Canonical Layer Inference for Deep Gaussian Processes via Symmetry-Breaking Augmentation

Yichen Zhu; Hengrui Luo

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arXiv · 2610.23842

Canonical Layer Inference for Deep Gaussian Processes via Symmetry-Breaking Augmentation

Yichen Zhu·Hengrui Luo

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Abstract

Deep Gaussian processes (DGPs) are flexible priors for compositional regression functions, but inference for intermediate layers faces questions. The basic obstruction is an adjacent-layer symmetry: applying an invertible transformation and its inverse to two adjacent layers leaves the composed regression map unchanged. Under explicit symmetry and prior-comparability conditions, we show that the posterior may not concentrate on any prescribed intermediate-layer representation, even when the composed regression function is estimable. We then introduce an augmented $ρ$-fractional posterior that reweights layer decompositions by a non-saturating identity-deviation penalty. The augmentation changes posterior odds between symmetry-related decompositions by an exponential penalty ratio. Under proper choices of the penalty and a prior mass condition, it preserves function-level fractional posterior contraction, while shrinking the redundant layers towards the identity mapping. Thus, the likelihood does not identify a unique ``true'' set of layers; the augmentation favors identity-biased parsimonious decompositions within an equivalence class of layers; uniqueness of a selected representative is not established.

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Yichen Zhu, Hengrui Luo. 2026-10-16. Canonical Layer Inference for Deep Gaussian Processes via Symmetry-Breaking Augmentation. https://arxiv.org/abs/2610.23842

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