = ((aus) ang A, S n}
(38)
This comes with natural involution that is given by (ai,))* = (at,)) for all (ai,) E Mn(4).
We use the following definition to create a C"-normn on M,)
Definition B..2. Suppose that Φ :A A - B is a linear map between "-algebras A and B. Then, for all positive integers n, we define the linear map ②(a). (A) > (B) to be p(¹)) ((ar,)) = (o(a:,)). The map g(a) is called a matrix amplification of Φb
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We create the -orm on (A) as follows. By Gelfand-Naimark II, we have a faithful "-representation A + B(H) for some Hilbert space H. This induces faithful "-representation x(*) (,))) B(H²) since one can show Rørdam
et al. (2000, Section 6.1) that M, (B(HH) is "-isomorphic to B(H*). Then we can define a function || (A) - C by
[I(xs))] = (x) ((at))) B(H))
(39)
which is C*-norm since x (²)) is injective and the norm on (((()) is C"-norm.
One can show the following inequality, whose proof is outlined in Rørdam et al. (2000, Exercise 1.13).
Lemma B.13. Let (Tiz)= E (B(H)) Then
[TaillBa I (ITTs))lM. S
|Tas BOx
(40)
Lemma B.13 implies that the image of (A) under x ("), which is "-isomorphic to M,) since (()) is injective, is
closed "-subalgebra of the C* -allarabraa ((H"). Hence Ma (A) is itself C* -algebarr. Moreover, the norm (A) defined in (39) is unique because a "-algebra admits at most one norm that makes it into a C"-algebra Lin (2001, Corollary 1.2.8).
Expressing this result as a theorem, we have that Theorem B.14. If A is a C* -algebra, then (A) is also a C C'lalgebra whose norm is defined in (39).
One can also show the following result.
Lemma B.15. Suppose that A is C* -alaeebbra and let n be any positive integer. Define a map o M,(A) + (C) O
by
(aas))
Σ
E.g O 0i
(41)
1,j=1
where Fag is the matrix unit with in the (i. -entry, and 0 otherwise. Then is a "-isomorphism.
We the "-isomorphism between (A) and Ma A to obtain the following isomorphism.
Proposition B.16. Suppose that a C*-algebra and let be any positive integer. Then there a unique Co-norm on the algebraic tensor product M, (C) @ A. Hence write M, (C) & A, and have that
a (C) O A = (A(A)
(42)
Proof. The -isomorphism together with the C* -norm given M,(()) defines the existence of C*-norm on MC A, namely
((Ar,) e all = [
(43)
(C) @ now "-isomorphic to a closed "-subalgebra of the C*talgebra M,, (A), namely Ma itself, hence is a C C*-aagebra. The result now immediate because a *-algebra admits at most one norm making into C* C'-aaeebra.
This means that we can use the i-isooorphism @ to identify elements of Mx(A) with elements of M, A, namely
(ai,) + Σ Eij & aa.
(44)
i,==
We now turn to the concept of universal C** -ll-ebaeras. We know that can create C* by starting with a *-algebra, finding C*-norn for it, and then completing the pair. We would like to be able to create C'* using sets of generators and relations. We follow the presentation that is given in Speicher Weber (2020, Chapter 6).
begin by creating a * "lagebrra as follows.
Definition B.17. Let = {æ, | i E I} be a set of symbols over an index set I. Let E* = {ap i 1} be another set of symbols that is disjoint from F., Then
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A non-commutative monomial in E E* is word of non-commuting symbols si Fmm for some positive integer such that i1 :mm
im E and 03 {+1] for all E [m].
A non-commutative polynomial in E U E is a formal linear combination non-commutative monomials EUUUUE E with coefficients in C.
We can define concatenation operation on non-commutative monomials in E' U E
(
(x))
I..II
1ig 1 91 α 8n
(45)
for any non-commutative monomials Ti ... non-commutative polynomials in
Cmn and T o
... o in EU E". This operation is extended linearly to to
E".
We can also define an involution operation on non-commutative monomials in EU E
(
im
(46)
where a," is if a and is + if a, = for all j [m]. This operation is extended antilinearly to non-commutative
polynomials in EUUE
Then, have the following definition.
Definition B.18. Let E,, E* be before. The free *-algebra the generator set E, P(E), is the of non-commutative polynomials in EUE" where addition and scalar multiplication the standard operations, and multiplication and involution are given by the concatenation and involution operations given in (45) and (46), respectively.
The -algebra P(E) is free in the sense that there are no relations amongst the polynomials in P(E).
For any subset A of polynomials in PEE), we can create the "-ideal ((R) that is generated by R. Considering P(E) and I(R) as algebras, we can certainly form the quotient algebra P(E)/I(R). Now, since P(E) and I(R) are "-algebras, the involution operation on P(E)/T(R) given by
(a + 1)* a" +1
(47)
for all P(E) is well-defined, and in this case P(E)/I(R) is also (quotient) *-algebra. Hence, have that Definition B.19. The universal "-algebra with generators E and relations R, A(E R), is the quotient *-algebra (R))
We now aim create -o-morm "-algebra, although this --algebra isn't quite A(E R). For this, first need the definition of C"-seminorm.
Definition B.20. A C*--seminom "-algebra A is map y A -> RR such that
1. y a seminorm, that is
?(x) for all A
(Ax) = (>/(2) for all æ e A and E 7(x y) (xx) + tv) for all x, E A
2. y is submultiplicative, that is, ((zy) I (B) for all x, y, e A, and
3. 7 satisfies the C'-identity, that is >(=xx) (2)
Note that C**-semiormm satisfying this additional property.
algebra A is a seminorm 7 A R such that if 7(x) then 0. Hence C* C*-norm is
We make the following definition.
Definition B.21. We define function [|-I)A(E))) A(E R) + R by
|| == sup(r(x) a C* -eminorm on A(E) R))
(48)
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It is possible that ||= (()))))))))) is infinity for some E A(E) R), so we only allow ourselves consider those E and R such that || A(EIR) < oo for all A(E R). In this case, one can show that || ((E))))))))) is a C -m-seminorm: for example, for all
I, E A(E R) and any C* semminmm on A(E R), we have that
(ay (1) S (e))
(49)
and so we see that (1-IAA is submultiplicative. Also, we see that
Lo far A(B R) | - 0}
(50)
is -didell. Hence, A(E | R) is quotient -laleera having a C*-norm | | that is obtained from the -*-seminorm A(E R) under the quotient map A(E R) A(E R)/I. Hence, have that
Definition B.22. The universal C"~algebra with generators and relations R, C*(E R), is the completion of A(E] R with respect to |= It is indeed -alagbbra,, by construction.
