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Stability of sorting based embeddings

Radu Balan, Efstratios Tsoukanis, Matthias Wellershoff

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 $α: \mathbb{R}^{M \times N} \to \mathbb{R}^D$ to the invariant map $β_Φ: V \to \mathbb{R}^{M \times N}$ given by sorting the coorbits $(\langle v, g φ_i \rangle_V)_{g \in G}$, where $(φ_i)_{i=1}^N \in V$, 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.

Stability of sorting based embeddings

Abstract

Consider a group of order acting unitarily on a real inner product space . We show that the sorting based embedding obtained by applying a general linear map to the invariant map given by sorting the coorbits , where , 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.
Paper Structure (9 sections, 19 theorems, 104 equations)

This paper contains 9 sections, 19 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Sorting-based embeddings
  3. Results
  4. Examples
  5. Continuous maps factor through sorting based embeddings
  6. Preliminaries for the proof of the main theorem
  7. Sorting coorbits
  8. Stabilisers of the group action
  9. The proof of the main theorem

Key Result

Theorem 1

The embedding $\gamma = \alpha \circ \beta_\Phi : V \to \mathbb{R}^D$ separates orbits if and only if it satisfies the bi-Lipschitz condition eq:biLipschitz_condition.

Theorems & Definitions (59)

  • Theorem 1: Main result
  • Theorem 2
  • Remark 3
  • Theorem 4
  • Remark 5
  • Remark 6: There exists an injective ReLU neural networks that is not bi-Lipschitz
  • Example 7: Permutation invariant representations
  • Example 8: Sign retrieval
  • Theorem 9: Kirszbraun--Valentine extension theorem
  • Corollary 10
  • ...and 49 more