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Deep Ridgelet Transform and Unified Universality Theorem for Deep and Shallow Joint-Group-Equivariant Machines

ICML 2025 · Learning theory and algorithms › Kernels and regression theory · approach: Neural tangent kernel analysis · 0 citations (OpenAlex)

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Derives a closed-form ridgelet transform (right inverse of the integral machine) for joint-group-equivariant machines via Schur's lemma, yielding a unified constructive universality theorem covering depth-n fully-connected nets, depth-n group-convolutional nets, and a new quadratic-form network.

Problem

Universal approximation theorems for deep networks have been proven differently from shallow ones, and closed-form (constructive) parameter distributions (ridgelet transforms) were known only for depth-2 networks.

Approach

Introduces joint-group-equivariant feature maps (equivariance acting jointly on data and parameter domains) and shows that irreducibility of the group representation on L^2(X;Y) plus Schur's lemma gives a ridgelet transform that right-inverts the integral machine for any depth, unifying shallow and deep universality.

Key results

• Proves that depth-n joint-equivariant machines are L^2-universal with an explicit ridgelet transform whenever the representation on L^2(X;Y) is irreducible and the composite operator is bounded.
• Shows the depth-n fully-connected network and depth-n group-convolutional network are universal under this framework, reproducing and extending depth-2 results.
• Derives universality of a new depth-2 quadratic-form network sigma(x^T A x + x^T b + c) on L^2(R^m).

Surprising findings

• Fully-connected networks are not group-equivariant but are joint-group-equivariant.
• The irreducibility requirement applies to the representation on the data side only, not the parameter side.

Limitations

• Reconstruction requires bounded M∘R and irreducibility, which must be verified case by case.
• Theory only; no experiments on practical networks are reported.
• The constant can vanish (e.g., polynomial activations for depth-2 fully-connected networks).

Experimental setup

Baselines: Cybenko universality Hornik universality Sonoda and Murata depth-2 ridgelet transform

Builds on: ridgelet transform integral representation of neural networks group-equivariant networks

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