The following lemma is very useful for showing the existence of the universal C* Calalgebra C²((B) (R).
Lemma B.23. Let E = {xx I E } (with E* understood) be set of generators and let R S P(E) be set of relations. If there is constant C > 0 such that the value o on any generator is less than or equal to C and all C* -e-ssmmnorms on A(E R). then || < 0o for all x E A(E R). Hence by the above construction C* C*
| R) exists.
Proof. Let be any C* -'-sseinorm on A(E R). Then, since any monomial in U E" has positive integer length, pick an arbitrary monomial in E U E having some arbitrary length m. Then, by the submultiplicative property, we have that the value of on this monomial is less than or equal to Cm", and so by taking the supremum over all ("-seminormss we see that any element of A(E R) has a norm that is bounded, as required.
One can show Speicher & Weber (2020, Proposition 6.7) that every universal C"lalgebra C*(E) R) satisfies the following so-called universal property.
Proposition B.24 (Universal Property). Let E {a, ie 13 be set of generators and let R S P(E) be a set of relations such that the universal C -allebbr C* | R) exists. Suppose that A is C* -algebra having a subset E' {ya i f} that satisfies the relations R. that is, all polynomials in R become zero when replace each 2; by y for all i 1. Then there exists unique" -homomorphism o C R) + A such that x, + Yi for all i E I. Example B.25. For n 2 2, we have -isomorphism of C'-aagebras
M, = C" (",,, 15 1 i,j S uhy = Ujss U1 (g.
(51)
Proof. Let (E R) = (u, 1 i < n "h, = Ugss ,. = ov. We first need to show that C*(E' R) exists, that is, that ||~||| < co for all x E A(E R). This is immediate from Lemma B.23 if we can show that there is some constant C' such that ( S for all 1 S i,j S n and for all C C"*sseminorms 7 on A(E R).
Let y be a -"-seminorm on A(E R). By the C'-identity and the relations R we have that
?2 = U5a) = (33.) = (6,,
= (4jss
(52)
Hence ((uss) is either 0 or for all j E [n] and so, again by the C*-identity and the relations Rt we have that
(uu,,)² (u = (15,,"),)) = (6(,)135) ((3g))
(53)
Hence ((u,)) is bounded above by for all i,j S n and all C*-seminorms 7 A(E R), required.
We now prove the * "iisomorphism M.(C)= & (* R). The matrix units Fi,- E M.(C) satisfy the relations Rt since Ei = ET_TT = Ey and Et, = 6z. Eij, hence, by the universal property. there exists a surjective "-homomorphism o: C R) + (C) such that u4,j + Ex, for all i,j E [n]. Looking at the relations R we see that the monomials in CE R) can be at most the elements Us.j, but since ø is surjective the monomials are exactly the elements u,,y. Hence, as a vector space, C* (FF) R) has dimension n² and so & is also injective. Hence & is a "-isomorphism, as required.
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Compact Matrix Quantum Group Equivariant Neural Networks
C. Missing Proofs and Supplementary Content
We include proofs of results that appeared in the main paper, and provide the full statement of Woronowicz-Tannaka--ee Duality (Woronowicz, 1988), the first part of which appeared in Theorem 5.8.
Proof of Proposition 3.3. Let ((n) C (T()) be compact matrix group. Consider ((G(n)), the C* -lgebra of continuous functions G(n) C, which is commutative. Firstly,
define functions M. G( - C for i,j [n|] such that (g) Di.j- By the Sene-Weiessstaass Theorem (Theorem B.6) and the compactness of ((n), the úij generate C(G(n))
and the matrices u = (ux,)) and u are invertible since u (g) = "(g")).
Moreover, we have that the comultiplication map A for C(G(n)) is given by
(g, h) (4.
(54)
since
Σ uskk s ukj (g = Σ wo (..K(g)un,(1)) Licht.. = (gh)es = (.ssggh)
(55)
that is, A coincides with matrix multiplication in G(n).
The definition of A that is given in (54) is a *-homomorphism because
A A(ui (2, (igua,
(56)
(57)
(g, h) (wx,
(58)
A(H,,))
(59)
and
A (gh) wir (g. h) A(u))" (g. h))
(60)
Note that in (57) we have used that C(G(n)) has a pointwise multiplication, and likewise, in (59), have used that C(G(()) C(G(n)) has a pointwise multiplication.
Proof of Proposition 4.5. We have that u @ w is representation of G(n) because
(() ((x,)) Σ Vi.a & Uz.J
Σ Wk.y & Wy.l =
-' Wk @ Uz
(61)
and ē is a representation of G(n) because
A(rt.) A(vas)*=
Σ Ui, @ Uki
Σ of @ rhs
(62)
where in both (61) and (62) we have used that is x-homomorphism.
Proof of Lemma 4.6. We have that
ã' (==,))))² = ((13,))) = (u,,)) = m²
(63)
and also
Tu = (uz,s)* = (Mi,)) ū
(64)
Hence
u. (u 1 (u yuu + ūū" 1 ū
(65)
as required.
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Compact Matrix Quantum Group Equivariant Neural Networks
Proof of Theorem 5.7. It is easy to show that (e satisfies the first four axioms of Definition 5.5. For example, the second axiom says that if di E (endR) (wr,, wi) and Φ2 E FundRepa(r) (101 Wm), then have that
(Ø2 o Φ1 ou 22 Sw: Φ1 u Sww (62 o 01)
(66)
and 02 01 (Pnd) (Wa, Wm)
For the fifth axiom, we need to show that the operator R 1 Ei= 6 @ er in both FundRepatn)(e (0, oo) and PundRePecm) (Ø, so).
For the first case, have that
R E (2n - R = (u@T)R
(67)
By applying both sides to we have that (67) holds and only if
e 8 e
ue @ ūe:-
ej 1
= Σ j.k=1
eg 8 ck O (umn)}a
(68)
j.k=1
Hence, by equating coefficients, we see that R FundRepG(n) (0, oo) if and only if uu = 1. For the second case, similar calculation shows that R E FundRepaam) (0, o0) if and only if ūu" = 1. Since and ā are both unitary, we obtain the result.
As promised in Remark 5.9, we provide the full statement of Noronowicz-Tamntt--kiii Duality (Woronowicz, 1988) here, the first part of which already appeared in Theorem 5.8.
