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Topology Structure Optimization of Reservoirs Using GLMY Homology

Yu Chen, Shengwei Wang, Hongwei Lin

Abstract

Reservoir is an efficient network for time series processing. It is well known that network structure is one of the determinants of its performance. However, the topology structure of reservoirs, as well as their performance, is hard to analyzed, due to the lack of suitable mathematical tools. In this paper, we study the topology structure of reservoirs using persistent GLMY homology theory, and develop a method to improve its performance. Specifically, it is found that the reservoir performance is closely related to the one-dimensional GLMY homology groups. Then, we develop a reservoir structure optimization method by modifying the minimal representative cycles of one-dimensional GLMY homology groups. Finally, by experiments, it is validated that the performance of reservoirs is jointly influenced by the reservoir structure and the periodicity of the dataset.

Topology Structure Optimization of Reservoirs Using GLMY Homology

Abstract

Reservoir is an efficient network for time series processing. It is well known that network structure is one of the determinants of its performance. However, the topology structure of reservoirs, as well as their performance, is hard to analyzed, due to the lack of suitable mathematical tools. In this paper, we study the topology structure of reservoirs using persistent GLMY homology theory, and develop a method to improve its performance. Specifically, it is found that the reservoir performance is closely related to the one-dimensional GLMY homology groups. Then, we develop a reservoir structure optimization method by modifying the minimal representative cycles of one-dimensional GLMY homology groups. Finally, by experiments, it is validated that the performance of reservoirs is jointly influenced by the reservoir structure and the periodicity of the dataset.

Paper Structure

This paper contains 19 sections, 18 equations, 8 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Reservoir computing and reservoir optimization
  4. GLMY homology and its applications
  5. Preliminary
  6. Methodology
  7. Reservoir computing and its theoretical analysis
  8. Optimizing reservoir structure by persistent GLMY homology
  9. Experiments and Discussions
  10. Reservoir Size Sensitivity and Reservoir Dynamics Analysis
  11. Datasets and their topology
  12. Initial reservoir generation and its optimization
  13. Memory capacity analysis
  14. Prediction performance analysis
  15. Discussion
  16. ...and 4 more sections

Figures (8)

  • Figure 1: Illustration of the flowchart of our method and experimental procedures.
  • Figure 2: (a) Example of a one-dimensional persistence diagram. (b) An illustration of the confidence set (red area) of (a).
  • Figure 3: Structure of Reservoir Computing Model
  • Figure 4: Illustration of compatibly modifying cycles into rings. (a) A digraph has two rings on the left and right. The middle cycle (red) is not a ring. (b) By changing the direction of $AD$, the middle cycle becomes a ring $ABCD$ compatible with the other rings. (c) The case where the middle cycle cannot be changed into a ring without destroying the other two rings, since either $AB$ or $DC$ would need to have its direction changed.
  • Figure 5: Reservoir size sensitivity analysis. Here we analyze reservoir sizes ranging from 100 to 500 and calculate the orthogonality measurement (OM) before and after optimization. The results presented from left to right correspond to Random, Small-world, and Scale-free initialization, respectively.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Definition 1