Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generic and Isometric Embeddings in Reservoir Computers

Allen G Hart

TL;DR

This work addresses when reservoir computers realize generalized synchronization (GS) that faithfully embeds the source attractor. It leverages Whitney's embedding theorem ($N > 2q$) to prove that generic GS maps are embeddings, and Nash's embedding theorem ($N \ge 2q+1$) to show the existence of isometric GS, with explicit constructions in the linear case. A key insight is that generic reservoir realizations with universal approximation (e.g., ESNs) will almost surely yield embedding GS, and that, in the linear regime, any embedding GS is isomorphic to an isometric GS. These results provide a theoretical basis for distortion-free representation of nonlinear time series by reservoirs, enabling robust learning, invariant discovery, and geometry-preserving analysis of dynamical systems.

Abstract

We prove that a generic reservoir system admits a generalized synchronization that is a topological embedding of the input system's attractor. We also prove that for sufficiently high reservoir dimension (given by Nash's embedding theorem) there exists an isometric embedding generalized synchronization. The isometric embedding can be constructed explicitly when the reservoir system and source dynamics are linear.

Generic and Isometric Embeddings in Reservoir Computers

TL;DR

This work addresses when reservoir computers realize generalized synchronization (GS) that faithfully embeds the source attractor. It leverages Whitney's embedding theorem () to prove that generic GS maps are embeddings, and Nash's embedding theorem () to show the existence of isometric GS, with explicit constructions in the linear case. A key insight is that generic reservoir realizations with universal approximation (e.g., ESNs) will almost surely yield embedding GS, and that, in the linear regime, any embedding GS is isomorphic to an isometric GS. These results provide a theoretical basis for distortion-free representation of nonlinear time series by reservoirs, enabling robust learning, invariant discovery, and geometry-preserving analysis of dynamical systems.

Abstract

We prove that a generic reservoir system admits a generalized synchronization that is a topological embedding of the input system's attractor. We also prove that for sufficiently high reservoir dimension (given by Nash's embedding theorem) there exists an isometric embedding generalized synchronization. The isometric embedding can be constructed explicitly when the reservoir system and source dynamics are linear.

Paper Structure

This paper contains 6 sections, 9 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Generic Embeddings
  3. Discussion
  4. Isometric Embeddings
  5. Linear Reservoir Systems
  6. Conclusion

Key Result

Theorem 2.4

Let $M$ be a smooth manifold of dimension $q$. If $N > 2q$, then a generic map in $C^1(M,\mathbb{R}^N)$ is an embedding. Equivalently, the subset of embeddings $E \subset C^1(M,\mathbb{R}^N)$ is open and dense.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: Whitney Whitney1936Lee2013
  • Theorem 2.5
  • proof
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 4.1
  • ...and 19 more