Theorem C.1. [Woronowicz-Tanune Taiike--annaaBBBaa Krein Duality] Let be a two-coloured representation category. Then there exists a unique compact matrix quantum group (G(n), u) such that FundRepa(s) = C. If -{rig 1 i,j j S n} is a set of generators for the free " -algebra P(E). then the --deell Iα()) S P(E) that determines G(n) is given by
span {l @wr sw 611, E C(w,, wr), I [ni]' [m]*}
(69)
In Definition 7.4, we also said that we would provide definitions of the following three C-(bi)linear operations on two- coloured set partition diagrams
composition:
piill (n) p (n) -
mr (n)
(70)
tensor product: @ : mn (n) x P." (n) -
pum -m (n)
(71)
involution: Pwu (n) - pivk (n)
(72)
We define these operations as follows.
Definition C.2 (Composition). Let dz, E (n) and dxs E P (n). First, we concatenate the diagrams, written daz d by putting dx, below d,g. concatenating the edges in the middle row of vertices, and then removing all connected components that lie entirely in the middle row of the concatenated diagrams. Let c(d,, da,) be the number of connected components that are removed from the middle row dæg o dzy. Then the composition is defined, using infix notation, as
d dry n cldz des ) dmy)
(73)
Example C.3. If dryg is the two-coloured (o
partition diagram
(74)
and dr, is the two-coloured
partition diagram
(75)
23

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Compact Matrix Quantum Group Equivariant Neural Networks
then have that dz2 dm is the
-partition diagram
(76)
and so dxz dæ, is the diagram (76) multiplied by since connected component was removed from the middle row of d dzy,
Definition C.4 (Tensor Product). Let dxy E Pwg (n) and d Par (n). Then dx, @ dr is defined to be the (wx Wgg wy Wm) parttparition diagram obtained by horizontally placing dr to the left of drg without any overlapping of vertices.
Example C.5. If dx, is the two-coloured (o )) partition diagram
(77)
and d7 is the two-coloured (o 0 parrtition diagram
(78)
then we have that dx & dr is the (o
partition diagram
(79)
Definition C.6 (Involution). Let d, E mg (n). Then d E Pw (n) is defined by reflecting d, in the horizontal axis.
Example C.7. Recalling the two-coloured for ease)
oprrtiitoon diagram d, from Example 7.3 (reprinted below
(80)
have that d is the (+ o partition diagram
(81)
With this, can now provide the following proof Proposition 7.5.
Proof of Proposition 7.5. P(n) is strict monoidal category because the bifunctor (horizontal operation) on objects reduces the concatenation operation words, which is associative, and the bifunctor morphisms the tensor product of linear combinations of two-coloured partition diagrams that is given in Definition C.4, which is associative by definition.
(nn) is C-linear because the morphism space between any two objects is by definition a vector space, and the composition of morphisms is C-bilinear by definition. For the same reason, the bifunctor is also C-bilinear.
Proof of Theorem 7.9, We prove this theorem in series of steps. We first consider the image Cp(n) of the two-coloured partition category (n)) under d, + Dr.
Step We need to show that Cp (n: actually a category.
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Compact Matrix Quantum Group Equivariant Neural Networks
The objects and the morphisms have been defined in the statement of the theorem. For every triple of objects, the composition rule is given by matrix multiplication, which is associative. Furthermore, the identity morphism for every object wa of length & is the n x n² identity matrix. Note also that the unit object is Ø, the empty word.
Step 2: We need to show that ④: P(n) + Cp(n) is a functor.
θ maps the objects of ((n) to the objects of Cp(n) since θ maps each word wy to itself, by definition.
It is enough to show that (f)) (g f) on arbitrary basis elements of arbitrary morphism spaces where the codomain of f is the domain of g because the morphism spaces vector spaces.
Let f = drg be a (wa; wi) partiiion diagram, and let g - drg be a (wi, wm parttitoon diagram. Then, by Definition
C.2, we have that
dxz dm n des > d
(82)
where dr be the composition dr, dr, expressed as a (uk, Wmn) (om)-pape -partition diagram. Then
(g f) = (d dz,) = n clld Dea
(83)
We also have that
(()) Dr, Do1
Σ
0T..I.K)EI, Er,K
Σ
I Er,,J
(84)
[Elm|m,KK|luj
JElM!"
Σ
0ra.11.K)) EI, EK.J
(85)
rE|p] [Elj] KElp)),JE[M
For fixed I, consider
Σ KE(n
(86)
This equal
add2 dnny (
(87)
Indeed, for fixed I, J, (x..(T,J) is 1 and only if both 6 (1, and 0 KJJ) are 1 for some K E [n]] since d is the composition dx2 dæ,, The number of such K is determined only by the vertices that appear connected components that are removed from the middle row of dm, d since, for fixed J, only these vertices can be freely chosen if we want both (1,K(I and (k,.). to be 1. However, since the entries in each connected component must be the same for both (T. and (KK)) to be 1, this implies that the number of K [n] such that both (I,K( and & (x,,(K are is n c(day, dr,),
Hence (85) becomes
Σ
oldr Aay (T. ( = n (ddgg (r1) Dayg
(88)
JE(M]"
and so, by (83), we have that O(g) Ogg 1,, as required.
For each object wi in P(n), the identity morphism w is the two-coloured (Wk, w partition diagram d, where is the set partition
{L, + 1 2, 2, ... 2,2kk
(89)
of [2k). and its image under d, + Dar is the n x n² identity matrix, required.
Step 3:
We need to show that Cp(n) is strict C-linear monoidal category.
We first consider the tensor product of arbitrary basis elements of arbitrary morphism spaces. Let d, be (Wy, (r))-parittion diagram, and let d, be (wm, wyy wy)- -pariiiio diagram. Then, defining d, to be the (w, Wgg wi wm) parttiton diagram, where w is the set partition æ 7 of + m + k + q). we see that
25

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Compact Matrix Quantum Group Equivariant Neural Networks
Da @ D=
Σ
BI, BI,J
80 Xe(n)
Σ
1x,,
(90)
JE(N)
Σ
(91)
(1.0)e|) th (J.)))( =
(92)
since u -((()))))))
The bifunctor on objects of (n) reduces to the concatenation operation words, which is associative. Also, since the bifunctor on morphisms in Cp (n) is the tensor product of linear combinations of images of two-coloured partition diagrams, is associative because the tensor product of images of two-coloured partition diagrams is associative, both by the above calculation and because taking unions of set partitions is associative. Hence, Cp(n) (n) strict monoidal category.
Cp (n) C-linear because the morphism space between any two objects is by definition vector space, and the composition of morphisms is C-bilinear by definition. For the same reason, the bifunctor is also C-bilinear.
Step 4: We need to show that θ P(n) + Cp(n) is strict C-linear monoidal functor.
1. θ preserves the tensor product on objects, since ((wx) = wi for any object in P(n), and the tensor product in both categories is given by the concatenation of words.
