Echo State Property upon Noisy Driving Input

Figure 5

Reservoir states of two-dimensional ESNs with zero input and with only the white Gaussian noise. Evolution of reservoir states with different initial states for one white Gaussian noise input against the spectral radius of the internal weight matrix (left), 1.1 (center), and reservoirs states against (right) in the last 100 steps (201 ≤ k ≤ 300; green shaded area for the first and second columns): (a) reservoir states without noise; (b) reservoir states driven by white Gaussian input noise with SNR = 4.0; (c) with SNR = 1.0; (d) SNR = 1.0. Each panel of the first and second columns displays 100 reservoir states starting from random initial states, while that of the third column includes 1000 different states (100 randomized reservoir states × 10 randomized noise generations at each SNR). The internal weight matrix was chosen to W = , for all simulations, where is a positive constant to rescale the spectral radius . The noise levels are the same as MNIST tasks in Section 3.3. The last 100 states are colored dark green if they are bounded with tolerance , i.e., , and otherwise light green.

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