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Compact Matrix Quantum Group Equivariant Neural Networks

ICML 2025 · Learning theory and algorithms › Matrix and tensor methods · approach: Geometry of low-rank and SPD structure · 0 citations (OpenAlex)

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Derives compact matrix quantum group equivariant neural networks via Woronowicz-Tannaka-Krein duality and characterises the linear equivariant weight matrices for easy compact matrix quantum groups defined by set partitions.

Problem

Group equivariant neural networks cannot handle data living in non-commutative geometry, since C*-algebras of continuous functions on compact matrix groups are commutative; a broader notion of quantum group symmetry is needed.

Approach

Define neural networks equivariant to compact matrix quantum groups by building layers from two-coloured representation categories (via Woronowicz-Tannaka-Krein duality), and characterise equivariant linear weight matrices for easy quantum groups as spanning sets indexed by two-coloured set partition diagrams.

Key results

• Characterises equivariant linear weight matrices for easy compact matrix quantum groups via spanning sets of partition diagrams.
• Obtains new weight-matrix characterisations for hyperoctahedral group H_n, bistochastic group B_n, and unitary group U(n).
• Shows every compact matrix group equivariant neural network is a compact matrix quantum group equivariant neural network.

Limitations

• Purely theoretical; equivariant non-linear layers are not characterised, so networks cannot yet be implemented.
• No experiments demonstrate practical performance.
• Characterisation restricted to easy compact matrix quantum groups.

Experimental setup

Builds on: Woronowicz compact matrix quantum groups Tannaka-Krein duality group equivariant neural networks

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