2. It is enough to show that @(f) & O(g) Of @ g) on arbitrary basis elements of arbitrary morphism spaces as the morphism spaces are vector spaces. This result is now immediate from the calculation given in (90).
3. It is clear that sends the unit object @ in ((n)) to Ø, which is the unit object in Cp(n).
4. This immediate because the map d + D C-linear by Definition 7.7.
Step 5: We need to show that Cp(n) is a two-coloured representation category.
The tensor product, concatenation and identity axioms have already been shown in previous steps. For the involution, Dr an element of Homc, (wx, wi) and is the image of (wk, wt) ppartiiin diagram d, under O then
(D+)* Σ
Σ
DT-
(93)
JE(M)"
As Dz- is the image of the (wi, wk) partition diagram d=- under θ, this implies that (Dz)* is in Homc, (wi, wk), as required.
Finally, we see that the map R: : 1 + Σi= e, @ e is in both Home, (m) (Ø, oo) and Hom (m) (Ø, o0) since is the image under dz + D, of the top-row pair partition diagram corresponding to the set partition {1,2] of {1,2] superimposed, on the one hand, with the word w2 c,, and on the other hand, with the word W2 = @c.
Step 6: We need to show that d,, Dx defines strict C-linear monoidal functor θ K(n) Cx((n) such that Ci((n) is two-coloured representation category.
This is now immediate by Definition 7.6, the definition of a two-coloured category of partitions, and since the functor on (nn) is strict C-linear monoidal whose image is a two-coloured representation category. In particular, the morphism spaces for K[(n) are closed under composition, tensor product and involution, and so these are inherited by their image under O. Also, the map R : + Σ e, & e; is in Homo (m) (0, oo) and Hom (m) (Ø, so) since the top row pair partition diagram corresponding to the set partition (1, of (1,2) superimposed either with with is in both Homx(,)) oe) and Homg()) o.).
D. Additional Examples
We provide additional examples and content for the examples that were given in Section 8. We begin with one-coloured partition categories, which have been fully characterised by Raum Weber (2016).
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Compact Matrix Quantum Group Equivariant Neural Networks
D.I. One-Coloured Partition Categories
One-coloured partition categories can be divided into four cases: group, non-crossing, half-liberated, and the rest. The two most important cases the group and non-crossing ones: refer the reader either to Banica al. (2010) or to Gromada (2020) for results relating to the two other cases. The corresponding compact matrix quantum groups G(n) are known in the literature as easy orthogonal compact matrix quantum groups, or sometimes Banica-Speicher quantum groups, because S, S G(n) C (n)
Group: Banica & Speicher (2009) provided a full characterisation: there are only six compact matrix groups such that S, S G()) S
Onn)*. Consequently, by Corollary 6.3, we obtain characterisations of six compact matrix group equivariant neural networks, of which two the symmetric group S, (Ravanbakhsh et al., 2017; Maron et al., 201 2019;; Crump, 2024; Godfrey et al., 2023) and the orthogonal group ((n) (Pearce-Crump, 2023a) were known previously to the machine learning community. Said differently, we have found characterisations of the weight matrices that appear in four new compact matrix group equivariant neural networks, two of which we discussed in Section 8. We state the result of Banica & Speicher (2009) below.
Theorem D.1 (Banica & Speicher (2009) [TTherem 2.8]). There are only six easy orthogonal compact matrix groups such that Sn S G(n) S O(n)+ the groups and the spanning sets that determine the weight matrices of Theorem 7.10 are given below. In what follows, when we refer to the image of a (ks 1) parrtittionn diagram, we mean the image under the map d + Dr given in Definition! 7.7.
The symmetric group Sn: the image of all (K:I)-partition diagrams.
The orthogonal group (n)) the image of all (k. -pprrttiinn diagrams whose blocks come in pairs.
The hyperoctahedral group Hn (the symmetry group of the hypercube)): the image of all (k. -partiiitn diagrams having blocks of even size.
The bistochastic group Bn (the group of orthogonal matrices having sum in each row and in each column): the image of all (ks, 1) parrtittio diagrams having blocks of size one or two.
The modified symmetric group S' = Zz x Sn: the image of all (k., Papatiitoon diagrams where the number of blocks of odd size is even.
The modified bistochastic group B¹ = Zz x Bui the image all (ks, -parrtttionn diagrams having an even number of blocks of size one and any number of blocks of size two.
Remark D.2. We know from Godfrey et al. (2023) that we can improve upon the spanning set for the symmetric group Sn given in Theorem D.I by removing all (k: ()-partitoon diagrams that have more than n blocks from the set of all (ks, 1--artiitonn diagrams before taking their image. The resulting set of matrices the diagram basis for Sn.
Non-Crossing: In order to state the classifications for the non-crossing case, we first need the following definition.
Definition D.3. A set partition diagram dx corresponding a partition of [L [1+ k] is said be crossing if there exist four integers x1 < 22 23 < xΛ < l + k satisfying:
1. æi and Jg are the same block
2. and dt are in the same block, and
3. æi and 22 are not the same block.
Otherwise, dz is said to be non-crossing.
The six easy orthogonal compact matrix groups that were found by Banica & Speicher (2009) were liberated by the same authors to create six compact matrix quantum groups which are true quantum groups - that fall into the non-crossing case. Liberation means that crossings are not allowed in any of the set partition diagrams. Weber (2013) found a seventh using this method. We call these compact matrix quantum groups free since their categories of partitions K((n) only contain non non-crossing set partition diagrams. We have the following result.
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Compact Matrix Quantum Group Equivariant Neural Networks
2
2
2
2
O
O
3
3
2
2
2
0
0
O
0
0
0
0
0
Figure I. The fifteen (2, (2rtiipntitiin diagrams.
Theorem D.4. There only seven free easy orthogonal compact matrix quantum groups. We list the quantum groups and the spanning sets that determine the weight matrices of Theorem 7.10 below.
The first six are the symmetric quantum group Stt, the orthogonal quantum group O(n)*, the hyperoctahedral quanium group H the bistochastic quantum group B the modified symmettric quantum group St+ and the modified bistochastic quantum group Bx+,, whose spanning sets are determined by the same sets of set partition diagrams that were given in Theorem D. for their "sister" groups, but with all crossing sel partition diagrams removed.
The seventh is the freely modified bistochastic quantum group B# To obtain its corresponding spanning set, we consider the same set of set partition diagrams as for B but now temporarily label each vertex with colour from {o, }} in an alternating fashion, starting with in the top right corner and moving counterclockwise, ultimately finishing in the bottom right corner. We only retain those set partition diagrams where the blocks of size two pair one with one and take its image under dæ + Da, removing the colours in the process.
