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Position: $C^*$-Algebraic Machine Learning $-$ Moving in a New Direction

ICML 2024 · Learning theory and algorithms › Kernels and regression theory · approach: Kernel regression under shift · 0 citations (OpenAlex)

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Position paper proposing C*-algebraic machine learning, arguing C*-algebras (Hilbert C*-modules, C*-algebra nets) can unify kernel methods and neural networks, with theoretical illustrations and open challenges.

Problem

Modern ML involves structured data, multiple models/tasks, and limited samples, which existing complex/vector-valued kernels and networks handle poorly; the paper asks whether C*-algebra generalizations can unify learning strategies.

Approach

Generalize inner products, RKHSs (to RKHMs over Hilbert C*-modules), kernel mean embeddings, and neural network parameters from complex/vector values to C*-algebra-valued objects, so a single C*-algebra net represents infinitely many networks indexed by points of a compact space, with convexity in the averaging measure.

Key results

• Proves that a C*-algebra net with linear activations is a degree-L polynomial in the variables v_l(z), so its expressiveness in z grows with depth even without nonlinear activations.
• Shows that for C*-algebra nets over C(Z) with Z = W^N, any real-valued network is representable, and the averaged net is convex in the probability measure P (loss convex in P).

Surprising findings

• Linear-activation C*-algebra nets still gain nonlinearity in the input variable z as depth grows.

Limitations

• Riesz representation and orthogonal complements do not always exist in Hilbert C*-modules, complicating theory.
• Elements of C*-algebras may lack inverses, so Hilbert-space algorithms do not transfer directly.
• No software library exists and computational cost grows with discretization dimension d.

Experimental setup

Builds on: reproducing kernel Hilbert space Hilbert C*-module kernel mean embedding C*-algebra net

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