Stability of sorting based embeddings

2024/10/07 by Radu Bălan, Balan, Radu, Efstratios Tsoukanis +3 · 2 citations
Computer Science · #Cellular Automata and Applications #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Machine Learning (cs.LG)

paper · pdf · doi:10.48550/arxiv.2410.05446

openalex publication_date 2024/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Consider a group G of order M acting unitarily on a real inner product space V. We show that the sorting based embedding obtained by applying a general linear map α: ℝ^{M × N} → ℝ^D to the invariant map β_Φ: V → ℝ^{M × N} given by sorting the coorbits (⟨ v, g ϕ_i ⟩_V)_{g ∈ G}, where (ϕ_i)_i=1^N ∈ V, satisfies a bi-Lipschitz condition if and only if it separates orbits. Additionally, we note that any invariant Lipschitz continuous map (into a Hilbert space) factors through the sorting based embedding, and that any invariant continuous map (into a locally convex space) factors through the sorting based embedding as well.

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