In fact, we obtain stronger classification for the non-crossing case due to the following theorem. Theorem D.5 (Banica & Speicher (2009) [Theorem 3.8)). The spanning sets for the seven compact matrix quantum groups in the non-crossing case are bases when n 2 4.
Remark D.6. By Theorem 7.10, note that each easy compact matrix quantum group (G(n), u) that appears in Theorems D.1 and D.4 is defined by the two-coloured category partitions ([nn) that associated with That is, Banica & Speicher (2009) actually showed that there only six one-coloured category of partitions that contain the swap partition diagram, that is, the ppartiin diagram
(94)
and together with Weber (2013) they showed that there are only seven one-coloured category of partitions that non- crossing. With these restrictions on the possible one-coloured category of partitions in place they could immediately apply Woronowiez--Tannaa---Mee duality to construct the easy compact matrix quantum groups from each of these one-coloured category of partitions. In fact, we have, again by Theorem 7.10, that each compact matrix quantum group is precisely the universal Ctt -algeraa
C(G(n)) =
i S Dowe u. 20m D for all d, E ([n) wy))
(95)
Example D.7. We calculate the weight matrices that appear in the compact matrix quantum group equivariant neural networks for all of the easy orthogonal compact matrix quantum groups that appear in the group and non-crossing case when
k - 2. Recall from Theorems D. and D.4 that the weight matrices are formed as weighted linear combination of the image under d, + D2 of some subset of all of the possible (2, partiitio diagrams.
There are B(4) = 15 (2, partiitoo diagrams in total, where B(4) is the fourth Bell number. They are shown in Figure 1. We number the diagrams from = 1 to 15, going from left to right and then top to bottom, assigning a weight w; to each diagram.
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Compact Matrix Quantum Group Equivariant Neural Networks
Procedure 2: How to Calculate the (I, -enntry of each Equivariant Spanning Set Matrix D, from ((C")Sk,
8k u u@wx) + ((C")ot, , ( for an Easy Compact Matrix Quantum Group (G(n),u).
We assume that d,, is two-coloured (wa, wi) partition diagram in the two-coloured category of partitions ([(n) that uniquely determines (n( u). We perform the following steps:
1. Place the indices I the top row of d and the indices J on the bottom row of dg.
2. If all of the vertices in each block in d have been overlaid with the same number, then the (I,J) entry of Dr
is 1, otherwise is 0.
Figure 2 shows the images of the fifteen (2, 2ppatiiti diagrams under the map d, + Dr.. We apply the characterisations of Theorems D.I and D.4 to obtain the following results.
For the group case, we have that
The Symmetric Group Sn: The weight matrix is weighted linear combination of the images of all fifteen (2 partition diagrams.
The Orthogonal Group O(n): The weight matrix a weighted linear combination of the images of diagrams six, seven and eight.
The Hyperoctahedral Group Hn: The weight matrix is (coincidentally) the same as for the orthogonal group in this case, since the (2, -parttiani diagrams with blocks of even size are the same the (2, -prttiito diagrams whose blocks come in pairs.
The Bistochastic Group Ba: The weight matrix a weighted linear combination of the images of diagrams six through fifteen, inclusive.
The Modified Symmetric Group S The weight matrix is (coincidentally) the same as for the symmetric group, since the number of blocks of odd size in any (2, -partiitoo diagram is either 0, 2 or 4.
The Modified Bistochastic Group B The weight matrix is (coincidentally) the same as for the bistochastic group, since the number of blocks of odd size any (2, 2) partiiton diagram is either 0, 2 or 4.
and for the non-crossing case, we have that
The Symmetric Quantum Group S The weight matrix is weighted linear combination of the images of all diagrams except for seven, which is the only crossing diagram.
The Orthogonal Quantum Group Onn)+: The weight matrix is weighted linear combination of the images of diagrams six and eight.
The Hyperoctahedral Quantum Group H The weight matrix the same as for 0(n)+
The Bistochastic Quantum Group : The weight matrix is a weighted linear combination of the images of diagrams six and eight through fifteen, inclusive.
The Modified Symmetric Quantum Group Sr The weight matrix is the same as for S
The Modified Bistochastic Quantum Group Bt:: The weight matrix is the same for B
The Freely Modified Bistochastic Quantum Group B# +: We colour in the diagrams that appear in the weight matrix for Bi,, and retain only the diagrams where the blocks size two pair one with one *. Consequently, we remove diagrams eleven and fourteen from the set, and so the weight matrix a weighted linear combination of the images of diagrams six, eight, nine, ten, twelve, thirteen and fifteen.
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Compact Matrix Quantum Group Equivariant Neural Networks
dx
DT
dx
Dx
1.1
0
1,
0
0
0
9,1
0
0
1.2
2.1
2.0
5.1
12
2.1
12
0
1.2
90
22
3
1,3
0
0
0
2.1
93
18
0
1,2
0
0
0
1,2
0
0
22
0
2.1
0
3
1
2
2,1
2.2
122
21
1.1
9
1,2
1
0
2.1
0
0
9,2
0
3
2,1
12
0
1.1
1
0
0
3
2
2.1
2.9
1.1
1,8
2.1
1.1
12
0
1.2
2.1
0
2.2
3
1.1
0
0
Figure 2. For we display images under the map d, D,, the (2, 2) partition diagrams.
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Compact Matrix Quantum Group Equivariant Neural Networks
D.2. Two-Coloured Partition Categories
Two-coloured partition categories are much richer than one-coloured partition categories because the generators ui of the corresponding compact matrix quantum groups are no longer self- aelf-adjoint They were first investigated by Freslon & Weber (2016). Unfortunately, at the time of writing, full characterisation of these categories is unknown. However, Tarrago & Weber (2016; 201 8) classified all two-coloured partition categories in the group and non-crossing case. In particular, there are seven series of two-coloured partition categories in the group case and twelve series of two-coloured partition categories in the non non-crossing case. The corresponding compact matrix quantum groups ((n)) are known as easy unitary compact matrix quantum groups, since they satisty S, S (in) C U(n)+. We focus on the characterisation for only the two most important series: the unitary group (n) and the unitary quantum group Unz) referring the reader to Tarrago & Weber (2016, Theorem 5.3) for the characterisations of the other series.
Tarrago & Weber (2016; 2018) showed that the two-coloured category of partitions for the unitary group (n) has, for fixed words We and w/ morphism space that is spanned by all two-coloured (W,, wy) pyy-tparttionn diagrams whose blocks come in pairs such that if two vertices of a block are in the same row, then they have different colours, otherwise they have the same colours. Tarrago & Weber (2016; 2018) also showed that the two-coloured category of partitions for the unitary quantum group (n))+ has, for words w/ and wi, a morphism space that is spanned by all non-crossing two-coloured (uy, p diagrams satisfying the same conditions as those for the unitary group. As before, we obtain the n' x n weight matrices of Theorem 7.10 for these compact matrix quantum groups by taking the images of the two-coloured partition diagrams that span the respective morphism spaces under the map dæ + Dx given in Definition 7.7.
