A Lifting Framework for Constructing Group-Invariant Embeddings

We study the problem of constructing group-invariant embeddings that faithfully represent data modulo group symmetries, a task that arises naturally in signal processing, physics, and machine learning. A central challenge is to design embeddings that are simultaneously orbit-separating, stable, and computationally tractable. To address this, we develop a general lifting framework for constructing such embeddings. The key idea is to start from a group-invariant embedding defined on a low-dimensional reduced space, and lift it to the ambient space by composing it with a finite family of parameterized linear maps, followed by an aggregation step that produces a global embedding. This framework provides a unified perspective that connects classical problems such as phase retrieval and permutation-invariant embeddings. We demonstrate the effectiveness of this framework in the finite group setting. In this setting, we establish general sufficient conditions for orbit separation and prove that any orbit-separating lifting embedding is automatically bi-Lipschitz. We further extend the bi-Lipschitz result to sparse regimes, and show that, when applied to phase retrieval, it yields an equivalence between uniqueness and stability for real sparse phase retrieval.