Symmetry kills the square in a multifunctional reservoir computer

Area of Science:

• Computational neuroscience
• Nonlinear dynamics
• Machine learning

Background:

• Reservoir computers (RCs) are powerful tools for time-series analysis.
• Symmetry in RC design can limit their learning capabilities.
• Mirror-attractors, inverted copies of solutions, can arise and interfere with RC performance.

Purpose of the Study:

• To investigate the impact of data symmetries on the formation of square readout matrices in RCs.
• To explore the concept of multifunctionality in RCs trained to handle symmetric attractors.
• To understand the relationship between RC internal connection spectral radius and square readout matrix emergence.

Main Methods:

• Analytical proof of symmetry constraints on square readout matrix existence.
• Numerical simulations training RCs to reconstruct coexisting Lorenz attractors and their mirror-attractors.
• Analysis of RC performance under varying attractor positions and spectral radius values.

Main Results:

• Certain training data symmetries analytically forbid the existence of the square readout matrix.
• The square readout matrix emerges numerically when attractor positions are slightly altered, even with overlapping regions.
• The square readout matrix reappears at large spectral radius values before the edge of chaos is reached.

Conclusions:

• Data symmetry is a critical factor limiting the effectiveness of square readout matrices in RCs.
• RC multifunctionality, particularly in handling symmetric attractors, is sensitive to subtle changes in initial conditions.
• The spectral radius of internal connections plays a significant role in the emergence of the square readout matrix and RC stability.