This introduces the rather interesting point for the unitary group U (n). If we pick two words wy and wy, then the n' x n weight matrix corresponding to these words that appears in a compact matrix quantum group equivariant neural network for the unitary group U (n) is the C-linear span of the image of all (Wy, wi) wp)-ptrtiiion diagrams that live in the two-coloured partition category for U (m). However, if we consider compact matrix group equivariant neural networks for U(n) instead, as the words can only be formed from white points, then not only must we have that k equals I, by the characterisation of the two-coloured partition category for U (n). but also that the n² x n² weight matrix is the image of all permutations in the symmetric group Ss. expressed as (k, -arttiiio diagrams. We see that this is the classic version of Schur Weyl duality. Example D.8. We calculate the weight matrices that appear in compact matrix quantum group equivariant neural networks for the unitary group Un) and unitary quantum group U Un)) when A I == 2.
The easiest way to do this is to consider the one-coloured partitions that come in pairs and then superimpose colours onto each row of vertices to see which two-coloured partitions match the classification result.
The one-coloured (2, p-artitionnn diagrams that come in pairs are
2
2
1
2
(96)
4
3
We label the diagrams as 1 to 3 from left to right. Note that the matrices obtained under dx + D,r are the same no matter the colours on the vertices, hence we can reuse the matrices for diagrams six, seven and eight from Figure 2 in order to calculate the weight matrices.
We now consider pairs of words. As there are only 4 words of length 2, this implics that there are 16 such pairs of words. We obtain the following result for the weight matrices for (2), where for each cell, we take weighted linear combinations of the matrices D, for each set partition diagram in the cell.
(T, J
0o
2, 3
Ø
Ø
2,3
+
Ø
1,3
1,2
Ø
(97)
80
Ø
1,2
1,3
Ø
2,3
2.3
..
Similarly, for U, the weight matrices are weighted linear combinations of the images of the following set partition
31

---

Compact Matrix Quantum Group Equivariant Neural Networks
diagrams, where have removed the second diagram from (97).
(1,
J
oo
.o
..
3
Ø
Ø
3
Ø
1,3
1
Ø
(98)
Ø
1
1,3
Ø
3
Ø
Ø
3
Note that the top left hand cell (97) provides a classification the weight matrix appearing in the compact matrix group equivariant neural network for (2) when n 2.
32


Reviews:
[{"review_id": "bNtJUJrFhr", "reviewer": "Reviewer_sFDU", "summary": "The authors present the definition of a class of equivariant neural network which are equivariant with respect to a more general definition of group, namely of a quantum matrix group. The strategy relies as describing the non-commutative geometry by using non-commutative C* algebras. The authors show that they explicit derive weight representation for a class of simple quantum groups.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their comments. We are pleased that they found the paper to be “very well written despite its technical nature” and that they did not encounter any issues with our theoretical contributions. We are also pleased that the reviewer is excited about the potential applications of our work to generative modelling on non-commutative spaces: we would be very keen to see if the ML community is able to build upon the formalism that we have proposed in our paper to demonstrate such applications in practice.\n\nWe welcome the feedback to try to make the paper more accessible to a more general audience. This has naturally been quite a challenging task given the high technical requirements that are needed to be able to understand its content – as the reviewer has noted, we have included a number of examples of the construction both in the main paper and the appendix and we have also supplemented them in the appendix with comprehensive background sections on all the material that is needed to understand the work. However, we will look for additional ways to help enhance the paper’s clarity, impact, and intuitive accessibility in a revised edition of our paper.\n\nWe wish to highlight that although our contribution is predominantly a theoretical one, an empirical test for the feasibility of equivariance in the weight matrices is not required, since the weight matrices are derived mathematically and are proven to be equivariant – this is Theorem 7.10 in our paper. Our work establishes a theoretical foundation that enables future empirical studies, providing a framework for researchers to explore its applications in real-world machine learning tasks.\n\nThe reviewer would like us “to convey the message that the formalism is indeed useful to solve concrete problems in various fields.” We wish to emphasise that the particular constructions behind our formalism have already appeared directly in real-world applications, which we have highlighted in our response to Reviewer sFDU. We note that in these applications, only quantum groups (and not groups) can model the relevant symmetries. We have also noted in our response to Reviewer Admb that in AI, for data in the form of a set or a graph, modelling quantum symmetries with our networks could offer advantages over classical approaches that use group equivariant neural networks.\n\nFinally, on the reviewer’s main question about $SO(3)$ and $SO(n)$:\n\n- In order for equivariant networks for $SO(3)$ (or any $SO(n)$) to fit into the proposed construction, we need the layers to be a tensor power or $\\mathbb{R}^n$ or $\\mathbb{C}^n$ – this is by the design of the proposed construction in Section 6. However, the characterisation of these networks has already been shown by Pearce-Crump in [1], so their work fits perfectly into our construction as a special case. We will add a comment on this point in the main paper itself.\n\n[1] Pearce-Crump, E. (2023). Brauer’s Group Equivariant Neural Networks. Proceedings of the 40th International Conference on Machine Learning, PMLR 202:27461-27482, 2023.", "claims_and_evidence": "See below.", "methods_and_evaluation_criteria": "-", "theoretical_claims": "There are several theoretical claims. I am not very familiar with the field but I did not encounter any issues.", "experimental_designs_or_analyses": "While there are some examples at the end of the paper, I do not seem to have found any experiment involving learning from data. I think this would be very important to justify the feasibility of the proposed method, since the claim is that it defines the weight matrices for a linear layer.", "relation_to_broader_scientific_literature": "To make the paper more accessible to a more general audience, and to convey the message that the formalism is indeed useful to solve concrete problems in various fields, would be good to review how some more standard work using (non-commutative) group theory or algebras can be understood from the perspective of the presented formalism, for instance\n\n- Le et al, \"Parameterized hypercomplex graph neural networks for graph classification\" \n- De Bortoli et al, \"Riemannian score-based generative modelling\"\n\nIt would be interesting to see if there is any application or extension of the proposed formalism for generative modeling on non-commutative spaces (eg., SO(N) here)\n\n- Bertolini et al, \"Generative Modeling on Lie Groups via Euclidean Generalized Score Matching\"\n- Zhu et al, \"Trivialized Momentum Facilitates Diffusion Generative Modeling on Lie Groups\"", "essential_references_not_discussed": "-", "other_strengths_and_weaknesses": "The paper is very well written despite its technical nature.", "other_comments_or_suggestions": "-", "questions_for_authors": "- Given that SO(3) (or any SO(N) )is a non-commutative group, how do the traditional equivariant network fit into the proposed construction?