Identification with finite environments for nonparametric representation-level invariance

Determine whether identification in representation-level invariance learning is possible with any finite number of environments when the representation map Φ: ℝ^d → ℝ^r is allowed to be nonparametric. Specifically, prove or refute the conjecture that identification is impossible for all finite |E| when Φ belongs to a nonparametric function class, in contrast to the linear case where at least r environments are necessary even under sufficient heterogeneity and known r.

Background

The paper contrasts variable-selection-level invariance with representation-level invariance (as in IRM), noting that identification requirements can be substantially different. For linear representation learning of dimension r, the authors remark that at least r distinct environments are necessary for identification, even assuming sufficient heterogeneity.

They then raise a broader question for nonparametric representation classes: whether any finite number of environments could ever suffice to identify the representation Φ and the corresponding invariant predictor. This conjecture targets the fundamental limits of environment-based identification in nonparametric settings and directly impacts the feasibility and design of representation-level invariance methods.