Expressive Power of Implicit Models: Rich Equilibria and Test-Time Scaling

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Abstract:Implicit models, an emerging model class, compute outputs by iterating a single parameter block to a fixed point. This architecture realizes an infinite-depth, weight-tied network that trains with constant memory, significantly reducing memory needs for the same level of performance compared to explicit models. While it is empirically known that these compact models can often match or even exceed the accuracy of larger explicit networks by allocating more test-time compute, the underlying mechanism remains poorly understood. We study this gap through a nonparametric analysis of expressive power. We provide a strict mathematical characterization, showing that a simple and regular implicit operator can, through iteration, progressively express more complex mappings. We prove that for a broad class of implicit models, this process lets the model's expressive power scale with test-time compute, ultimately matching a much richer function class. The theory is validated across four domains: image reconstruction, scientific computing, operations research, and LLM reasoning, demonstrating that as test-time iterations increase, the complexity of the learned mapping rises, while the solution quality simultaneously improves and stabilizes.

Subjects:

Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Representation Theory (math.RT); Machine Learning (stat.ML)

Cite as: arXiv:2510.03638 [cs.LG]

(or arXiv:2510.03638v4 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2510.03638

arXiv-issued DOI via DataCite

Submission history

From: Jialin Liu [view email] [v1] Sat, 4 Oct 2025 02:49:22 UTC (389 KB) [v2] Fri, 28 Nov 2025 21:40:19 UTC (493 KB) [v3] Sun, 1 Mar 2026 02:46:03 UTC (493 KB) [v4] Tue, 31 Mar 2026 02:07:23 UTC (493 KB)