\n-", "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": "", "extra_scores": {}}, {"review_id": "s0pW9xitHS", "reviewer": "Reviewer_Admb", "summary": "- This work addresses the limitation that group equivariant neural networks cannot learn from the data with a non-commutative geometry. \n\n- Specifically, it derives the existence of compact matrix quantum group equivariant neural network, a new type of equivariant neural network that encodes symmetries described by compact matrix quantum groups\n\n- It characterizes the weight matrices that appear in compact matrix quantum group equivariant neural networks for the easy compact matrix quantum groups, obtaining characterizations of equivariant weight matrices for some compact matrix groups that have not appeared in the previous literature.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate the reviewer’s thoughtful and encouraging critique of our work. We are delighted that they recognise the “original and significant” contributions that are contained in our paper, and that they found the paper to have a “top presentation quality”. We felt that the reviewer really understood the purpose of our paper and, in particular, how our work “addresses the limitation that group equivariant neural networks cannot learn from data with a non-commutative geometry.”\n\nTo answer the reviewer’s questions:\n\n- The training of our proposed networks for the easy compact matrix quantum groups with the standard backpropagation algorithm works in exactly the same way as for a standard MLP: the weight matrices have numerical entries (weights) in the same way as standard MLPs – the only difference is that the requirement to achieve equivariance (for a particular easy compact matrix quantum group) induces a certain amount of weight tying in the weight matrices themselves (and we have shown exactly how these weight matrices are constructed in Theorem 7.10 of our paper).\n- While we anticipate that major applications of our networks will be in physics, we also see potential in AI domains. As we have discussed in our response to Reviewer sFDU, the underlying constructions of our formalism have already appeared in real-world applications where only quantum groups (and not groups) can model the relevant symmetries. Within AI, our work provides a quantum-symmetric analogue of established models such as Deep Sets [1] and Invariant and Equivariant Graph Networks [2], since, for example, $S_n^+$ is the quantum symmetric group on a set of $n$ objects, and $S_n^+$ can also be used to model the quantum symmetries of a graph with $n$ nodes. Hence, for data in the form of a set or a graph, modelling quantum symmetries with our networks could offer advantages over classical approaches that use group equivariant neural networks.\n\n[1] Zaheer, M., Kottur, S., Ravanbakhsh, S., Poczos, B., Salakhutdinov, R. R., and Smola, A. J. (2017). Deep Sets. In Advances in Neural Information Processing Systems.\n\n[2] Maron, H., Ben-Hamu, H., Shamir, N., and Lipman, Y. (2019). Invariant and Equivariant Graph Networks. In International Conference on Learning Representations.", "claims_and_evidence": "All claims are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "It is a pure theoretical work, and the proposed compact matrix quantum group equivariant neural network can handle data with non-commutative geometries in theory.", "theoretical_claims": "I merely checked all proofs for the claims and theorems in the main body of the paper, and found no errors.", "experimental_designs_or_analyses": "There is no experimental design or analysis entailed in this work.", "relation_to_broader_scientific_literature": "The main result of the paper is the derivation a new class of neural networks, termed compact matrix quantum group equivariant neural networks, that allows for learning data with intrinsically non-commutative characteristics.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "- The paper has a top presentation quality.\n\n- The contribution of the paper is limited to the theoretical level, but still original and significant in that it can potentially give rise to neural networks that address data with non-commutative features.", "other_comments_or_suggestions": "I don't have any comments or suggestions.", "questions_for_authors": "- Could you please elaborate how to train a compact matrix quantum group equivariant neural network with standard backpropagation algorithm in the case of, for example, easy compact matrix quantum groups?\n\n- Are there any scenarios in the AI domain where quantum group equivariant neural network could potentially be superior to group equivariant neural networks?", "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": "", "extra_scores": {}}, {"review_id": "0hSR95QDiY", "reviewer": "Reviewer_ADLY", "summary": "The authors propose a new type of equivariant neural networks for handling symmetries described by compact matrix quantum groups and obtain new weight matrices for these groups. The new methods are motivated by the study of non-commutative geometry's symmetries (i.e., quantum symmetries) appearing in quantum groups.\n\nTo be more precise, they use Woronowicz–Tannaka–Krein duality to construct neural networks equivariant to quantum symmetric data and characterize all the weight matrices needed for those models.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their comments. We are pleased that they recognise that our work “introduces a new class of equivariant neural networks for compact quantum groups with applications in physics, as explained in the paper.” We are also glad that they found the claims to be supported with appropriate evidence, and that we gave a comprehensive literature review.\n\nWhile our work is necessarily technical due to its solid mathematical foundation, we have taken deliberate steps to enhance its accessibility. This includes providing concrete examples within the main text and appendix as well as a comprehensive appendix with all of the necessary background material to aid the general reader’s understanding.\n\nTo answer the reviewer’s questions:\n\n- on line 143, we mean that the compact matrix group A is the universal C*-algebra $C^*(u_{i,j})$ having no relations (as in Definition B.22) – we will clarify this in an amended version of the paper. \n- the computational complexity of Procedure 1 depends on several factors, including the specific compact matrix quantum group being considered, the implementation of the algorithm, and how the two-coloured diagrams are generated. Because of these variables, it is not possible to give a single, unified complexity measure. The complexity will generally be influenced by the size of the matrices and the specific structure of the compact matrix quantum group in question.", "claims_and_evidence": "The claims are supported with appropriate evidence as far as I could understand the material", "methods_and_evaluation_criteria": "Yes, no issues regarding this matter", "theoretical_claims": "I checked the theoretical claims and didn't find any issues as far as I understood the material", "experimental_designs_or_analyses": "No experiments", "relation_to_broader_scientific_literature": "No issues regarding this matter", "essential_references_not_discussed": "The authors did a comprehensive literature review.", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "pros:\n\n- the paper introduce a new class of equivariant nerual networks for compact quantum groups with applications in physics (explained in the paper)\n\ncons:\n\n- the material is super technical and less available to most part of the community", "questions_for_authors": "- line 143: what do you mean by having elements generating the algebra? can you clarify this a bit in the paper \n\n\n - what is the computational complexity of running Procedure 1?", "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": "", "extra_scores": {}}, {"review_id": "bnI9jC2FED", "reviewer": "Reviewer_smma", "summary": "The authors consider the problem of extending group equivariant neural networks to compact quantum groups.  Compact quantum groups are generalization of commutative groups.  In commutative groups, the complex-valued functions on the group form a commutative C*-algebra.  In contrast, the functions on Compact quantum groups form a noncommutative C*-algebra.  Non-commutative C*-algebras arise in many applications, including quantum mechanics and statistical mechanics applications, and Non-commutative C*-algebras have a non-commutative geometry associated to them.\n\n\n## update after rebuttal\n\nThank you for these potential applications. I have adjusted my score accordingly, but will defer to the other reviewers' judgment as I am not very familiar with the applications of this subject area.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their critique of our work. We are glad that the reviewer has recognised our main theoretical contribution (namely, constructing a neural network for easy compact matrix quantum groups and characterising its weight matrices), and that they felt that we explained non-commutative geometry well, which was not a trivial task given its technical nature.\n\nThe reviewer questions why one would want to construct such a neural network, as well as its potential applications. We wish to emphasise that the particular constructions behind our formalism have already appeared directly in real-world applications. In addition to the tasks that we listed in the introduction to our paper, we note that quantum symmetric groups $S_n^+$ have been studied in relation to the Potts model, a type of physical model in statistical mechanics, by Goshwami and Hossain [1]; in elementary particle physics on $n$ half-spin particles by Woronowicz [2] (where the author notes on page 120 that interchanging particles is no longer given by the swap operator, and in effect states that the symmetry object is $S_n^+$ – this is discussed further in Weber [3, Section 3.2]); and they have also appeared in the study of non-local games in quantum information theory by Lupini et al. [4]. We have also noted in our response to Reviewer Admb that in AI, for data in the form of a set or a graph, modelling quantum symmetries with our networks could offer advantages over classical approaches that use group equivariant neural networks. We feel that we could form a concise argument along these lines to provide the more intuitive explanations and motivation for the theoretical results contained within our paper that the reviewer would like to see.\n\nThe reviewer also asks us to give a more direct comparison and specify the improvements over prior work. Our work builds upon a body of work in group equivariance where the authors derive weight tying schemes to guarantee equivariance to a particular group, such as [5, 6, 7, 8, 9]; However, the key difference and major contribution of our work is that we guarantee equivariance to compact matrix quantum groups, which generalise groups in some sense, namely by also deriving weight tying schemes for certain compact matrix quantum groups (the “easy” ones). Studying equivariance to compact matrix quantum groups and obtaining a characterisation of these weight matrices is our main theoretical improvement over these previous works since quantum group equivariance has not been explored before in the machine learning literature. The ability to model such equivariance is needed because, as the previously stated applications demonstrate, only quantum groups (and not groups) can model the relevant symmetries in these tasks.\n\nFinally, on the subject of experiments, we would like to emphasise that although our contribution is predominantly a theoretical one, we recognise that experiments on real-world data would provide valuable insights. Given that this is the first work on quantum group equivariance, we have struggled to find an appropriate dataset that we could apply these networks to (and consequently no benchmarks exist, hence even if we had an appropriate dataset we would not be able to demonstrate an improvement over any previous methods since they don’t exist).  However, we look forward to demonstrating the broader potential of our methods in future work.\n\n[1] Debashish Goswami and S. K. Asfaq Hossain (2022). Quantum symmetry on Potts model. J. Math. Phys. 63, 043504.\n\n[2] Stanisław Lech Woronowicz, Twisted SU(2) Group. An Example of a Non-Commutative Differential Calculus. Publ. Res. Inst. Math. Sci. 23 (1987), no. 1, pp. 117–181\n\n[3] Weber, M. (2020). Quantum symmetry. Snapshots of modern mathematics from Oberwolfach, 5. \n\n[4] Lupini, M., Mančinska, L., Roberson, D.E.: Nonlocal games and quantum permutation groups. J. Funct. Anal. 279(5), 108592 (2020)\n\n[5] Zaheer, M., Kottur, S., Ravanbakhsh, S., Poczos, B., Salakhutdinov, R. R., and Smola, A. J. (2017). Deep Sets. In Advances in Neural Information Processing Systems.\n\n[6] Maron, H., Ben-Hamu, H., Shamir, N., and Lipman, Y. (2019). Invariant and Equivariant Graph Networks. In International Conference on Learning Representations. \n\n[7] Ravanbakhsh, S., Schneider, J., and Poczos, B. (2017). Equivariance Through Parameter-Sharing. In Proceedings of the 34th International Conference on Machine Learning. \n\n[8] Godfrey, C., Rawson, M. G., Brown, D., and Kvinge, H. (2023) Fast computation of permutation equivariant layers with the partition algebra. In ICLR 2023 Workshop on Physics for Machine Learning. \n\n[9] Pearce-Crump, E. (2023) Brauer’s Group Equivariant Neural Networks. In Proceedings of the 40th International Conference on Machine Learning.", "claims_and_evidence": "The authors do a nice job explaining noncommutative geometry, and constructing a neural network for what the authors refer to as “easy” Compact quantum groups.  However, the authors should better explain why one might want to construct such a neural network:   What are the applications of Compact quantum groups and noncommutative geometrys, and what are the applications of such a neural network?", "methods_and_evaluation_criteria": "It is a bit unclear to me what would be the application of the authors' proposed neural networks, and what the main previous works they are improving on (even theoretical improvements).   Moreover, it would be helpful to provide experiments, as this would allow comparison to the performance of previous work, and might also provide a real-world example where an improvement can be shown.  This would make it easier to evaluate the significance of the paper.", "theoretical_claims": "I did not carefully check the proofs in the appendix.", "experimental_designs_or_analyses": "It would be helpful to provide experiments, as this would allow comparsison to the performance of previous work, and might also provide a real-world example where an improvement can be shown.  This would make it easier to evaluate the significance of the paper.", "relation_to_broader_scientific_literature": "It is a bit unclear to me what would be the application of the authors' proposed neural networks, and what the main previous works they are improving on (even theoretical improvements).  Explaining this better would make it easier to effectively evaluate the paper.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Please see answers to the above questions.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "It would be helpful if the authors could explain why one might want to construct such a neural network:   What are the applications of Compact quantum groups and noncommutative geometrys, and what are the applications of such a neural network?  Also, it would be helpful to give a more direct comparison and specify the improvements over prior work, if at all possible.", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The submission received unanimous positive reviews. Reviewers praised the presentation quality and originality. The main weaknesses raised were about 1) accessibility to wider audience, 2) lack of experiments, and 3) lack of practical applications. 1) and 2) are somewhat expected from purely theoretical papers, and 3) was addressed in the rebuttal to smma.
Decision: Accept (poster)